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+1
+Two-Dimensional Golay Complementary Array
+Sets With Arbitrary Lengths for
+Omnidirectional MIMO Transmission
+You-Qi Zhao, Cheng-Yu Pai, Zhen-Ming Huang, Zilong Liu, Senior
+Member, IEEE, and Chao-Yu Chen, Member, IEEE
+Abstract
+This paper presents a coding approach for achieving omnidirectional transmission of certain common
+signals in massive multi-input multi-output (MIMO) networks such that the received power at any
+direction in a cell remains constant for any given distance. Specifically, two-dimensional (2D) Golay
+complementary array set (GCAS) can be used to design optimal massive MIMO precoding matrix so as
+to achieve omnidirectional transmission due to its complementary autocorrelation property. In this paper,
+novel constructions of new 2D GCASs with arbitrary array lengths are proposed. Our key idea is to
+carefully truncate the columns of certain larger arrays generated by 2D generalized Boolean functions.
+Finally, the power radiation patterns and numerical results are provided to verify the omnidirectional
+property of the GCAS-based precoding. The error performances of the proposed precoding scheme are
+presented to validate its superiority over the existing alternatives.
+Index Terms
+This work was supported by the Ministry of Science and Technology, Taiwan, R.O.C., under Grant MOST 109–2628–E–006–
+008–MY3 and MOST 111–2218–E–305–002.
+You-Qi Zhao and C.-Y. Pai are with the Department of Engineering Science, National Cheng Kung University, Tainan 701,
+Taiwan, R.O.C. (e-mail: n98081505@gs.ncku.edu.tw).
+Z.-M. Huang is with the Institute of Computer and Communication Engineering, National Cheng Kung University, Tainan
+701, Taiwan, R.O.C. (e-mail: n98101012@gs.ncku.edu.tw).
+Zilong Liu is with the School of Computer Science and Electronic Engineering, University of Essex, United Kingdom (e-mail:
+zilong.liu@essex.ac.uk).
+C.-Y. Chen is with the Department of Electrical Engineering and the Institute of Computer and Communication Engineering,
+National Cheng Kung University, Tainan 701, Taiwan, R.O.C. (e-mail: super@mail.ncku.edu.tw).
+January 4, 2023
+DRAFT
+arXiv:2301.01225v1  [cs.IT]  3 Jan 2023
+
+2
+Generalized Boolean function (GBF), Golay complementary array pair (GCAP), Golay comple-
+mentary array set (GCAS), omnidirectional precoding (OP), uniform rectangular array (URA).
+I. INTRODUCTION
+Complementary pairs/sets of sequences have attracted a sustained research interest owing to
+their zero aperiodic correlation sums properties. To be specific, a Golay complementary pair
+(GCP) refers to a pair of equal-length sequences whose summation of aperiodic autocorrelations
+is zero except at the zero time-shift [1]. Such a concept was extended to Golay complementary
+set (GCS) with constituent sequences of more than 2 by Tseng and Liu in [2]. Furthermore, a
+maximum collection of GCSs is called a set of complete complementary code (CCC) [3] if any
+two different GCSs have zero aperiodic cross-correlation sums for all time-shifts. In the literature,
+GCSs and CCCs have been widely used for radar sensing [4], channel estimation [5], precoding
+for massive multi-input multi-output (MIMO) [6], peak-to-average power ratio (PAPR) reduction
+in orthogonal frequency division multiplexing (OFDM) [7]–[13], interference-free multicarrier
+code division multiple access [14]–[17], and many other applications [18], [19].
+Recently, there is a surge of research attention to study two-dimensional (2D) Golay com-
+plementary array sets (GCASs) [18]-[23], each having zero aperiodic autocorrelation sums
+property for two directions of shifts (compared to conventional GCSs and CCCs with time-
+shifts only). An important application of the 2D GCASs is for omnidirectional transmission in
+MIMO communication systems with a uniform rectangular array (URA) configuration [20], [21].
+In massive MIMO systems, some common messages (e.g., reference signals, synchronization
+signals, control signals, etc.) need to be power-uniformly broadcasted to all the angles within
+the whole cell. In this paper, we consider space-time block code (STBC) for the harvesting
+of the diversity gain. At the base station (BS), the STBC encoded symbols are assigned to
+several streams and then mapped onto the antenna arrays in URA by certain 2D GCASs assisted
+precoding matrices to achieve uniform power radiation at any angle.
+On the other hand, since a large number of antennas are considered in massive MIMO
+systems, a huge pilot overhead may be needed to acquire the channel state information (CSI). As
+pointed out in [22], this can be alleviated by omnidirectional precoding (OP) based transmission.
+For uniform linear arrays (ULAs), Zadoff-Chu (ZC) sequences were adopted to satisfy the
+requirements of the omnidirectional property. However, [22] only considered the omnidirectional
+January 4, 2023
+DRAFT
+
+3
+transmission in certain directions. Later in [6], GCSs and CCCs based OP matrices were proposed
+to meet the requirement of omnidirectional transmission across all directions.
+In [20], [21], [23], [24], 2D GCASs were employed for precoding matrices in URAs by
+applying interleaving and Kronecker-product to existing 1D sequences or 2D arrays. As a result,
+the array sizes of 2D GCASs are only feasible for certain lengths. A construction of 2D GCASs
+of array size pn × pm was proposed in [25] by using permutation ploynomials (PPs) functions
+and 2-level autocorrelation sequences, where p is a prime number, m, n are two positive integers,
+and p, m, n > 0. Furthermore, a unifying construction framework for 2D GCASs was developed
+in [26] by a multivariate polynomial matrix from certain seed para-unitary (PU) matrices. In
+[27], [28], Pai and Chen proposed direct constructions of 2D Golay complementary array pairs
+(GCAPs) and GCASs with array size 2n × 2m from 2D generliazed Boolean functions (GBFs)
+[29] where n, m are integers and n, m ≥ 2. 2D GCAP can be regarded as a case of 2D GCAS
+when the set size is equal to 2. Moreover, Pai et al. [30] proposed a direct construction of 2D
+CCCs with array size 2n×2m, which have ideal autocorrelations and cross-correlations. Later, Liu
+et al. [31] proposed a construction of GCASs with array size pn ×pm by using 2D multivariable
+functions, where p is a prime number, n, m are integers, and n, m ≥ 2. Based on [27], [32]
+developed a direct construction of GCASs with set size 4 and array size 2n × (2m−1 + 2v) by
+using 2D GBFs, where n, m, v are positive number with n, m ≥ 2, and 0 ≤ v ≤ m − 1.
+The aforementioned research efforts are generally driven by the need of highly flexible array
+sizes of 2D GCASs. Motivated by this, we aim for generating new GCASs with arbitrary array
+lengths. The key idea of our proposed constructions is to carefully truncate some columns of
+the certain larger arrays generated by 2D GBFs. Thus, our proposed GCASs can be applied to
+URAs with various array sizes. In addition, the proposed GCASs can be directly generated from
+2D GBFs without the requirements of any specific sequences or tedious sequence operations. In
+Table I, we compare the existing parameters of 2D GCASs with our proposed ones.
+The remainder of this paper is defined as follows. Section II discusses notations, definitions,
+system models, and the omnidirectional transmission in MIMO systems. Section III describes
+our proposed constructions of 2D GCASs. Section IV shows the power radiation pattern and bit
+error rate (BER) performance based on our proposed 2D GCASs precoding. Finally, Section V
+presents the conclusion.
+January 4, 2023
+DRAFT
+
+4
+TABLE I
+A COMPARISON OF CONSTRUCTIONS FOR 2D GCASS
+Construction
+Parameters
+Approaches
+[26, Th. 5]
+(N, N n, N m), N, n, m > 0
+Seed PU matrices
+[26, Th. 7]
+(2k, 2kn, 2km), n, m, k > 0
+[25, Th. 4]
+(p, pn, pm), prime p, n, m > 0
+PPs and 2-level
+autocorrelation sequences
+[25, Th. 6]
+(pk, pkn, pkm), prime p, k, n, m > 0
+[31, Th. 1]
+(pk1
+1 pk2
+2 , pn
+1 , pm
+2 ), primes p1, p2
+2D multivariable functions
+[31, Th. 2]
+(pk, pn, pm), prime p, n + m ≥ k > 0
+[27], [28], [30]
+(2k, 2n, 2m), n, m ≥ k > 0, and k > 0
+2D GBFs
+[32]
+(4, 2n, 2m−1 + 2v), n, m ≥ 2, and k > 0
+Th. 1
+(2k+1, 2n, 2m−1 + �k−1
+α=1 dα2m−k+α−1 + d02v),
+k < m, 0 ≤ v ≤ m − k, dα ∈ {0, 1}
+Th. 2
+(2k+1, 2n, 2m−1 + �k−1
+α=1 dα2π1(m−k+α)−1 + d02v),
+k < m, 0 ≤ v ≤ m − k, dα ∈ {0, 1}
+II. PRELIMINARIES AND DEFINITIONS
+A. Notations
+Throughout this paper, we present the notations in the following:
+• (a)i refers to the i-th element of the vector a.
+• (A)i,j denotes the (i, j)-th element of the array A.
+• (·)H refers to the conjugate transpose.
+• diag(A) refers to the column vector composed of the main diagonal of A.
+• (·)∗ refers to the complex conjugation of an element.
+• (·)T refers to the transpose.
+• vec(·) express stacking one column of the matrix into one another column.
+• 1 is a vector whose elements are all 1.
+• Let ξ = e2π√−1/q.
+• In this paper, q is an even number.
+Let X and Y be two arrays of size L1 × L2. Then X and Y can be stated as
+X = (Xg,i), Y = (Yg,i),
+(1)
+where g = 0, 1, · · · , L1 − 1 and i = 0, 1, · · · , L2 − 1.
+January 4, 2023
+DRAFT
+
+5
+Definition 1: Given two arrays X and Y of size L1 × L2, the 2D aperiodic cross-correlation
+function (AACF) is defined by
+ρ (X, Y; u1, u2) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+L1−1−u1
+�
+g=0
+L2−1−u2
+�
+i=0
+Yg+u1,i+u2X∗
+g,i, 0 ≤ u1 < L1,
+0 ≤ u2 < L2;
+L1−1−u1
+�
+g=0
+L2−1−u2
+�
+i=0
+Yg+u1,iX∗
+g,i−u2, 0 < u1 < L1,
+−L2 < u2 < 0;
+L1−1−u1
+�
+g=0
+L2−1−u2
+�
+i=0
+Yg,iX∗
+g−u1,i−u2, −L1 < u1 < 0,
+−L2 < u2 < 0;
+L1−1+u1
+�
+g=0
+L2−1−u2
+�
+i=0
+Yg,i+u2X∗
+g−u1,i, −L1 < u1 < 0,
+0 < u2 < L2.
+(2)
+When X = Y , then it is called 2D aperiodic autocorrelation function (AACF) and denoted
+by ρ(X; u1, u2). If taking L1 = 1, two 2D arrays X and Y are degraded as a 1-D sequence
+X = Xi for i = 0, 1, · · · , L2 − 1 and Y = Yi for i = 0, 1, · · · , L2 − 1, respectively. Then the
+1-D AACF of 1-D sequence X is related by
+ρ(X; u) =
+�
+�
+�
+�
+�
+�
+�
+L2−1−u
+�
+i=0
+Xi+uX∗
+i ,
+0 ≤ u ≤ L2 − 1;
+L2−1+u
+�
+i=0
+XiX∗
+i−u,
+−L2 + 1 ≤ u < 0.
+(3)
+In this paper, q-PSK modulation is employed. Thus, x and y denote q-ary arrays and (1) is
+expressed as
+X = (Xg,i) = (ξxg,i) = ξx;
+Y = (Yg,i) = (ξyg,i) = ξy,
+(4)
+where x = (xg,i), y = (yg,i), and xg,i, yg,i ∈ Zq = {0, 1, · · · , q−1} for 0 ≤ g < L1, 0 ≤ i < L2.
+Consider a set of N L-length sequences can be represented as
+C = {X0, X1, · · · , XN−1}
+where
+Xn = (Xn,0, Xn,1, · · · , Xn,L−1)
+for n = 0, 1, · · · , N − 1.
+January 4, 2023
+DRAFT
+
+6
+Definition 2: [19] If a set C consisting of N sequences of length L satisfies
+N−1
+�
+k=0
+ρ(Xk; u) =
+�
+�
+�
+�
+�
+NL,
+u = 0;
+0,
+u ̸= 0,
+(5)
+then the set C is called a Golay complementary set of size N, denoted by (N, L)-GCS. The
+GCP can be regarded as a special case of the GCS by setting N = 2.
+Definition 3: For a GCP (X0, X1), if another GCP (Y0, Y1) meets the following condition:
+ρ(X0, Y0; u) + ρ(X1, Y1; u) = 0, for all u,
+(6)
+then the two GCPs are called the Golay complementary mate of each other.
+Definition 4: A pair of arrays X and Y of array size L1 × L2 is called a 2D Golay
+complementary array pair if
+ρ(X; u1, u2) + ρ(Y ; u1, u2) =
+�
+�
+�
+�
+�
+2L1L2,
+u1 = u2 = 0;
+0,
+u1 ̸= 0 or u2 ̸= 0.
+(7)
+Definition 5: Let the array set G = {X0, X1, · · · , XN−1} where each array in set G is of
+size L1 × L2. If the array set G satisfies
+N−1
+�
+k=0
+ρ(Xk; u1, u2) =
+�
+�
+�
+�
+�
+NL1L2,
+u1 = u2 = 0;
+0,
+u1 ̸= 0 or u2 ̸= 0,
+(8)
+the set G is called the Golay complementary array set of set size N denoted by (N, L1, L2)-
+GCAS where L2 is defined as the length of the GCAS. If N = 2, the GCAS G is degraded as
+a GCAP.
+B. Generalized Boolean Functions
+A 2D generalized Boolean function (GBF) f in n + m binary variables y1, y2, · · · , yn,
+x1, x2, · · · , xm, is a function mapping: Zn
+2 ×Zm
+2 → Zq, where xi, yg ∈ {0, 1} for i = 1, 2, · · · , m
+and g = 1, 2, · · · , n. A monomial of degree r is given by any product of r distinct variables
+among y1, y2, · · · , yn, x1, x2, · · · , xm. For instance, x1x3y1y2 is a monomial of degree 4. Next,
+the variables z1, z2, · · · , zn+m are defined as
+zl =
+�
+�
+�
+�
+�
+yl
+if 1 ≤ l ≤ n;
+xl−n
+if n < l ≤ m + n,
+(9)
+January 4, 2023
+DRAFT
+
+7
+which are useful for our proposed constructions. For a 2D GBF with n + m variables, the 2D
+Zq-valued array
+f =
+�
+�
+�
+�
+�
+�
+�
+f0,0
+f0,1
+· · ·
+f0,2m−1
+f1,0
+f1,1
+· · ·
+f1,2m−1
+...
+...
+...
+...
+f2n−1,0
+f2n−1,1
+· · ·
+f2n−1,2m−1
+�
+�
+�
+�
+�
+�
+�
+(10)
+of size 2n×2m is given by letting fg,i = f((g1, g2, · · · , gn), (i1, i2, · · · , im)), where (g1, g2, · · · , gn)
+and (i1, i2, · · · , im) are binary vector representations of integers g = �n
+h=1 gh2h−1 and i =
+�n
+j=1 ij2j−1, respectively.
+Example 1: Taking q = 4, n = 2, and m = 3 for example, the 2D GBF is given as f =
+3z5z4 + z2z3 + 2z2. Then the array f of size 4 × 8 corresponding to f can be obtained, i.e.,
+f =
+�
+�
+�
+�
+�
+�
+�
+0
+0
+0
+0
+0
+0
+3
+3
+0
+0
+0
+2
+1
+1
+3
+3
+2
+3
+2
+3
+2
+3
+1
+2
+2
+3
+2
+3
+2
+3
+1
+2
+�
+�
+�
+�
+�
+�
+�
+.
+(11)
+The GBF f can be rewritten as f = 3x3x2 + y2x1 + 2y2. In this paper, we consider the array
+size ̸= 2n × 2m. Hence, we define the truncated array f (L) corresponding to the 2D GBF f by
+ignoring the last 2m − L columns of the corresponding array f.
+Example 2: Following the same notations given in Example 1, the truncated array f (6) is
+given by
+f (6) =
+�
+�
+�
+�
+�
+�
+�
+0
+0
+0
+0
+0
+0
+0
+0
+0
+2
+1
+1
+2
+3
+2
+3
+2
+3
+2
+3
+2
+3
+2
+3
+�
+�
+�
+�
+�
+�
+�
+.
+(12)
+For simplicity, we use f to stand for f (L) when L is known.
+C. System Model
+Considering downlink transmission from a BS to UEs where each has one single antenna,
+we suppose that the number of antennas at the BS is M = L1 × L2, i.e., the URA consists of
+L1 rows and L2 columns. Fig. 1 illustrates the diagram of data downlink transmission. For an
+January 4, 2023
+DRAFT
+
+8
+Fig. 1. Diagram of data transmission through STBC encoding and omnidirectional precoding.
+L1 × L2 URA, the steering matrix A(ϕ, θ) at the direction (ϕ, θ) with the (g, i)-th entry can be
+expressed as
+(A(ϕ, θ))g,i =e−j 2π
+λ gdy sin ϕ sin θ−j 2π
+λ idx sin ϕ cos θ,
+for g = 0, 1, . . . , L1 − 1, i = 0, 1, . . . , L2 − 1,
+θ ∈ [0, 2π], ϕ ∈ [0, π/2],
+(13)
+where dx and dy denote the vertical antenna and horizontal antenna inter-element spacings of
+the URA, respectively, and λ denotes the carrier wavelength. To enhance the spatial diversity
+and communication reliability, the STBC signal transmission scheme is used. The N ×M STBC
+is given by
+S ≜
+�
+�
+�
+�
+�
+�
+�
+s0(0)
+s0(1)
+· · ·
+s0(M − 1)
+s1(0)
+s1(1)
+· · ·
+s1(M − 1)
+...
+...
+...
+...
+sN−1(0)
+sN−1(1)
+· · ·
+sN−1(M − 1)
+�
+�
+�
+�
+�
+�
+�
+∈ CN×M
+(14)
+January 4, 2023
+DRAFT
+
+So(t)
+Xo(t) 
+7
+Data
+Omni-
+Space-time
+UE1
+Si(t)
+directional
+xi(t)
+block coding
+y(t)
+...
+precoding
+V
+SN-1(t)
+XL;L2
+-1(t)
+y
+d
+d
+(t)
+xo(t)
+x
+Omnidirectional
+precoding
+UE19
+where CN×M refers to the N-by-M complex space and sn(t) denotes the (n, t)-th element of
+the STBC at time instant t for t = 0, 1, · · · , M − 1. We define the precoding matrix Wn of size
+L1 × L2. The encoded symbols is given by
+x(t) = (x0(t), x1(t), · · · , xL1L2−1(t))T = vec
+�N−1
+�
+n=0
+Wn · sn(t)
+�
+, for t = 0, 1, · · · , M − 1,
+(15)
+which are transmitted by the L1L2 antennas of the URA. In the light-of-sight (LOS) channel
+without multipaths, the received signal at the direction (ϕ, θ) can be written as
+y(t) =
+N−1
+�
+n=0
+�
+vec(A(ϕ, θ))Tvec(Wn)
+�
+· sn(t) + η(t), t = 0, . . ., M − 1,
+(16)
+where η(t) is the additive Gaussian white noise (AWGN) at time instant t.
+D. Omnidirectional Precoding Matrices Based on 2D Arrays
+In this subsection, we list two necessary requirements for the design of OP matrices. Then,
+we will connect these two requirements with the conditions of 2D arrays.
+Requirement 1 (R1): Omnidirectional transmission.
+We consider the MIMO system with URA. Following (16), the received power E at the angle
+(ϕ, θ) is represented as
+E =
+N−1
+�
+n=0
+��[vec(A(ϕ, θ))Tvec(Wn)]
+��2 .
+(17)
+Therefore, to satisfy the omnidirectional transmission in the whole cell, (17) must be constant
+for all ϕ and θ.
+Requirement 2 (R2): Equal average power on each antenna.
+To enhance the efficiency of the power amplifier, the average transmission power on all L1×L2
+antennas is required to be equal. We define
+W = (vec(W0), vec(W1), · · · , vec(WN−1)) ,
+(18)
+where the array size of W is L1L2 × N. Hence, (15) can be rewritten as
+X = (x(0), x(1), · · · , x(M − 1)) = W S.
+(19)
+January 4, 2023
+DRAFT
+
+10
+Let s(t) be the t-th column of S. Throughout this paper, we assume E
+�
+s(t)s(t)H�
+=IN. The
+transmitted signal on the (l1, l2)-th antenna is (W s)l2L1+l1. The average power on the (l1, l2)-th
+antenna can be expressed as
+E
+�
+|(W s)l2L1+l1|2�
+=
+�
+W E
+�
+s(t)s(t)H�
+W H�
+l2L1+l1,l2L1+l1
+= (W W H)l2L1+l1,l2L1+l1.
+(20)
+Therefore, the condition to guarantee equal power on each antenna is equivalent to
+diag(W W H) = N1.
+(21)
+Next, we will derive two sufficient conditions on the precoding matrices to fulfill requirements
+R1 and R2.
+Lemma 1: [21] For an L1 × L2 URA, if the precoding matrices W0, W1, · · · , WN−1 of size
+L1 × L2 form an (N, L1, L2)-GCAS, then the omnidirectional transmission is achieved.
+Lemma 2: For an L1×L2 URA, if the precoding matrices W0, W1, · · · , WN−1 of size L1×L2
+are unimodular, then the average power on each antenna is equal.
+Proof: In order to meet the requirement for equal average power on each antenna, the
+precoding matrix W must satisfy (21). We let wi = vec(Wi), for i = 0, 1, · · · , N − 1. Then,
+diag
+�
+W W H�
+=
+�N−1
+�
+i=0
+|(wi)0|2 ,
+N−1
+�
+i=0
+|(wi)1|2 , · · · ,
+N−1
+�
+i=0
+|(wi)L1L2−1|2
+�T
+= N1
+(22)
+since we have
+|(wi)n|2 = 1,
+for i = 0, 1, · · · , N − 1 and n = 0, 1, · · · , L1L2 − 1.
+(23)
+According to (21), the requirement (R2) is fulfilled.
+In the sequel, the design of OP matrices W0, W1, · · · , WN−1 are based on Lemma 1 and
+Lemma 2. That is, our goal is to construct unimodular GCASs with flexible sizes.
+III. GCASS WITH FLEXIBLE ARRAY SIZE
+In this section, two constructions of 2D GCASs with arbitrary array lengths based on 2D
+GBFs will be proposed. By recalling the function mapping in (9), we present our first theorem
+in the following.
+January 4, 2023
+DRAFT
+
+11
+Theorem 1: For any integers q, m, n ≥ 2, and k < m, v is an integer satisfies 0 ≤ v ≤ m−k
+and let π be a permutation of {1, 2, · · · m + n − k} satisfying {zπ(1), zπ(2), · · · , zπ(v+n)} =
+{z1, z2, · · · , zv+n}. The 2D generalized Boolean function can be written as
+f = q
+2
+�m+n−k−1
+�
+l=1
+zπ(l)zπ(l+1)
+�
++
+m+n
+�
+s=1
+pszs + p0
+(24)
+where ps ∈ Zq. The array set
+G =
+�
+f + q
+2
+k
+�
+α=1
+λαzm+n−k+α + q
+2λk+1zπ(1) : λα ∈ {0, 1}
+�
+(25)
+is a q-ary (2k+1, 2n, 2m−1 + �k−1
+α=1 dα2m−k+α−1 + d02v)-GCAS where dα ∈ {0, 1}.
+Proof: Without loss of generality, we consider L1 = 2n and L2 = 2m−1+�k−1
+α=1 2m−k+α−1+ 2v.
+We need to show that
+�
+c∈G
+L1−1−u1
+�
+g=0
+L2−1−u2
+�
+i=0
+�
+ξcg+u1,i+u2−cg,i�
+= 0
+(26)
+for 0 ≤ u1 < 2n, 0 ≤ u2 < 2m−1 + �k−1
+α=1 2m−k+α−1 + 2v and (u1, u2) ̸= (0, 0). Then we let
+h = g + u1 and j = i + u2 for any integers g and i. We also let (g1, g2, · · · , gn),(i1, i2, · · · , im),
+(h1, h2, · · · , hn), and (j1, j2, · · · , jm) be the binary representations of g, i, h, and j, respectively.
+For the ease of presentation, we denote
+al =
+�
+�
+�
+�
+�
+gl
+for 1 ≤ l ≤ n;
+il−n for n < l ≤ n + m;
+bl =
+�
+�
+�
+�
+�
+hl
+for 1 ≤ l ≤ n;
+jl−n for n < l ≤ n + m;
+(27)
+In what follows, we consider four cases to show that the above formula holds.
+Case 1: If aπ(1) ̸= bπ(1), we can find that c′ = c + (q/2)zπ(1) for any arrayc ∈ G satisfying
+ch,j − cg,i − c′
+h,j+c′
+g,i = q
+2(aπ(1) − bπ(1)) ≡ q
+2
+(mod q).
+(28)
+Therefore, we have
+ξch,j−cg,i + ξc′
+h,j−c′
+g,i = 0.
+(29)
+Case 2: If am+n−k+α ̸= bm+n−k+α, we can find that c′ = c + (q/2)zm+n−k+α for any array
+c ∈ G. Similar to Case 1, we have
+ξch,j−cg,i + ξc′
+h,j−c′
+g,i = 0.
+(30)
+January 4, 2023
+DRAFT
+
+12
+Case 3: If aπ(1) = bπ(1) and am+n−k+α = bm+n−k+α for α = 1, 2, · · · , k. Suppose that α′
+is the largest integer satisfying am+n−k+α′ = bm+n−k+α′ = 0 for α′ ≤ k. Then we assume β
+is the smallest integer which satisfies aπ(β) ̸= bπ(β). Let a′ and b′ be integers distinct from a
+and b, respectively, only in one position π(β − 1). In other words, a′
+π(β−1) = 1 − aπ(β−1) and
+b′
+π(β−1) = 1 − bπ(β−1). If 1 ≤ π(β − 1) ≤ n, by using the above definition, we have
+cg′,i − cg,i
+= q
+2
+�
+aπ(β−2)g′
+π(β−1) − aπ(β−2)gπ(β−1) + g′
+π(β−1)aπ(β)
+−gπ(β−1)aπ(β)
+�
++ pπ(β−1)g′
+π2(β−1) − pπ(β−1)gπ(β−1)
+≡ q
+2(aπ(β−2) + aπ(β)) + pπ(β−1)(1 − 2gπ(β−1))
+(mod q).
+(31)
+where a′
+π(β−1) = g′
+π(β−1) and aπ(β−1) = gπ(β−1). Since aπ(β−2) = bπ(β−2) and aπ(β−1) = bπ(β−1),
+we have
+ch,j − cg,i − ch′,j + cg′,i
+≡ q
+2(aπ(β−2) − bπ(β−2) + aπ(β) − bπ(β))
++ pπ(β−1)(2hπ(β−1) − 2gπ(β−1))
+≡ q
+2(aπ(β) − bπ(β)) ≡ q
+2
+(mod q)
+(32)
+implying ξch,j−cg,i/ξch′,j−cg′,i = −1. We can also obtain
+ξch,j−cg,i + ξch′,j−cg′,i = 0.
+(33)
+If n < π(β − 1) ≤ n + m, note that a′
+π(β−1) = i′
+π(β−1)−n and aπ(β−1) = iπ(β − 1) − n according
+to (27). Following the similar argument as given above, we can get ξch,j−cg,i + ξch,j′−cg,i′ = 0.
+Case 4: If aπ(1) = bπ(1) and am+n−k+α = bm+n−k+α = 1 for α = 1, 2, · · · , k. We assume β is
+the smallest integer such that aπ(β) ̸= bπ(β). Since as = bs = 0 for s = v+n+1, v+n+2, · · · , m+
+n−k, we can obtain π(β) ≤ v+n implying π(β −1) ≤ v+n. If 1 ≤ π(β−1) ≤ n, by following
+the similar argument as given above, we have ξch,j−cg,i +ξch′,j−cg′,i = 0. If n < π(β −1) ≤ v+n,
+we have ξch,j−cg,i + ξch,j′−cg,i′ = 0. From Cases 1 to 4, the theorem can be proved.
+Remark 1: The parameter 2m−1+�k−1
+α=1 dα2m−k+α−1+d02v of the proposed GCASs in Theorem
+1 can be any arbitrary length since m, k, v are flexible and dα ∈ {0, 1}.
+Example 3: Taking q = 2, m = 6, n = 2, k = 1, and v = 0, we let π = (1, 2, 3, 4, 5, 6, 7).
+The generalized Boolean function is f = z1z2 + z2z3 + z3z4 + z4z5 + z5z6 + z6z7 = x1x2 +
+x2x3 + x3x4 + x4x5 + y1y2 + y2x1 by setting pk = 0 for k = 0, 1, . . . , m + n. The array set
+January 4, 2023
+DRAFT
+
+13
+TABLE II
+THE CONSTRUCTED (4, 4, 33)-GCAS IN EXAMPLE 3
+c0 =
+�
+�
+�
+�
+�
+�
+�
+0
+1
+1
+1
+1
+0
+1
+1
+1
+0
+0
+0
+1
+0
+1
+1
+1
+0
+0
+0
+0
+1
+0
+0
+1
+0
+0
+0
+1
+0
+1
+1
+1
+1
+0
+0
+0
+0
+1
+0
+0
+0
+1
+1
+1
+0
+1
+0
+0
+0
+1
+1
+1
+1
+0
+1
+1
+0
+1
+1
+1
+0
+1
+0
+0
+1
+1
+0
+0
+0
+0
+1
+0
+0
+0
+1
+1
+1
+0
+1
+0
+0
+0
+1
+1
+1
+1
+0
+1
+1
+0
+1
+1
+1
+0
+1
+0
+0
+0
+1
+0
+0
+0
+0
+1
+0
+0
+0
+1
+1
+1
+0
+1
+0
+0
+0
+1
+1
+1
+1
+0
+1
+1
+0
+1
+1
+1
+0
+1
+0
+0
+1
+�
+�
+�
+�
+�
+�
+�
+c1 =
+�
+�
+�
+�
+�
+�
+�
+0
+0
+1
+0
+1
+1
+1
+0
+1
+1
+0
+1
+1
+1
+1
+0
+1
+1
+0
+1
+0
+0
+0
+1
+1
+1
+0
+1
+1
+1
+1
+0
+1
+1
+1
+0
+1
+0
+0
+0
+1
+0
+0
+1
+0
+0
+0
+0
+1
+0
+0
+1
+0
+1
+1
+1
+0
+0
+0
+1
+0
+0
+0
+0
+1
+1
+1
+1
+0
+1
+0
+0
+0
+1
+0
+0
+1
+0
+0
+0
+0
+1
+0
+0
+1
+0
+1
+1
+1
+0
+0
+0
+1
+0
+0
+0
+0
+1
+0
+1
+1
+0
+1
+0
+0
+0
+1
+0
+0
+1
+0
+0
+0
+0
+1
+0
+0
+1
+0
+1
+1
+1
+0
+0
+0
+1
+0
+0
+0
+0
+1
+1
+�
+�
+�
+�
+�
+�
+�
+c2 =
+�
+�
+�
+�
+�
+�
+�
+0
+1
+1
+1
+1
+0
+1
+1
+1
+0
+0
+0
+1
+0
+1
+1
+1
+0
+0
+0
+0
+1
+0
+0
+1
+0
+0
+0
+1
+0
+1
+1
+1
+1
+0
+0
+0
+0
+1
+0
+0
+0
+1
+1
+1
+0
+1
+0
+0
+0
+1
+1
+1
+1
+0
+1
+1
+0
+1
+1
+1
+0
+1
+0
+0
+1
+0
+1
+1
+1
+1
+0
+1
+1
+1
+0
+0
+0
+1
+0
+1
+1
+1
+0
+0
+0
+0
+1
+0
+0
+1
+0
+0
+0
+1
+0
+1
+1
+1
+0
+1
+1
+1
+1
+0
+1
+1
+1
+0
+0
+0
+1
+0
+1
+1
+1
+0
+0
+0
+0
+1
+0
+0
+1
+0
+0
+0
+1
+0
+1
+1
+0
+�
+�
+�
+�
+�
+�
+�
+c3 =
+�
+�
+�
+��
+�
+�
+�
+0
+0
+1
+0
+1
+1
+1
+0
+1
+1
+0
+1
+1
+1
+1
+0
+1
+1
+0
+1
+0
+0
+0
+1
+1
+1
+0
+1
+1
+1
+1
+0
+1
+1
+1
+0
+1
+0
+0
+0
+1
+0
+0
+1
+0
+0
+0
+0
+1
+0
+0
+1
+0
+1
+1
+1
+0
+0
+0
+1
+0
+0
+0
+0
+1
+1
+0
+0
+1
+0
+1
+1
+1
+0
+1
+1
+0
+1
+1
+1
+1
+0
+1
+1
+0
+1
+0
+0
+0
+1
+1
+1
+0
+1
+1
+1
+1
+0
+1
+0
+0
+1
+0
+1
+1
+1
+0
+1
+1
+0
+1
+1
+1
+1
+0
+1
+1
+0
+1
+0
+0
+0
+1
+1
+1
+0
+1
+1
+1
+1
+0
+0
+�
+�
+�
+�
+�
+�
+�
+0
+200
+2
+400
+40
+600
+20
+800
+0
+1000
+0
+-20
+-2
+-40
+Fig. 2. The summation of autocorrelations of constituent arrays in the GCAS in Example 3.
+G = {f, f + x8, f + y1, f + x8 + y1} is a GCAS of size 4 and the array size is 4 × 33. We let
+G = {c0, c1, c2, c3} and list the constituent arrays in Table II. Fig. 2 shows the AACF sum of
+set G is zero at shift u1 ̸= 0 or u2 ̸= 0. Thus, we can find that array set G is a (4, 4, 33)-GCAS.
+January 4, 2023
+DRAFT
+
+14
+Next, we introduce a lemma which illustrates a construction of (4, 2n, 2m−1 +2v)-GCAS from
+2D GBFs.
+Lemma 3: [32, Th. 1] For nonnegative integers m, n, and v with 0 ≤ v < m − 1, let π1 be
+a permutation of {1, 2, · · · , m − 1} and π2 be a permutation of {1, 2, · · · , n}. The 2D GBF is
+given by
+f =q
+2
+�m−2
+�
+k=1
+xπ1(k)xπ1(k+1) +
+n−1
+�
+k=1
+yπ2(k)yπ2(k+1) + xπ1(m−1)xm + xmyπ2(1)
+�
++
+m
+�
+l=1
+plxl +
+n
+�
+s=1
+κsys + p0
+(34)
+where pl, κs ∈ Zq. Then the array set
+G =
+�
+f, f + q
+2xπ1(1), f + q
+2yπ2(n), f + q
+2xπ1(1) + q
+2yπ2(n)
+�
+is a (4, 2n, 2m−1 + 2v)-GCAS.
+Since the set size of the GCAS from Lemma 3 is limited to 4, we propose a general
+construction of 2D GCASs with more flexible array sizes and set sizes which can include Lemma
+3 as a special case.
+Theorem 2: For any integers q, m, n ≥ 2, and k < m, v is an integer satisfies 0 ≤ v ≤ m−k.
+Assume that π1 is a permutation of {1, 2, · · · m} and π2 is a permutation of {1, 2, · · · n}. The
+2D generalized Boolean function can be written as
+f =q
+2
+�m−k−1
+�
+l=1
+xπ1(l)xπ1(l+1) +
+n−1
+�
+s=1
+yπ2(s)yπ2(s+1) + xπ1(m)yπ2(n)
+�
++
+m−k
+�
+l=1
+µlxπ1(l)xπ1(m) +
+m
+�
+l=1
+plxk +
+n
+�
+s=1
+κsys + p0
+(35)
+where µl, pl, κs, ∈ Zq. The array set
+G =
+�
+f + q
+2
+k−1
+�
+α=1
+λαxπ1(m−k+α) + q
+2λkyπ2(1) + q
+2λk+1xπ1(1) : λα ∈ {0, 1}
+�
+(36)
+is a q-ary (2k+1, 2n, 2m−1 + �k−1
+α=1 dα2π1(m−k+α)−1 + d02v)-GCAS where dα ∈ {0, 1} if the
+following three conditions hold.
+(C1) {π1(1), π1(2), · · · , π1(v)} = {1, 2, · · · , v} if v > 0;
+(C2) π1(m − k + α) < π1(m − k + α + 1) for 1 ≤ α ≤ k − 1 where π1(m) = m;
+(C3) For 1 ≤ α ≤ k − 1 and 2 ≤ β ≤ m − k, if π1(β) < π1(m − k + α), then π1(β − 1) <
+π1(m − k + α).
+January 4, 2023
+DRAFT
+
+15
+Proof: Similarly, we consider L1 = 2n and L2 = 2m−1 + �k−1
+α=1 2π1(m−k+α)−1 + 2v. Then
+we would like to prove that
+�
+C
+ρ(C; u1, u2) =
+�
+c∈G
+L1−1−u1
+�
+g=0
+L2−1−u2
+�
+i=0
+�
+ξcg+u1,i+u2−cg,i�
+= 0
+(37)
+for 0 ≤ u1 < 2n, 0 ≤ u2 < 2m−1 + �k−1
+α=1 2π1(m−k+α)−1 + 2v and (u1, u2) ̸= (0, 0). From (4)
+we can find that
+c = q
+2
+�m−k−1
+�
+l=1
+xπ1(l)xπ1(l+1) +
+n−1
+�
+s=1
+yπ2(s)yπ2(s+1) + xπ1(m)yπ2(n)
+�
++
+m−k
+�
+l=1
+µlxπ1(l)xπ1(m) +
+m
+�
+l=1
+plxl +
+n
+�
+s=1
+κsys + p0 · 1.
+(38)
+Then we let h = g + u1 and j = i + u2 for any integers g and i. Next, we discuss seven cases
+to complete the proof.
+Case 1: Assuming u1 > 0, u2 ≥ 0, and gπ2(1) ̸= hπ2(1), we can find an array c′ = c +
+(q/2)yπ2(1) ∈ G for any array c ∈ G. Therefore, we can obtain
+ch,j − cg,i − c′
+h,j+c′
+g,i = q
+2(gπ2(1) − hπ2(1)) ≡ q
+2
+(mod q)
+(39)
+Since gπ2(1) ̸= hπ2(1), we have
+ξch,j−cg,i/ξc′
+h,j−c′
+g,i = ξ
+q
+2 = −1.
+(40)
+Thus,
+ξch,j−cg,i + ξc′
+h,j−c′
+g,i = 0.
+(41)
+Case 2: If u1 > 0, u2 ≥ 0, and gπ2(1) = hπ2(1). Let β be the smallest integer such that
+gπ2(β) ̸= hπ2(β). We define g′ and h′ are two integers which are distinct from g and h only in
+one position π2(β − 1), respectively. Then, similar to Case 2 of Theorem 1, we have
+ξch,j−cg,i + ξch′,j−cg′,i = 0.
+(42)
+Case 3: We suppose im ̸= jm, u1 = 0 and u2 > 0. We let g′ be an integer distinct from
+i only in one position, i.e., g′
+π2(n) = 1 − gπ2(n). Similar to Case 3 of Theorem 1, we have
+ξcg,j−cg,i + ξcg′,j−cg′,i = 0.
+Case 4: If u1 = 0, u2 > 0, and iπ1(1) ̸= jπ1(1) or iπ1(m−k+α) ̸= jπ1(m−k+α), we can find an
+array c′ = c + (q/2)xπ1(1) ∈ G or c′ = c + (q/2)xπ1(m−k+α) for any array c ∈ G. Similar to
+Case 1, we can obtain ξcg,j−cg,i + ξc′
+g,j−c′
+g,i = 0.
+January 4, 2023
+DRAFT
+
+16
+Case 5: Suppose u1 = 0, u2 > 0, iπ1(1) = jπ1(1), and iπ1(m−k+α) = jπ1(m−k+α) for all
+α = 1, 2, · · · , k. Suppose that α′ is the largest non-negative integer satisfying iπ1(m−k+α′) =
+jπ1(m−k+α′) = 0. Then we assume β is the smallest integer which satisfies iπ1(β) ̸= jπ1(β).
+Here, we have is = js = 0 for s = π1(m − k + α′) + 1, π1(m − k + α′) + 2, . . . , m − 1, and
+s ̸= π1(m − k + α) for α = α′ + 1, α′ + 2, . . . , k. Hence, it implies π1(β) < π1(m − k + α′)
+and π1(β − 1) < π1(m − k + α′) according to the condition (C-3). Let i′ and j′ be integers that
+differ from i and j, respectively, in the position π1(β − 1). Similar to Case 2, we have
+ξcg,j−cg,i + ξcg,j′−cg,i′ = 0.
+(43)
+Case 6: Suppose u1 = 0, u2 > 0, iπ1(1) = jπ1(1), and iπ1(m−k+α) = jπ1(m−k+α) = 1 for all
+α = 1, 2, · · · , k. Then we assume β is the smallest integer which satisfies iπ1(β) ̸= jπ1(β). Since
+is = js = 0 for s = v + 1, v + 2, · · · , m − k and s ̸= π1(m − k + α) for α = 1, 2, . . . , k − 1, we
+can obtain π1(β) ≤ v implying π1(β − 1) ≤ v. Similar to Case 2, we have
+ξcg,j−cg,i + ξcg,j′−cg,i′ = 0.
+(44)
+From Cases 1 to 6, the theorem can be proved.
+Remark 2: Taking σ2(l) = π2(n − l + 1) for l = 1, 2, . . . , n and π1(m − k + α) = m − k + α
+for α = 1, 2, . . . , k in Theorem 2, (34) can be represented as
+f =q
+2
+�m−k−1
+�
+k=1
+xπ1(k)xπ1(k+1) +
+n−1
+�
+k=1
+yσ2(k)yσ2(k+1) + xmyσ2(1)
+�
++
+m−k
+�
+l=1
+µlxπ1(l)xm
++
+m
+�
+l=1
+plxl +
+n
+�
+s=1
+κsys + p0
+(45)
+where pl, κs ∈ Zq. We can find that the result of Lemma 3 is a special case of Theorem 2 by
+simply setting k = 1, µm−1 = q
+2, and µl = 0 for l = 1, · · · , m − 2.
+Example 4: Taking q = 2, m = 5, n = 2, k = 2, and v = 0, we let π1 = (1, 2, 4, 3, 5) and
+π2 = (1, 2). The generalized Boolean function is f = x1x2 + x2x4 + y1y2 + x5y1 by setting
+pl, κs = 0. The array set G is a GCAS of size 8 when the truncated size L1 = 4 L2 = 21. We
+let G = {c0, c1, · · · , c7} and list the constituent arrays in Table III. Also, their AACF sum is
+shown as Fig. 3.
+IV. SIMULATION RESULTS
+In this section, we present the numerical results including the power radiation pattern and
+BER performance by using our proposed 2D GCASs for massive MIMO systems with URA.
+January 4, 2023
+DRAFT
+
+17
+TABLE III
+THE CONSTRUCTED (8, 4, 21)-GCAS IN EXAMPLE 4
+c0 =
+�
+�
+�
+�
+�
+�
+�
+0
+0
+0
+1
+0
+1
+1
+1
+0
+0
+1
+0
+1
+0
+1
+1
+0
+1
+1
+1
+0
+0
+0
+0
+1
+0
+1
+1
+1
+0
+0
+1
+0
+1
+0
+1
+1
+0
+1
+1
+1
+0
+0
+1
+0
+0
+0
+0
+1
+0
+0
+1
+1
+1
+1
+1
+1
+0
+0
+0
+1
+0
+0
+1
+0
+1
+1
+1
+1
+0
+1
+1
+0
+0
+0
+0
+0
+0
+1
+1
+1
+0
+1
+1
+�
+�
+�
+�
+�
+�
+�
+c1 =
+�
+�
+�
+�
+�
+�
+�
+0
+0
+0
+1
+1
+0
+0
+0
+0
+0
+1
+0
+0
+1
+0
+0
+0
+1
+1
+1
+1
+0
+0
+0
+1
+1
+0
+0
+0
+0
+0
+1
+0
+0
+1
+0
+0
+0
+1
+1
+1
+1
+0
+1
+0
+0
+1
+1
+0
+1
+0
+1
+1
+1
+0
+0
+0
+1
+0
+0
+1
+0
+1
+1
+0
+1
+1
+0
+0
+1
+0
+1
+0
+0
+0
+1
+1
+1
+0
+1
+1
+0
+1
+0
+�
+�
+�
+�
+�
+�
+�
+c2 =
+�
+�
+�
+�
+�
+�
+�
+0
+0
+0
+1
+0
+1
+1
+1
+0
+0
+1
+0
+1
+0
+1
+1
+1
+0
+0
+0
+1
+0
+0
+0
+1
+0
+1
+1
+1
+0
+0
+1
+0
+1
+0
+1
+1
+1
+0
+0
+0
+1
+0
+1
+0
+0
+0
+0
+1
+0
+0
+1
+1
+1
+1
+1
+1
+0
+1
+1
+0
+1
+1
+1
+0
+1
+1
+1
+1
+0
+1
+1
+0
+0
+0
+0
+0
+0
+1
+0
+0
+1
+0
+0
+�
+�
+�
+�
+�
+�
+�
+c3 =
+�
+�
+�
+�
+�
+�
+�
+0
+0
+0
+1
+1
+0
+0
+0
+0
+0
+1
+0
+0
+1
+0
+0
+1
+0
+0
+0
+0
+0
+0
+0
+1
+1
+0
+0
+0
+0
+0
+1
+0
+0
+1
+0
+0
+1
+0
+0
+0
+0
+0
+1
+0
+0
+1
+1
+0
+1
+0
+1
+1
+1
+0
+0
+0
+1
+1
+1
+0
+1
+0
+1
+0
+1
+1
+0
+0
+1
+0
+1
+0
+0
+0
+1
+1
+1
+0
+0
+0
+1
+0
+1
+�
+�
+�
+�
+�
+�
+�
+c4 =
+�
+�
+�
+�
+�
+�
+�
+0
+0
+0
+1
+0
+1
+1
+1
+0
+0
+1
+0
+1
+0
+1
+1
+0
+1
+1
+1
+0
+1
+1
+1
+0
+1
+0
+0
+0
+1
+1
+0
+1
+0
+1
+0
+0
+1
+0
+0
+0
+1
+0
+1
+0
+0
+0
+0
+1
+0
+0
+1
+1
+1
+1
+1
+1
+0
+0
+0
+1
+0
+0
+0
+1
+0
+0
+0
+0
+1
+0
+0
+1
+1
+1
+1
+1
+1
+0
+0
+0
+1
+0
+0
+�
+�
+�
+�
+�
+�
+�
+c5 =
+�
+�
+�
+�
+�
+�
+�
+0
+0
+0
+1
+1
+0
+0
+0
+0
+0
+1
+0
+0
+1
+0
+0
+0
+1
+1
+1
+1
+1
+1
+1
+0
+0
+1
+1
+1
+1
+1
+0
+1
+1
+0
+1
+1
+1
+0
+0
+0
+0
+0
+1
+0
+0
+1
+1
+0
+1
+0
+1
+1
+1
+0
+0
+0
+1
+0
+0
+1
+0
+1
+0
+1
+0
+0
+1
+1
+0
+1
+0
+1
+1
+1
+0
+0
+0
+1
+0
+0
+1
+0
+1
+�
+�
+�
+�
+�
+�
+�
+c6 =
+�
+�
+�
+�
+�
+�
+�
+0
+0
+0
+1
+0
+1
+1
+1
+0
+0
+1
+0
+1
+0
+1
+1
+1
+0
+0
+0
+1
+1
+1
+1
+0
+1
+0
+0
+0
+1
+1
+0
+1
+0
+1
+0
+0
+0
+1
+1
+1
+0
+0
+1
+0
+0
+0
+0
+1
+0
+0
+1
+1
+1
+1
+1
+1
+0
+1
+1
+0
+1
+1
+0
+1
+0
+0
+0
+0
+1
+0
+0
+1
+1
+1
+1
+1
+1
+0
+1
+1
+0
+1
+1
+�
+�
+�
+�
+�
+�
+�
+c7 =
+�
+�
+�
+�
+�
+�
+�
+0
+0
+0
+1
+1
+0
+0
+0
+0
+0
+1
+0
+0
+1
+0
+0
+1
+0
+0
+0
+0
+1
+1
+1
+0
+0
+1
+1
+1
+1
+1
+0
+1
+1
+0
+1
+1
+0
+1
+1
+1
+1
+0
+1
+0
+0
+1
+1
+0
+1
+0
+1
+1
+1
+0
+0
+0
+1
+1
+1
+0
+1
+0
+0
+1
+0
+0
+1
+1
+0
+1
+0
+1
+1
+1
+0
+0
+0
+1
+1
+1
+0
+1
+0
+�
+�
+�
+�
+�
+�
+�
+A. Power Radiation Pattern
+According to (16), the power radiation pattern �N−1
+n=0
+��[vec(A(ϕ, θ))Tvec(Wn)]
+��2 can be ob-
+tained. We first consider the massive MIMO system equipped with a URA of size 4 × 33,
+January 4, 2023
+DRAFT
+
+18
+0
+200
+2
+400
+40
+600
+20
+800
+0
+1000
+0
+-20
+-2
+-40
+Fig. 3. The summation of autocorrelations of constituent arrays in the GCAS in Example 4.
+i.e., L1 = 4 and L2 = 33. We take the GCS G = {c0, c1, c2, c3} listed in Table II to
+generate the precoding matrices {W0, W1, W2, W3} = {(−1)c0, (−1)c1, (−1)c2, (−1)c3} with
+the omnidirectional property. The power radiation pattern of the GCAS-based scheme with array
+size 4 × 33 is perfectly omnidirectional as illustrated in Fig 4(a).
+For the purpose of comparison, we also show the power radiation patterns of the precoding
+matrices based on Zadoff-Chu sequences and random-matrices whose elements are randomly
+generated from “+1” and “−1”. The ZC-based precoder consists of four 4 × 33 precoding
+matrices, which are obtained based on a ZC sequence of length 4 and a ZC sequence of length
+33 [21]. Fig. 4(b) illustrates the power radiation pattern of the ZC-based precoder. We can
+find that its power radiation pattern is not omnidirectional. The random-matrix-based precoder
+consists of four 4 × 33 precoding matrices. The elements in the random-matrix-based precoding
+matrices are generated by selecting the elements from {1, −1} with equal probability. Fig. 4(c)
+describes the power radiation pattern of the random matrix-based precoder. We can observe that
+the power radiation pattern is not omnidirectional.
+Next, we consider the massive MIMO system equipped with a URA of size 4 × 21, i.e.,
+January 4, 2023
+DRAFT
+
+19
+(a) GCAS-based precoding.
+(b) ZC-based precoding.
+(c) Random-matrix-based precoding.
+Fig. 4. Power radiation pattern with 4 × 33 URA and 4 × 4 STBC.
+L1 = 4 and L2 = 21. We use the GCS G = {c0, c1, · · · , c7} listed in Table III for the precoding
+matrix {W0, W1, · · · , W7} = {(−1)c0, (−1)c1, · · · , (−1)c7}. The power radiation pattern of the
+GCAS-based scheme with array size 4×21 is described in Fig. 5(a). The perfect omnidirectional
+property can be observed. We also see that the power radiation patterns of the ZC-based precoder
+and the random-matrix precoder shown in Fig. 5(b) and Fig. 5(c) are not omnidirectional. The
+ZC-based precoding matrices are obtained by a ZC sequence of length 4 and ZC sequence of
+21 [21].
+B. Bit Error Rate Performance
+In this subsection, we present the BER performance of our proposed 2D GCAS-based schemes.
+We first consider the massive MIMO system equipped with a URA of size 4×33. We let N = 4
+January 4, 2023
+DRAFT
+
+X-axis-0.500.51Sy-ax1sX-axis550.50.51Sy-ax.
+1S0.3X-axis00.50.5Sy-ax.
+1S20
+(a) GCAS-based precoding.
+(b) ZC-based precoding.
+(c) Random-matrix-based precoding.
+Fig. 5. Power radiation pattern with 4 × 21 URA and 8 × 8 STBC.
+and then the 4 × 4 orthogonal real STBC be presented as
+S =
+�
+�
+�
+�
+�
+�
+�
+s0
+−s1
+−s2
+−s3
+s1
+s0
+s3
+−s2
+s2
+−s3
+s0
+s1
+s3
+s2
+−s1
+s0
+�
+�
+�
+�
+�
+�
+�
+,
+(46)
+where s0, s1, s2, s3 are binary phase shift keying (BPSK) modulated symbols. We employ the
+maximum likelihood (ML) decoding here. For each realization, the elevation and the azimuth
+angles are uniformly distributed at random between [0, π/2] and [0, 2π], respectively. For com-
+parison, the ZC-based precoder and random-matrix-based precoder are the same as mentioned in
+Section IV-A. The BER performances of three different schemes are depicted in Fig. 6. We can
+find that the 2D GCAS-based scheme outperform the others. At BER of 10−4, there are 1.6 dB
+and 3.6 dB gains over the ZC-based scheme and the random-matrix-based scheme, respectively.
+January 4, 2023
+DRAFT
+
+X-axis-0.500.51Sy-ax1sX-axis-0.500.51Sy-ax1s0.8X-axis-0.500.510.2Sy-06.ax1s21
+-2
+0
+2
+4
+6
+8
+10
+12
+14
+SNR (dB)
+10-8
+10-6
+10-4
+10-2
+100
+BER
+GCAS-based precoding (N=4)
+ZC-based precoding (N=4)
+Random-matrix-based precoding (N=4)
+Fig. 6. BER performance of the different schemes for a 4 × 33 URA.
+Next, we consider the massive MIMO system equipped with a URA of size 4 × 21. We
+consider 8 × 8 STBC and the 8 × 8 orthogonal real STBC is given by
+S =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+s0
+s1
+s2
+s3
+s4
+s5
+s6
+s7
+−s1
+s0
+s3
+−s2
+s5
+−s4
+−s7
+s6
+−s2
+−s3
+s0
+s1
+s6
+s7
+−s4
+−s5
+−s3
+s2
+−s1
+s0
+s7
+−s6
+s5
+−s4
+−s4
+−s5
+−s6
+−s7
+s0
+s1
+s2
+s3
+−s5
+s4
+−s7
+s6
+−s1
+s0
+−s3
+s2
+−s6
+s7
+s4
+−s5
+−s2
+s3
+s0
+−s1
+−s7
+−s6
+s5
+−s4
+−s3
+s2
+s1
+s0
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+(47)
+where s0, s1, · · · , s7 are BPSK modulated symbols. We also take the ZC-based precoding and
+random-matrix-based precoding for comparison. The BER performance comparison for these
+three different schemes is depicted in Fig. 7. At BER of 10−4, there are 0.2 dB and 1.8 dB
+gains over the ZC-based scheme and the random-matrix-based scheme, respectively. As a result,
+the 2D GCASs are good candidates as precoding matrices for omnidirectional transmission in
+January 4, 2023
+DRAFT
+
+22
+-2
+0
+2
+4
+6
+8
+10
+12
+14
+SNR (dB)
+10-8
+10-6
+10-4
+10-2
+100
+BER
+GCAS-based precoding (N=8)
+ZC-based precoding (N=8)
+Random-matrix-based precoding (N=8)
+Fig. 7. BER performance of the different schemes for a 4 × 21 URA.
+massive MIMO systems.
+V. CONCLUSION
+In this paper, constructions of 2D GCASs with flexible array sizes have been proposed in
+Theorems 1 and 2. Our constructions can be obtained directly from 2D GBFs without the aid
+of special sequences. Besides, our proposed GCASs have flexible array sizes which can fit
+more antenna configuration. Furthermore, Theorem 2 can include the results in [32] as a special
+case. Simulation results showed that the omnidirectional transmission can be achieved when
+the precoding matrices are based on the proposed GCASs. The BER performance due to their
+omnidirectional power radiation patterns, the ZC-based scheme and random-matix-based have
+inferior performances because their power radiation patterns both are not ideally omnidirectional.
+Although Theorems 1 and 2 can provide direct constructions of 2D GCASs, the first dimension
+has size L1 limited to 2n. Therefore, the future work includes the extension of constructions of
+2D GCASs of which both dimensions have non-power-of-two sizes.
+January 4, 2023
+DRAFT
+
+23
+REFERENCES
+[1] M. J. E. Golay, “Complementary series,” IRE Trans. Inf. Theory, vol. IT-7, pp. 82–87, Apr. 1961.
+[2] C.-C. Tseng and C. L. Liu, “Complementary sets of sequences,” IEEE Trans. Inf. Theory, vol. IT-18, no. 5, pp. 644–652,
+Sep. 1972.
+[3] N. Suehiro and M. Hatori, “N-shift cross-orthogonal sequences,” IEEE Trans. Inf. Theory, vol. 34, no. 1, pp. 143–146,
+Jan. 1988.
+[4] A. Pezeshki, A. R. Calderbank, W. Moran, and S. D. Howard, “Doppler resilient Golay complementary waveforms,” IEEE
+Trans. Inf. Theory, vol. 54, no. 9, pp. 4254–4266, Sep. 2008.
+[5] P. Spasojevic and C. N. Georghiades, “Complementary sequences for ISI channel estimation,” IEEE Trans. Inf. Theory,
+vol. 47, no. 3, pp. 1145–1152, Mar. 2001.
+[6] D. Su, Y. Jiang, X. Wang, and X. Gao, “Omnidirectional precoding for massive MIMO with uniform rectangular array—part
+I: complementary codes-based schemes,” IEEE Trans. Signal Process., vol. 67, no. 18, pp. 4761–4771, Sep. 2019.
+[7] Z. Liu, Y. Li, and Y. L. Guan, “New constructions of general QAM Golay complementary sequences,” IEEE Trans. Inf.
+Theory, vol. 59, no. 11, pp. 7684–7692, Nov. 2013.
+[8] R. van Nee, “OFDM codes for peak-to-average power reduction and error correction,” in Proc. IEEE Global Telecommun.
+Conf., London, U.K., Nov. 1996, pp. 740–744.
+[9] J. A. Davis and J. Jedwab, “Peak-to-mean power control in OFDM, Golay complementary sequences, and Reed-Muller
+codes,” IEEE Trans. Inf. Theory, vol. 45, no. 7, pp. 2397–2417, Nov. 1999.
+[10] K. G. Paterson, “Generalized Reed-Muller codes and power control in OFDM modulation,” IEEE Trans. Inf. Theory,
+vol. 46, no. 1, pp. 104–120, Jan. 2000.
+[11] K.-U. Schmidt, “Complementary sets, generalized Reed-Muller codes, and power control for OFDM,” IEEE Trans. Inf.
+Theory, vol. 53, no. 2, pp. 808–814, Feb. 2007.
+[12] C.-Y. Chen, C.-H. Wang, and C.-C. Chao, “Complementary sets and Reed-Muller codes for peak-to-average power ratio
+reduction in OFDM,” in Proc. 16th Int. Symp. AAECC, LNCS 3857, Las Vegas, NV, Feb. 2006, pp. 317–327.
+[13] C.-Y. Chen, “Complementary sets of non-power-of-two length for peak-to-average power ratio reduction in OFDM,” IEEE
+Trans. Inf. Theory, vol. 62, no. 12, pp. 7538–7545, Dec. 2016.
+[14] Z. Liu, Y. L. Guan, and H. H. Chen, “Fractional-delay-resilient receiver design for interference-free MC-CDMA
+communications based on complete complementary codes,” IEEE Trans. Wireless Commun., vol. 14, no. 3, pp. 1226–
+1236, Mar. 2015.
+[15] H.-H. Chen, J.-F. Yeh, and N. Suehiro, “A multicarrier CDMA architecture based on orthogonal complete complementary
+codes for new generations of wideband wireless communications,” IEEE Commun. Mag., vol. 39, pp. 126–134, Oct. 2001.
+[16] N. Suehiro, “A signal design without co-channel interference for approximately synchronized CDMA systems,” IEEE J.
+Sel. Areas Commun., vol. 12, pp. 837–841, Jun. 1994.
+[17] S.-M. Tseng and M. R. Bell, “Asynchronous multicarrier DS-CDMA using mutually orthogonal complementary sets of
+sequences,” IEEE Trans. Commun., vol. 48, no. 1, pp. 53–59, Jan. 2000.
+[18] J. M. Groenewald and B. T. Maharaj, “MIMO channel synchronization using Golay complementary pairs,” in Proc.
+AFRICON 2007, Windhoek, South Africa, Sep. 2007, pp. 1–5.
+[19] S. Boyd, “Multitone signals with low crest factor,” IEEE Trans. Circuits Syst., vol. CAS-33, no. 10, pp. 1018–1022, Oct.
+1986.
+[20] A.-A. Lu, X. Gao, X. Meng, and X.-G. Xia, “Omnidirectional precoding for 3D massive MIMO with uniform planar
+arrays,” IEEE Trans. Wireless Commun., vol. 19, no. 4, pp. 2628–2642, Apr. 2020.
+January 4, 2023
+DRAFT
+
+24
+[21] F. Li, Y. Jiang, C. Du, and X. Wang, “Construction of Golay complementary matrices and its applications to MIMO
+omnidirectional transmission,” IEEE Trans. Signal Process., vol. 69, pp. 2100–2113, Mar. 2021.
+[22] X. Meng, X. Gao, and X.-G. Xia, “Omnidirectional precoding based transmission in massive MIMO systems,” IEEE Trans.
+Commun., vol. 64, no. 1, pp. 174–186, Jan. 2016.
+[23] Y. Jiang, F. Li, X. Wang, and J. Li, “Autocorrelation complementary matrices,” in Proc. 53rd Asilomar Conference on
+Signals, Systems, and Computers, Pacific Grove, CA, USA, Nov. 2019, pp. 1596–1600.
+[24] S. Matsufuji, R. Shigemitsu, Y. Tanada, and N. Kuroyanagi, “Construction of complementary arrays,” in Proc. Joint 1ST
+Workshop on Mobile Future Symp. Trends Commun. (SympoTIC), Bratislave, Slovakia, Oct. 2004, pp. 78–81.
+[25] Z. Wang and G. Gong, “Constructions of complementary sequence sets and complete complementary codes by 2-level
+autocorrelation sequences and permutation polynomials,” May 2020. [Online]. Available: https://arxiv.org/abs/2005.05825
+[26] Z. Wang, D. Ma, G. Gong, and E. Xue, “New construction of complementary sequence (or array) sets and complete
+complementary codes,” IEEE Trans. Inf. Theory, vol. 67, no. 7, pp. 4902–4928, Jul. 2021.
+[27] C.-Y. Pai and C.-Y. Chen, “Constructions of two-dimensional Golay complementary array pairs based on generalized
+Boolean functions,” in Proc. IEEE Int. Symp. Inf. Theory, Los Angeles, California, USA, Jun. 2020, pp. 2931–2935.
+[28] C.-Y. Pai and C.-Y. Chen, “Two-dimensional Golay complementary array pairs/sets with bounded row and column sequence
+PAPRs,” IEEE Trans. Commun., vol. 70, no. 6, pp. 3695–3707, Jun. 2022.
+[29] Z. Liu, Y. L. Guan, and U. Parampalli, “New complete complementary codes for peak-to-mean power control in multi-
+carrier CDMA,” IEEE Trans. Commun., vol. 62, pp. 1105–1113, Mar. 2014.
+[30] C.-Y. Pai, Z. Liu, Y.-Q. Zhao, Z.-M. Hunag, and C.-Y. Chen, “Designing two-dimensional complete complementary codes
+for omnidirectional transmission in massive MIMO systems,” in Proc. IEEE Int. Symp. Inf. Theory, Espoo, Finland, Jun.
+2022, pp. 1699–1704.
+[31] T. Liu, X. Men, Y. Li, and X. Chen, “Constructions of 2-D Golay complementary array sets for MIMO omnidirectional
+transmission,” IEEE Commun. Lett., pp. 1459 – 1463, Jul. 2022.
+[32] B. Shen, Y. Yang, and R. Ren, “Three constructions of Golay complementary array sets,” Adv. Math. Commun., Oct. 2022.
+January 4, 2023
+DRAFT
+

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+arXiv:2301.05087v1  [math.DG]  12 Jan 2023
+SCALAR CURVATURE RIGIDITY OF CONVEX
+POLYTOPES
+SIMON BRENDLE
+Abstract. We prove a scalar curvature rigidity theorem for convex
+polytopes. The proof uses the Fredholm theory for Dirac operators on
+manifolds with boundary.
+A variant of a theorem of Fefferman and
+Phong plays a central role in our analysis.
+1. Introduction
+Let Ω be a compact polytope in Rn with non-empty interior. We may
+write Ω = �
+α∈A{uα ≤ 0}, where A is a finite set and the uα are linear
+functions in Rn. For each α ∈ A, we denote by Nα ∈ Sn−1 the outward-
+pointing unit normal vector to the halfspace {uα ≤ 0} with respect to the
+Euclidean metric.
+Let g be a Riemannian metric which is defined on an open set containing
+Ω. For each α ∈ A, we denote by να the outward-pointing unit normal
+vector to the halfspace {uα ≤ 0} with respect to the metric g. We will make
+the following assumption:
+Matching Angle Hypothesis. If x is point in ∂Ω and α1, α2 ∈ A satisfy
+uα1(x) = uα2(x) = 0, then ⟨να1, να2⟩ = ⟨Nα1, Nα2⟩ at the point x. Here, the
+inner product ⟨να1, να2⟩ is computed with respect to the metric g, and the
+inner product ⟨Nα1, Nα2⟩ is the standard inner product in Rn.
+Theorem 1.1. Suppose that n ≥ 3 is an odd integer, and Ω is a compact
+polytope in Rn with non-empty interior. Let g be a Riemannian metric which
+is defined on an open set containing Ω and has nonnegative scalar curvature
+at each point in Ω. For each α ∈ A, we assume that the mean curvature of
+the hypersurface {uα = 0} with respect to g is nonnegative at each point in
+Ω ∩ {uα = 0}. Moreover, we assume that the Matching Angle Hypothesis is
+satisfied. Then the Ricci tensor of g vanishes at each point in Ω.
+Theorem 1.1 also holds in the even-dimensional case. This can be seen
+by considering the Cartesian product Ω × [0, 1] ⊂ Rn+1.
+Scalar curvature comparison theorems for polytopes were first studied in
+seminal work of Gromov [6],[7],[8]. Li [9] has used minimal surface techniques
+to prove a scalar curvature comparison theorem for certain polytopes in
+The author was support by the National Science Foundation under grant DMS-2103573
+and by the Simons Foundation. The author acknowledges the hospitality of T¨ubingen
+University, where part of this work was carried out.
+1
+
+2
+SIMON BRENDLE
+dimension 3. Wang, Xie, and Yu [10] have proposed a different approach to
+this problem which is based on the study of Dirac operators on manifolds
+with corners.
+In this paper, we describe another approach to this problem. As in [10], we
+employ a spinor approach. In contrast to [10], we work with boundary value
+problems for Dirac operators on smooth domains, which are well understood
+thanks to the work of B¨ar and Ballmann [1],[2].
+In the following, we outline the main steps involved in the proof of The-
+orem 1.1. We approximate a given convex polytope Ω by a one-parameter
+family of smooth convex domains Ωλ. On each domain Ωλ, we solve the
+Dirac equation for an m-tuple of spinors s = (s1, . . . , sm) with a suitable
+local boundary condition. To prove the existence of a solution satisfying
+that particular boundary condition, we use the Fredholm theory developed
+by B¨ar and Ballmann [1],[2] together with the homotopy invariance of the
+Fredholm index. Having constructed an m-tuple of harmonic spinors on Ωλ
+satisfying this boundary condition, we apply a Weitzenb¨ock formula, and
+integrate over Ωλ. The resulting integral formula contains a term involving
+the scalar curvature, as well as a boundary term. Unfortunately, it is not
+clear if the boundary term has a favorable sign. We are able to control the
+boundary integral by adapting a theorem due to Fefferman and Phong [4].
+2. A boundary value problem for the Dirac operator on a
+smooth domain
+Let m = 2[ n
+2 ] denote the dimension of the space of spinors on Rn. Let
+{E1, . . . , En} denote the standard basis of Rn. Throughout this section, we
+fix an orthonormal basis {¯s1, . . . , ¯sm} of the space of spinors on flat Rn.
+We define ωaαβ = ⟨Ea · ¯sα, ¯sβ⟩ for a = 1, . . . , n and α, β = 1, . . . , m. The
+matrices ω1, . . . , ωn ∈ End(Cm) are skew-Hermitian, so that ωaαβ = −ωaβα.
+Moreover, ωaωb + ωbωa = −2δab id. In other words,
+m
+�
+β=1
+(ωaαβ ωbβγ + ωbαβ ωaβγ) = −2δab δαγ.
+We begin by stating a basic algebraic fact which will be needed later.
+Lemma 2.1. Assume that n is odd. Then there is no non-zero element of
+End(Cm) which anti-commutes with ωa ∈ End(Cm) for each a = 1, . . . , n.
+Proof. We recall the definition of the spin representation in odd dimen-
+sions. Let {E1, . . . , En} denote the standard basis of Rn. For k = 1, . . . , [n
+2 ],
+we define wk = E2k−1 − iE2k ∈ Cn. The spinor space is defined as the ex-
+terior algebra Λ∗W, where W = span{wk : k = 1, . . . , [n
+2 ]} ⊂ Cn. For each
+k ∈ {1, . . . , [n
+2 ]}, we define a linear map Pk ∈ End(��∗W) by
+Pk(wj1 ∧ . . . ∧ wjr) = wk ∧ wj1 ∧ . . . ∧ wjr.
+
+SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES
+3
+Moreover, for each k ∈ {1, . . . , [n
+2 ]}, we define a linear map Qk ∈ End(Λ∗W)
+by
+Qk(wj1 ∧ . . . ∧ wjr) = 0,
+Qk(wk ∧ wj1 ∧ . . . ∧ wjr) = wj1 ∧ . . . ∧ wjr
+for k /∈ {j1, . . . , jr}. Then PkPl + PlPk = QkQl + QlQk = 0 and PkQl +
+QlPk = δkl id for k, l ∈ {1, . . . , [n
+2 ]}. Finally, we define a linear map S ∈
+End(Λ∗W) so that
+S(wj1 ∧ . . . ∧ wjr) =
+�
+wj1 ∧ . . . ∧ wjr
+if r is even
+−wj1 ∧ . . . ∧ wjr
+if r is odd.
+Clearly, PkS + SPk = 0, QkS + SQk = 0, and S2 = id.
+Consequently,
+there is a natural algebra homomorphism from the Clifford algebra ClC(n)
+to End(Λ∗W) which maps wk to i
+√
+2 Pk, ¯wk to i
+√
+2 Qk, and En to iS. It is
+well known (see [5], Lemma 20.9) that
+span{Pk1 · · · PkrQl1 · · · Qls : r + s is even}
+= End(ΛevenW) ⊕ End(ΛoddW)
+and
+span{Pk1 · · · PkrQl1 · · · Qls : r + s is odd}
+= Hom(ΛevenW, ΛoddW) ⊕ Hom(ΛoddW, ΛevenW).
+We claim that there is no non-zero element of End(Λ∗W) which anti-commutes
+with Pk, Qk, S for each k ∈ {1, . . . , [n
+2 ]}.
+Suppose that L ∈ End(Λ∗W)
+is such an element.
+Since L anti-commutes with S, it follows that L ∈
+Hom(ΛevenW, ΛoddW)⊕Hom(ΛoddW, ΛevenW). Since L anti-commutes with
+Pk, Qk for each k ∈ {1, . . . , [n
+2 ]}, it follows that L anti-commutes with every
+element of Hom(ΛevenW, ΛoddW)⊕Hom(ΛoddW, ΛevenW). This implies that
+L = 0. This completes the proof of Lemma 2.1.
+Assume that Ω is a domain in Rn with smooth boundary ∂Ω = Σ. Let g
+be a Riemannian metric on Ω. We denote by ν the outward-pointing unit
+normal vector field with respect to the metric g. Let ∇ denote the spin
+connection. The Dirac operator is defined by
+Ds =
+n
+�
+i=1
+ei · ∇eis,
+where {e1, . . . , en} is a local orthonormal frame on Ω. The boundary Dirac
+operator DΣ is given by
+DΣs =
+n−1
+�
+i=1
+ν · ei · ∇eis + 1
+2 H s
+at each point on Σ, where {e1, . . . , en−1} is a local orthonormal frame on Σ.
+In the remainder of this section, we consider the Dirac operator act-
+ing on m-tuples of spinors with a suitable local boundary condition of
+
+4
+SIMON BRENDLE
+Lopatinsky-Shapiro type. To formulate the boundary condition, we assume
+that N : Σ → Sn−1 is a given smooth map.
+Definition 2.2. Consider an m-tuple of spinors s = (s1, . . . , sm). At each
+point on Σ, we define
+(χs)α = −
+n
+�
+a=1
+m
+�
+β=1
+⟨N, Ea⟩ ωaαβ ν · sβ
+and
+(Bs)α =
+n−1
+�
+i=1
+n
+�
+a=1
+m
+�
+β=1
+⟨dN(ei), Ea⟩ ωaαβ ei · sβ,
+where {e1, . . . , en−1} is a local orthonormal frame on Σ.
+Lemma 2.3. The map χ is self-adjoint. Moreover, χ2 is the identity.
+Proof. Suppose that s = (s1, . . . , sm) and t = (t1, . . . , tm) are two m-
+tuples of spinors. We compute
+(χ2s)α =
+n
+�
+a,b=1
+m
+�
+β,γ=1
+⟨N, Ea⟩ ⟨N, Eb⟩ ωaαβ ωbβγ ν · ν · sγ
+= −
+n
+�
+a,b=1
+m
+�
+β,γ=1
+⟨N, Ea⟩ ⟨N, Eb⟩ ωaαβ ωbβγ sγ
+= −1
+2
+n
+�
+a,b=1
+m
+�
+β,γ=1
+⟨N, Ea⟩ ⟨N, Eb⟩ (ωaαβ ωbβγ + ωbαβ ωaβγ) sγ
+=
+n
+�
+a,b=1
+m
+�
+γ=1
+⟨N, Ea⟩ ⟨N, Eb⟩ δab δαγ sγ
+= sα.
+Moreover,
+m
+�
+α=1
+⟨(χs)α, tα⟩ = −
+n
+�
+a=1
+m
+�
+α,β=1
+⟨N, Ea⟩ ωaαβ ⟨ν · sβ, tα⟩
+= −
+n
+�
+a=1
+m
+�
+α,β=1
+⟨N, Ea⟩ ωaβα ⟨sβ, ν · tα⟩
+=
+m
+�
+β=1
+⟨sβ, (χt)β⟩.
+This completes the proof of Lemma 2.3.
+
+SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES
+5
+Lemma 2.4. Assume that x ∈ Σ and ξ ∈ TxΣ. Then the map (s1, . . . , sm) �→
+(ν · ξ · s1, . . . , ν · ξ · sm) maps the eigenspace of χ with eigenvalue 1 to the
+eigenspace of χ with eigenvalue −1, and vice versa. In particular, the two
+eigenspaces have the same dimension.
+Proof. For each vector ξ ∈ TxΣ, the map
+(s1, . . . , sm) �→ (ν · ξ · s1, . . . , ν · ξ · sm)
+anti-commutes with χ. From this, the assertion follows.
+Lemma 2.5. The map B is self-adjoint. Moreover, χ and B commute.
+Proof. Let {e1, . . . , en−1} be a local orthonormal frame on Σ. Then
+m
+�
+α=1
+⟨(Bs)α, tα⟩ =
+n−1
+�
+i=1
+n
+�
+a=1
+m
+�
+α,β=1
+⟨dN(ei), Ea⟩ ωaαβ ⟨ei · sβ, tα⟩
+=
+n−1
+�
+i=1
+n
+�
+a=1
+m
+�
+α,β=1
+⟨dN(ei), Ea⟩ ωaβα ⟨sβ, ei · tα⟩
+=
+m
+�
+β=1
+⟨sβ, (Bt)β⟩.
+This shows that B is self-adjoint. Moreover,
+(χBs)α − (Bχs)α
+= −
+n−1
+�
+i=1
+n
+�
+a,b=1
+n
+�
+β,γ=1
+⟨N, Ea⟩ ⟨dN(ei), Eb⟩ ωaαβ ωbβγ ν · ei · sγ
++
+n−1
+�
+i=1
+n
+�
+a,b=1
+n
+�
+β,γ=1
+⟨dN(ei), Ea⟩ ⟨N, Eb⟩ ωaαβ ωbβγ ei · ν · sγ
+= −
+n−1
+�
+i=1
+n
+�
+a,b=1
+n
+�
+β,γ=1
+⟨N, Ea⟩ ⟨dN(ei), Eb⟩ (ωaαβ ωbβγ + ωbαβ ωaβγ) ν · ei · sγ
+= 2
+n−1
+�
+i=1
+n
+�
+a,b=1
+n
+�
+γ=1
+⟨N, Ea⟩ ⟨dN(ei), Eb⟩ δab δαγ ν · ei · sγ
+= 2
+n−1
+�
+i=1
+⟨N, dN(ei)⟩ ν · ei · sα
+= 0.
+Thus, χ and B commute. This completes the proof of Lemma 2.5.
+At this point, we recall a definition from linear algebra.
+
+6
+SIMON BRENDLE
+Definition 2.6. Let V and W be finite-dimensional vector spaces of the
+same dime, each of them equipped with an inner product. The trace norm
+of a linear map L : V → W is defined by ∥L∥tr = supQ tr(QL), where the
+supremum is taken over all linear isometries Q : W → V . Equivalently,
+∥L∥tr can be characterized as the sum of the singular values of L.
+It is easy to see from the definition that the trace norm satisfies the tri-
+angle inequality.
+Lemma 2.7. Suppose that s = (s1, . . . , sm) is an m-tuple of spinors. Then
+����
+m
+�
+α=1
+⟨(Bs)α, sα⟩
+���� ≤ ∥dN∥tr
+� m
+�
+α=1
+|sα|2
+�
+at each point x ∈ Σ. Here, ∥dN∥tr denotes the trace norm of the differential
+dN : TxΣ → TN(x)Sn−1. The tangent space TxΣ is equipped with the restric-
+tion of the inner product g, and the tangent space TN(x)Sn−1 is equipped
+with the restriction of the standard inner product on Rn.
+Proof. Fix a point x ∈ Σ. We can find an orthonormal basis {e1, . . . , en−1}
+of TxΣ so that dN(ei) = λi ˆEi, where { ˆE1, . . . , ˆEn−1} is an orthonormal ba-
+sis of TN(x)Sn−1 and λ1, . . . , λn−1 ≥ 0 denote the singular values of dN.
+Then
+m
+�
+α=1
+����
+n
+�
+a=1
+m
+�
+β=1
+⟨ ˆEi, Ea⟩ ωaαβ ei · sβ
+����
+2
+=
+n
+�
+a,b=1
+m
+�
+α,β,γ=1
+⟨ ˆEi, Ea⟩ ⟨ ˆEi, Eb⟩ ωaαβ ωbαγ ⟨ei · sβ, ei · sγ⟩
+= −
+n
+�
+a,b=1
+m
+�
+α,β,γ=1
+⟨ ˆEi, Ea⟩ ⟨ ˆEi, Eb⟩ ωaαβ ωbγα ⟨sβ, sγ⟩
+= −1
+2
+n
+�
+a,b=1
+m
+�
+α,β,γ=1
+⟨ ˆEi, Ea⟩ ⟨ ˆEi, Eb⟩ (ωaγα ωbαβ + ωbγα ωaαβ) ⟨sβ, sγ⟩
+=
+n
+�
+a,b=1
+m
+�
+β,γ=1
+⟨ ˆEi, Ea⟩ ⟨ ˆEi, Eb⟩ δab δγβ ⟨sβ, sγ⟩
+=
+m
+�
+α=1
+|sα|2
+
+SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES
+7
+for each i = 1, . . . , n − 1. Using the Cauchy-Schwarz inequality, we obtain
+����
+n
+�
+a=1
+m
+�
+α,β=1
+⟨ ˆEi, Ea⟩ ωaαβ ⟨ei · sβ, sα⟩
+����
+≤
+� m
+�
+α=1
+����
+n
+�
+a=1
+m
+�
+β=1
+⟨ ˆEi, Ea⟩ ωaαβ ei · sβ
+����
+2� 1
+2 � m
+�
+α=1
+|sα|2
+� 1
+2
+=
+m
+�
+α=1
+|sα|2
+for each i = 1, . . . , n − 1. Summation over i = 1, . . . , n − 1 gives
+����
+m
+�
+α=1
+⟨(Bs)α, sα⟩
+���� =
+����
+n−1
+�
+i=1
+n
+�
+a=1
+m
+�
+α,β=1
+⟨dN(ei), Ea⟩ ωaαβ ⟨ei · sβ, sα⟩
+����
+=
+����
+n−1
+�
+i=1
+λi
+�
+n
+�
+a=1
+m
+�
+α,β=1
+⟨ ˆEi, Ea⟩ ωaαβ ⟨ei · sβ, sα⟩
+�����
+≤
+� n−1
+�
+i=1
+λi
+� � m
+�
+α=1
+|sα|2
+�
+,
+as claimed.
+Proposition 2.8. Suppose that s = (s1, . . . , sm) and t = (t1, . . . , tm) are
+m-tuples of spinors. Then
+0 =
+�
+Σ
+m
+�
+α=1
+⟨DΣsα, (χt)α⟩ dσg +
+�
+Σ
+m
+�
+α=1
+⟨(χs)α, DΣtα⟩ dσg
++
+�
+Σ
+m
+�
+α=1
+⟨(Bs)α, tα⟩ dσg.
+Equivalently,
+0 =
+�
+Σ
+m
+�
+α=1
+⟨(As)α, (χt)α⟩ dσg +
+�
+Σ
+m
+�
+α=1
+⟨(χs)α, (At)α⟩ dσg,
+where A is defined by A = DΣ + 1
+2χB.
+Proof. Let {e1, . . . , en−1} be a local orthonormal frame on Σ. We define
+a tangential vector field Z on Σ by
+⟨Z, ei⟩ =
+n
+�
+a=1
+m
+�
+α,β=1
+⟨N, Ea⟩ ωaαβ ⟨ei · sβ, tα⟩
+
+8
+SIMON BRENDLE
+for i = 1, . . . , n − 1. Then
+divΣZ =
+n−1
+�
+i=1
+n
+�
+a=1
+m
+�
+α,β=1
+⟨N, Ea⟩ ωaαβ ⟨ei · ∇eisβ, tα⟩
++
+n−1
+�
+i=1
+n
+�
+a=1
+m
+�
+α,β=1
+⟨N, Ea⟩ ωaαβ ⟨ei · sβ, ∇eitα⟩
+−
+n
+�
+a=1
+m
+�
+α,β=1
+H ⟨N, Ea⟩ ωaαβ ⟨ν · sβ, tα⟩
++
+n−1
+�
+i=1
+n
+�
+a=1
+m
+�
+α,β=1
+⟨dN(ei), Ea⟩ ωaαβ ⟨ei · sβ, tα⟩
+= −
+n−1
+�
+i=1
+m
+�
+β=1
+⟨ei · ∇eisβ, ν · (χt)β⟩
++
+n−1
+�
+i=1
+m
+�
+α=1
+⟨ei · ν · (χs)α, ∇eitα⟩
++
+m
+�
+α=1
+H ⟨(χs)α, tα⟩ +
+m
+�
+α=1
+⟨(Bs)α, tα⟩
+=
+m
+�
+β=1
+⟨DΣsβ, (χt)β⟩ +
+m
+�
+α=1
+⟨(χs)α, DΣtα⟩ +
+m
+�
+α=1
+⟨(Bs)α, tα⟩.
+Integrating over Σ, we obtain
+0 =
+�
+Σ
+m
+�
+β=1
+⟨DΣsβ, (χt)β⟩ dσg +
+�
+Σ
+m
+�
+α=1
+⟨(χs)α, DΣtα⟩ dσg
++
+�
+Σ
+m
+�
+α=1
+⟨(Bs)α, tα⟩ dσg.
+This completes the proof of Proposition 2.8.
+Remark 2.9. It is well known that the boundary Dirac operator DΣ is
+formally self-adjoint. Moreover, it follows from Lemma 2.3 and Lemma 2.5
+that χB is self-adjoint. Consequently, the operator A = DΣ+ 1
+2χB is formally
+self-adjoint. Finally, Proposition 2.8 implies that A and χ anti-commute.
+
+SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES
+9
+Proposition 2.10. Suppose that s = (s1, . . . , sm) is an m-tuple of spinors.
+Then
+−
+�
+Ω
+m
+�
+α=1
+|Dsα|2 dvolg +
+�
+Ω
+n
+�
+α=1
+|∇sα|2 dvolg + 1
+4
+�
+Ω
+m
+�
+α=1
+R |sα|2 dvolg
+≤ 1
+2
+�
+Σ
+m
+�
+α=1
+⟨DΣs��, sα − (χs)α⟩ dσg + 1
+2
+�
+Σ
+m
+�
+α=1
+⟨sα − (χs)α, DΣsα⟩ dσg
+− 1
+2
+�
+Σ
+(H − ∥dN∥tr)
+� m
+�
+α=1
+|sα|2
+�
+dσg.
+Proof. By the Weitzenb¨ock formula, D2sα = −∆sα + 1
+4 R sα, where ∆
+denotes the connection Laplacian on the spinor bundle. Using the divergence
+theorem, we obtain
+−
+�
+Ω
+m
+�
+α=1
+|Dsα|2 dvolg +
+�
+Ω
+m
+�
+α=1
+|∇sα|2 dvolg + 1
+4
+�
+Ω
+m
+�
+α=1
+R |sα|2 dvolg
+=
+�
+Σ
+m
+�
+α=1
+⟨ν · Dsα, sα⟩ dσg +
+�
+Σ
+m
+�
+α=1
+⟨∇νsα, sα⟩ dσg
+=
+�
+Σ
+⟨DΣsα, sα⟩ dσg − 1
+2
+�
+Σ
+m
+�
+α=1
+H |sα|2 dσg.
+Applying Proposition 2.8 with s = t gives
+0 =
+�
+Σ
+m
+�
+α=1
+⟨DΣsα, (χs)α⟩ dσg +
+�
+Σ
+m
+�
+α=1
+⟨(χs)α, DΣsα⟩ dσg
++
+�
+Σ
+m
+�
+α=1
+⟨(Bs)α, sα⟩ dσg.
+This gives
+−
+�
+Ω
+m
+�
+α=1
+|Dsα|2 dvolg +
+�
+Ω
+n
+�
+α=1
+|∇sα|2 dvolg + 1
+4
+�
+Ω
+m
+�
+α=1
+R |sα|2 dvolg
+= 1
+2
+�
+Σ
+m
+�
+α=1
+⟨DΣsα, sα⟩ dσg + 1
+2
+�
+Σ
+m
+�
+α=1
+⟨sα, DΣsα⟩ dσg − 1
+2
+�
+Σ
+m
+�
+α=1
+H |sα|2 dσg
+= 1
+2
+�
+Σ
+m
+�
+α=1
+⟨DΣsα, sα − (χs)α⟩ dσg + 1
+2
+�
+Σ
+m
+�
+α=1
+⟨sα − (χs)α, DΣsα⟩ dσg
+− 1
+2
+�
+Σ
+m
+�
+α=1
+⟨(Bs)α, sα⟩ dσg − 1
+2
+�
+Σ
+m
+�
+α=1
+H |sα|2 dσg.
+Hence, the assertion follows from Lemma 2.7.
+
+10
+SIMON BRENDLE
+Corollary 2.11. Suppose that R ≥ 0 at each point in Ω and H ≥ ∥dN∥tr at
+each point on Σ. Then every m-tuple of harmonic spinors s = (s1, . . . , sm)
+with χs = s is parallel.
+Replacing N by −N, we can draw the following conclusion:
+Corollary 2.12. Suppose that R ≥ 0 at each point in Ω and H ≥ ∥dN∥tr at
+each point on Σ. Then every m-tuple of harmonic spinors s = (s1, . . . , sm)
+with χs = −s is parallel.
+Proposition 2.13. Suppose that Ω is a convex domain in Rn with smooth
+boundary ∂Ω = Σ.
+Let g be a Riemannian metric on Ω.
+Suppose that
+N : Σ → Sn−1 is a smooth map. Then the boundary condition χs = s is a
+D-elliptic boundary condition in the sense of B¨ar and Ballmann [2].
+Proof.
+We apply Corollary 3.18 in [2] with E′ = ker(id − χ) and
+E′′ = ker(id + χ).
+Lemma 2.4 implies that, for each point x ∈ Σ and
+each ξ ∈ TxΣ, the map (s1, . . . , sm) �→ (ν · ξ · s1, . . . , ν · ξ · sm) interchanges
+ker(id − χ) and ker(id + χ). Therefore, the boundary condition χs = s is a
+D-elliptic boundary condition in the sense of [2].
+Proposition 2.14. Assume that n ≥ 3 is an odd integer. Suppose that
+Ω is a convex domain in Rn with smooth boundary ∂Ω = Σ. Let g be a
+Riemannian metric on Ω. Suppose that N : Σ → Sn−1 is homotopic to
+the Gauss map of Σ with respect to the Euclidean metric. Then the Dirac
+operator with the boundary condition χs = s has Fredholm index at least 1.
+Proof. Since the Fredholm index is homotopy invariant, it suffices to
+prove the assertion in the special case when g is the Euclidean metric and
+N is the Gauss map of Σ with respect to the Euclidean metric.
+We first analyze the kernel of the Dirac operator with the boundary con-
+dition χs = s. Recall that ¯s1, . . . , ¯sm is a basis of spinors on flat Rn, and
+ωaαβ = ⟨Ea · ¯sα, ¯sβ⟩. Clearly, ¯s = (¯s1, . . . , ¯sm) is an m-tuple of harmonic
+spinors on Ω which satisfies the boundary condition χ¯s = ¯s. Therefore, the
+kernel has dimension at least 1.
+We next examine the cokernel.
+The cokernel can be identified with
+the space of all m-tuples of harmonic spinors s = (s1, . . . , sm) such that
+⟨ν · s, t⟩ = 0 for all points x ∈ Σ and all t ∈ ker(id − χ) (see [2], Example
+3.20). We claim that this space has dimension 0. To see this, suppose that
+s = (s1, . . . , sm) is an m-tuple of harmonic spinors such that ⟨ν · s, t⟩ = 0
+for all points x ∈ Σ and all t ∈ ker(id − χ). This implies s ∈ ker(id + χ) at
+each point on Σ. Since H = ∥dN∥tr at each point on Σ, Corollary 2.12 im-
+plies that s = (s1, . . . , sm) is parallel. In other words, s1, . . . , sm are constant
+spinors. Let us write sα = �m
+β=1 zαβ ¯sβ for some matrix z ∈ End(Cm). Since
+χs = −s at each point on Σ, it follows that the matrix z ∈ End(Cm) anti-
+commutes with the matrix �n
+a=1⟨N(x), Ea⟩ ωa ∈ End(Cm) for each point
+x ∈ Σ. It is easy to see that the Gauss map N : Σ → Sn−1 is surjective.
+
+SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES
+11
+Consequently, the matrix z ∈ End(Cm) anti-commutes with ωa ∈ End(Cm)
+for each a = 1, . . . , n. Since n is odd, Lemma 2.1 implies that z = 0, hence
+s = 0. This shows that the cokernel has dimension 0. This completes the
+proof of Proposition 2.14.
+3. Approximating a compact, convex polytope by smooth
+domains
+Let us consider a compact, convex polytope Ω ⊂ Rn with non-empty
+interior. We write Ω = �
+α∈A{uα ≤ 0}, where A is a finite set and the uα
+are linear functions in Rn. After eliminating redundant inequalities, we may
+assume that the following condition is satisfied.
+Assumption 3.1. For each α ∈ A, the set Ω ∩ {uα > 0} is non-empty.
+Let g be a Riemannian metric which is defined on an open set containing
+Ω. For each α ∈ A, ∇uα will denote the gradient of uα with respect to the
+metric g; |∇uα| will denote the norm of the gradient of uα with respect to
+the metric g; and να =
+∇uα
+|∇uα| will denote the outward-pointing unit normal
+vector to the halfspace {uα ≤ 0} with respect to the metric g. For each
+α ∈ A, we denote by Nα ∈ Sn−1 the outward-pointing unit normal vector
+to the halfspace {uα ≤ 0} with respect to the Euclidean metric.
+For each λ > 0, the function �
+α∈A eλuα is convex with respect to the
+Euclidean metric. Clearly, �
+α∈A eλuα > 1 on ∂Ω. Moreover, we can find
+large number λ0 such that infΩ
+�
+α∈A eλuα < 1 for each λ > λ0. For each
+λ > λ0, we define
+Ωλ =
+� �
+α∈A
+eλuα ≤ 1
+�
+.
+For each λ > λ0, Ωλ is a convex domain in Rn with smooth boundary
+Σλ = ∂Ωλ. The sets Ωλ form an increasing family of sets in the sense that
+Ωλ ⊂ Ωµ for λ0 < λ < µ. Moreover,
+�
+λ>λ0
+Ωλ =
+�
+α∈A
+{uα < 0}.
+Lemma 3.2. If λ is sufficiently large, then infΣλ
+�� �
+α∈A eλuα duα
+�� ≥ C−1
+for some large constant C which is independent of λ.
+Proof. We argue by contradiction. Suppose that the assertion is false.
+Then there exists a sequence of positive real numbers λl → ∞ and a se-
+quence of points xl ∈ Σλl such that
+�� �
+α∈A eλuα duα
+�� ≤ l−1 at the point
+xl. After passing to a subsequence, we may assume that the sequence xl
+converges to a point x0 ∈ Ω.
+Moreover, we may assume that, for each
+α ∈ A, the sequence eλluα(xl) converges to a nonnegative real number zα.
+Since �
+α∈A eλluα(xl) = 1 for each l, we know that �
+α∈A zα > 0.
+Let
+A0 := {α ∈ A : zα > 0}. Clearly, A0 is non-empty, and uα(x0) = 0 for all
+
+12
+SIMON BRENDLE
+α ∈ A0. Moreover, �
+α∈A0 zα duα = 0 at the point x0. On the other hand,
+since Ω is a convex set with non-empty interior, we can find a tangent vector
+ξ ∈ Tx0Ω such that duα(ξ) > 0 for all α ∈ A0. This is a contradiction. This
+completes the proof of Lemma 3.2.
+Lemma 3.3. If λ is sufficiently large, then infΣλ
+�� �
+α∈A eλuα |∇uα| Nα
+�� ≥
+C−1 for some large constant C which is independent of λ.
+Proof. We argue by contradiction. Suppose that the assertion is false.
+Then there exists a sequence of positive real numbers λl → ∞ and a se-
+quence of points xl ∈ Σλl such that
+�� �
+α∈A eλuα |∇uα| Nα
+�� ≤ l−1 at the
+point xl. After passing to a subsequence, we may assume that the sequence
+xl converges to a point x0 ∈ Ω. Moreover, we may assume that, for each
+α ∈ A, the sequence eλluα(xl) |∇uα(xl)| converges to a nonnegative real num-
+ber zα. Since �
+α∈A eλluα(xl) = 1 for each l, we know that �
+α∈A zα > 0.
+Let A0 := {α ∈ A : zα > 0}. Clearly, A0 is non-empty, and uα(x0) = 0
+for all α ∈ A0. Moreover, �
+α∈A0 zαNα = 0 at the point x0. On the other
+hand, since Ω is a convex set with non-empty interior, we can find a vector
+ξ ∈ Rn such that ⟨Nα, ξ⟩ > 0 for all α ∈ A0. This is a contradiction. This
+completes the proof of Lemma 3.3.
+The outward-pointing unit normal vector to the domain Ωλ with respect
+to the metric g is given by
+ν =
+�
+α∈A eλuα ∇uα
+�� �
+α∈A eλuα ∇uα
+�� =
+�
+α∈A eλuα |∇uα| να
+�� �
+α∈A eλuα |∇uα| να
+��.
+We define a map N : Σλ → Sn−1 by
+N =
+�
+α∈A eλuα |∇uα| Nα
+�� �
+α∈A eλuα |∇uα| Nα
+��.
+Lemma 3.4. The map N : Σλ → Sn−1 is homotopic to the Gauss map of
+Σλ with respect to the Euclidean metric.
+Proof. In the special case when g is the Euclidean metric, the map N
+coincides with the Gauss map of Σλ, and the assertion is trivially true. To
+prove the assertion in general, we deform the metric g to the Euclidean met-
+ric.
+Proposition 3.5. Let x ∈ Σλ. Let π : TxΩ → TxΩ denotes the orthogonal
+projection to the orthogonal complement of ν and P : Rn → Rn denotes
+the orthogonal projection to the orthogonal complement of N. Then H −
+
+SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES
+13
+∥dN∥tr ≥ Vλ, where the function Vλ : Σλ → R is defined by
+Vλ = λ
+�
+α∈A eλuα |∇uα|2 |π(να)|2
+�� �
+α∈A eλuα |∇uα| να
+��
+− λ
+�
+α∈A eλuα |∇uα|2 |π(να)| |P(Nα)|
+�� �
+α∈A eλuα |∇uα| Nα
+��
++
+�
+α∈A eλuα (∆uα − (D2uα)(ν, ν))
+�� �
+α∈A eλuα |∇uα| να
+��
+−
+�
+α∈A eλuα |∇(|∇uα|)| |P(Nα)|
+�� �
+α∈A eλuα |∇uα| Nα
+��
+.
+Proof. Let {e1, . . . , en−1} denote a local orthonormal frame on Σλ. The
+mean curvature of Σλ is given by
+H = λ
+�n−1
+i=1
+�
+α∈A eλuα ⟨∇uα, ei⟩2
+�� �
+α∈A eλuα ∇uα
+��
++
+�n−1
+i=1
+�
+α∈A eλuα (D2uα)(ei, ei)
+�� �
+α∈A eλuα ∇uα
+��
+= λ
+�
+α∈A eλuα |π(∇uα)|2
+�� �
+α∈A eλuα ∇uα
+��
++
+�
+α∈A eλuα (∆uα − (D2uα)(ν, ν))
+�� �
+α∈A eλuα ∇uα
+��
+= λ
+�
+α∈A eλuα |∇uα|2 |π(να)|2
+�� �
+α∈A eλuα |∇uα| να
+��
++
+�
+α∈A eλuα (∆uα − (D2uα)(ν, ν))
+�� �
+α∈A eλuα |∇uα| να
+��
+.
+If ξ is a tangent vector to Σλ, then
+dN(ξ)
+= λ
+�
+α∈A eλuα |∇uα| ⟨∇uα, ξ⟩ P(Nα)
+�� �
+α∈A eλuα |∇uα| Nα
+��
++
+�
+α∈A eλuα ⟨∇(|∇uα|), ξ⟩ P(Nα)
+�� �
+α∈A eλuα |∇uα| Nα
+��
+= λ
+�
+α∈A eλuα |∇uα|2 ⟨π(να), ξ⟩ P(Nα)
+�� �
+α∈A eλuα |∇uα| Nα
+��
++
+�
+α∈A eλuα ⟨∇(|∇uα|), ξ⟩ P(Nα)
+�� �
+α∈A eλuα |∇uα| Nα
+��
+.
+The trace norm of a linear transformation of the form ξ �→ ⟨X, ξ⟩ Y is given
+by |X| |Y |. Since the trace norm satisfies the triangle inequality, it follows
+that
+∥dN∥tr
+≤ λ
+�
+α∈A eλuα |∇uα|2 |π(να)| |P(Nα)|
+�� �
+α∈A eλuα |∇uα| Nα
+��
++
+�
+α∈A eλuα |∇(|∇uα|)| |P(Nα)|
+�� �
+α∈A eλuα |∇uα| Nα
+��
+.
+Putting these facts together, the assertion follows.
+In the following, we denote by Vλ,− = max{−Vλ, 0} the negative part of
+Vλ.
+Proposition 3.6. Suppose that the Matching Angle Hypothesis is satisfied.
+Then supΣλ Vλ,− ≤ o(λ) as λ → ∞.
+Proof. We argue by contradiction. Suppose that the assertion is false.
+Then there exists a sequence of positive real numbers λl → ∞ and a se-
+quence of points xl ∈ Σλl such that lim supl→∞ λ−1
+l
+Vλl(xl) < 0.
+After
+passing to a subsequence, we may assume that the sequence xl converges
+to a point x0 ∈ Ω.
+Moreover, we may assume that, for each α ∈ A,
+the sequence eλluα(xl) |∇uα(xl)| converges to a nonnegative real number zα.
+
+14
+SIMON BRENDLE
+Since �
+α∈A eλluα(xl) = 1 for each l, we know that �
+α∈A zα > 0.
+Let
+A0 := {α ∈ A : zα > 0}. Clearly, A0 is non-empty, and uα(x0) = 0 for
+all α ∈ A0. The Matching Angle Hypothesis implies that, at the point x0,
+⟨να1, να2⟩ = ⟨Nα1, Nα2⟩ for all α1, α2 ∈ A0. Let π : Tx0Ω → Tx0Ω denote
+the orthogonal projection to the orthogonal complement of �
+α∈A0 zανα,
+and let P : Rn → Rn denote the orthogonal projection to the orthogonal
+complement of �
+α∈A0 zαNα. For each β ∈ A0, we have
+|π(νβ)|2 = 1 −
+� �
+α∈A0 zανα, νβ
+�2
+�� �
+α∈A0 zανα
+��2
+= 1 −
+� �
+α∈A0 zαNα, Nβ
+�2
+�� �
+α∈A0 zαNα
+��2
+= |P(Nβ)|2
+at the point x0. Therefore, for each β ∈ A0, we obtain
+|π(νβ)|
+�� �
+α∈A0 zανα
+�� =
+|P(Nβ)|
+�� �
+α∈A0 zαNα
+��
+at the point x0. Using Proposition 3.5, we conclude that λ−1
+l
+Vλl(xl) → 0 as
+l → ∞. This is a contradiction.
+In the remainder of this section, we will estimate the Ls-norm Vλ,− on Σλ∩
+Br(p), where s ∈ [1, 3
+2) is a fixed exponent and Br(p) denotes a Euclidean
+ball of radius r. We begin by recalling a basic fact about the area of convex
+hypersurfaces in Rn.
+Lemma 3.7. Let Br(p) denote a Euclidean ball of radius r. Then the in-
+tersection Σλ ∩ Br(p) has area at most Crn−1.
+Proof. This follows from the fact that the hypersurface Σλ = ∂Ωλ is
+outward-minimizing with respect to the Euclidean metric.
+Definition 3.8. Consider three pairwise distinct elements α1, α2, α3 ∈ A.
+We denote by G(α1,α2,α3)
+λ
+the set of all points x ∈ Σλ with the property that
+uα1(x) ≥ uα2(x) ≥ uα3(x) and uα3(x) ≥ uα(x) for α ∈ A \ {α1, α2, α3}.
+Lemma 3.9. Assume that the mean curvature of the hypersurface {uα = 0}
+with respect to g is nonnegative at each point in Ω ∩ {uα = 0}. Let us fix an
+exponent s ∈ [1, 3
+2), and let Br(p) denote a Euclidean ball of radius r ≤ 1.
+If λr is sufficiently large, then
+�
+rs+1−n
+�
+G(α1,α2,α3)
+λ
+∩{uα2≤−λ− 7
+8 r
+1
+8 }∩Br(p)
+V s
+λ,−
+� 1
+s
+≤ Cλr e−(λr)
+1
+8
+for all pairwise distinct elements α1, α2, α3 ∈ A.
+Proof. Let us consider an arbitrary point x ∈ G(α1,α2,α3)
+λ
+with uα2(x) ≤
+−λ− 7
+8 r
+1
+8 .
+By definition of G(α1,α2,α3)
+λ
+, it follows that uα(x) ≤ −λ− 7
+8r
+1
+8
+for all α ∈ A \ {α1}.
+Using the identity �
+α∈A eλuα(x) = 1, we obtain
+uα1(x) ≥ −Cλ−1 e−(λr)
+1
+8 . Moreover, |ν − να1| ≤ C e−(λr)
+1
+8 and |N − Nα1| ≤
+
+SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES
+15
+C e−(λr)
+1
+8 at the point x. From this, we deduce that |π(να1)| ≤ C e−(λr)
+1
+8
+and |P(Nα1)| ≤ C e−(λr)
+1
+8 at the point x. Therefore,
+Vλ ≥ ∆uα1 − (D2uα1)(να1, να1)
+|∇uα1|
+− Cλ e−(λr)
+1
+8
+at the point x. Since uα1(x) ≥ −Cλ−1 e−(λr)
+1
+8 and uα(x) ≤ −λ− 7
+8r
+1
+8 for
+all α ∈ A \ {α1}, we can find a point y ∈ Ω such that uα1(y) = 0 and
+d(x, y) ≤ Cλ−1 e−(λr)
+1
+8 . By assumption, the mean curvature of the hyper-
+surface {uα1 = 0} at the point y is nonnegative. This implies
+∆uα1 − (D2uα1)(να1, να1)
+|∇uα1|
+≥ 0
+at the point y. Consequently,
+∆uα1 − (D2uα1)(να1, να1)
+|∇uα1|
+≥ −C d(x, y)
+at the point x. Thus, we conclude that
+Vλ(x) ≥ −Cλ e−(λr)
+1
+8
+for each point x ∈ G(α1,α2,α3)
+λ
+∩ {uα2 ≤ −λ− 7
+8r
+1
+8}.
+On the other hand,
+Σλ ∩ Br(p) has area at most Crn−1. Consequently,
+�
+rs+1−n
+�
+G(α1,α2,α3)
+λ
+∩{uα2≤−λ− 7
+8 r
+1
+8 }∩Br(p)
+V s
+λ,−
+� 1
+s
+≤ Cλr e−(λr)
+1
+8 .
+This completes the proof of Lemma 3.9.
+Lemma 3.10. Assume that the Matching Angle Hypothesis holds. Let us
+fix an exponent s ∈ [1, 3
+2), and let Br(p) denote a Euclidean ball of radius
+r ≤ 1. If λr is sufficiently large, then
+�
+rs+1−n
+�
+G(α1,α2,α3)
+λ
+∩{uα2≥−λ− 7
+8 r
+1
+8 }∩{uα3≤−λ− 3
+4 r
+1
+4 }∩Br(p)
+V s
+λ,−
+� 1
+s
+≤ C (λr)
+1
+8 − 7
+8s
+for all pairwise distinct elements α1, α2, α3 ∈ A.
+Proof. We distinguish two cases:
+Case 1: Suppose that Ω ∩ {uα1 = 0} ∩ {uα2 = 0} = ∅. By continuity, we
+can find a real number δ such that Ω ∩ {uα1 ≥ −δ} ∩ {uα2 ≥ −δ} = ∅. If λr
+is sufficiently large, then λ− 7
+8r
+1
+8 ≤ δ. This implies
+G(α1,α2,α3)
+λ
+∩ {uα2 ≥ −λ− 7
+8 r
+1
+8 }
+⊂ Σλ ∩ {uα1 ≥ −δ} ∩ {uα2 ≥ −δ} = ∅.
+Hence, the assertion is trivially true in this case.
+
+16
+SIMON BRENDLE
+Case 2: Suppose that Ω ∩ {uα1 = 0} ∩ {uα2 = 0} ̸= ∅. It follows from
+Assumption 3.1 that the hypersurfaces {uα1 = 0} and {uα2 = 0} intersect
+transversally.
+Let us consider an arbitrary point x ∈ G(α1,α2,α3)
+λ
+with uα2(x) ≥ −λ− 7
+8r
+1
+8
+and uα3(x) ≤ −λ− 3
+4r
+1
+4. Clearly, uα1(x) ≥ −λ− 7
+8r
+1
+8 by definition of G(α1,α2,α3)
+λ
+.
+Moreover, uα(x) ≤ −λ− 3
+4r
+1
+4 for all α ∈ A \ {α1, α2}. Consequently, we can
+find a point y ∈ Ω such that uα1(y) = uα2(y) = 0 and d(x, y) ≤ Cλ− 7
+8r
+1
+8.
+The Matching Angle Hypothesis implies ⟨να1, να2⟩ = ⟨Nα1, Nα2⟩ at the point
+y. Consequently, |⟨να1, να2⟩ − ⟨Nα1, Nα2⟩| ≤ C d(x, y) at the point x. From
+this, we deduce that
+|π(να1)|
+�� �
+α∈A eλuα |∇uα| να
+�� −
+|P(Nα1)|
+�� �
+α∈A eλuα |∇uα| Nα
+�� ≥ −C d(x, y) − C e−(λr)
+1
+4
+and
+|π(να2)|
+�� �
+α∈A eλuα |∇uα| να
+�� −
+|P(Nα2)|
+�� �
+α∈A eλuα |∇uα| Nα
+�� ≥ −C d(x, y) − C e−(λr)
+1
+4
+at the point x. Thus, we conclude that
+Vλ(x) ≥ −Cλ
+1
+8 r− 7
+8
+for each point x ∈ G(α1,α2,α3)
+λ
+∩ {uα2 ≥ −λ− 7
+8 r
+1
+8 } ∩ {uα3 ≤ −λ− 3
+4r
+1
+4}. By
+transversality, the set {0 ≥ uα1 ≥ −λ− 7
+8r
+1
+8} ∩ {0 ≥ uα2 ≥ −λ− 7
+8 r
+1
+8} ∩ Br(p)
+can be covered by C (λr)
+7(n−2)
+8
+Euclidean balls of radius λ− 7
+8 r
+1
+8 .
+More-
+over, the intersection of Σλ with each ball of radius λ− 7
+8r
+1
+8 has area at
+most C (λr)− 7(n−1)
+8
+rn−1. This shows that Σλ ∩ {uα1 ≥ −λ− 7
+8 r
+1
+8 } ∩ {uα2 ≥
+−λ− 7
+8 r
+1
+8 } ∩ Br(p) has area at most C (λr)− 7
+8 rn−1. Since
+G(α1,α2,α3)
+λ
+∩ {uα2 ≥ −λ− 7
+8r
+1
+8} ∩ Br(p)
+⊂ Σλ ∩ {uα1 ≥ −λ− 7
+8 r
+1
+8 } ∩ {uα2 ≥ −λ− 7
+8r
+1
+8} ∩ Br(p),
+it follows that
+�
+rs+1−n
+�
+G(α1,α2,α3)
+λ
+∩{uα2≥−λ− 7
+8 r
+1
+8 }∩{uα3≤−λ− 3
+4 r
+1
+4 }∩Br(p)
+V s
+λ,−
+� 1
+s
+≤ C (λr)
+1
+8 − 7
+8s .
+This completes the proof of Lemma 3.10.
+Lemma 3.11. Let us fix an exponent s ∈ [1, 3
+2), and let Br(p) denote a
+Euclidean ball of radius r ≤ 1. If λr is sufficiently large, then
+�
+rs+1−n
+�
+G(α1,α2,α3)
+λ
+∩{uα3≥−λ− 3
+4 r
+1
+4 }∩Br(p)
+V s
+λ,−
+� 1
+s
+≤ C (λr)1− 3
+2s
+for all pairwise distinct elements α1, α2, α3 ∈ A.
+
+SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES
+17
+Proof. We distinguish two cases:
+Case 1: Suppose that Ω ∩ {uα1 = 0} ∩ {uα2 = 0} ∩ {uα3 = 0} = ∅. By
+continuity, we can find a real number δ such that Ω ∩ {uα1 ≥ −δ} ∩ {uα2 ≥
+−δ} ∩ {uα3 ≥ −δ} = ∅. If λr is sufficiently large, then λ− 3
+4r
+1
+4 ≤ δ. This
+implies
+G(α1,α2,α3)
+λ
+∩ {uα3 ≥ −λ− 3
+4 r
+1
+4 }
+⊂ Σλ ∩ {uα1 ≥ −δ} ∩ {uα2 ≥ −δ} ∩ {uα3 ≥ −δ} = ∅.
+Hence, the assertion is trivially true in this case.
+Case 2: Suppose that Ω ∩ {uα1 = 0} ∩ {uα2 = 0} ∩ {uα3 = 0} ̸= ∅. It
+follows from Assumption 3.1 that the hypersurfaces {uα1 = 0}, {uα2 = 0},
+and {uα3 = 0} intersect transversally.
+Let us consider an arbitrary point x ∈ G(α1,α2,α3)
+λ
+with uα3(x) ≥ −λ− 3
+4r
+1
+4.
+Clearly,
+Vλ(x) ≥ −Cλ
+for all points x ∈ G(α1,α2,α3)
+λ
+∩ {uα3 ≤ −λ− 3
+4 r
+1
+4 }. By transversality, the set
+{0 ≥ uα1 ≥ −λ− 3
+4r
+1
+4}∩{0 ≥ uα2 ≥ −λ− 3
+4 r
+1
+4}∩{0 ≥ uα3 ≥ −λ− 3
+4 r
+1
+4 }∩Br(p)
+can be covered by C (λr)
+3(n−3)
+4
+Euclidean balls of radius λ− 3
+4 r
+1
+4 .
+More-
+over, the intersection of Σλ with each ball of radius λ− 3
+4r
+1
+4 has area at
+most C (λr)− 3(n−1)
+4
+rn−1. This shows that Σλ ∩ {uα1 ≥ −λ− 3
+4 r
+1
+4 } ∩ {uα2 ≥
+−λ− 3
+4 r
+1
+4 }∩{uα3 ≥ −λ− 3
+4 r
+1
+4 }∩Br(p) has area at most C (λr)− 3
+2 rn−1. Since
+G(α1,α2,α3)
+λ
+∩ {uα3 ≥ −λ− 3
+4 r
+1
+4 } ∩ Br(p)
+⊂ Σλ ∩ {uα1 ≥ −λ− 3
+4 r
+1
+4 } ∩ {uα2 ≥ −λ− 3
+4 r
+1
+4 } ∩ {uα3 ≥ −λ− 3
+4r
+1
+4} ∩ Br(p),
+it follows that
+�
+rs+1−n
+�
+G(α1,α2,α3)
+λ
+∩{uα3≥−λ− 3
+4 r
+1
+4 }∩Br(p)
+V s
+λ,−
+� 1
+s
+≤ C (λr)1− 3
+2s .
+This completes the proof of Lemma 3.11.
+Proposition 3.12. Assume that the mean curvature of the hypersurface
+{uα = 0} with respect to g is nonnegative at each point in Ω ∩ {uα = 0}
+and that the Matching Angle Hypothesis is satisfied. Let us fix an exponent
+s ∈ [1, 3
+2), and let Br(p) denote a Euclidean ball of radius r ≤ 1. If λr is
+sufficiently large, then
+�
+rs+1−n
+�
+Σλ∩Br(p)
+V s
+λ,−
+� 1
+s
+≤ C (λr)−1 + C (λr)
+1
+8 − 7
+8s + C (λr)1− 3
+2s .
+
+18
+SIMON BRENDLE
+Proof. Combining Lemma 3.9, Lemma 3.10, and Lemma 3.11, we con-
+clude that
+�
+rs+1−n
+�
+G(α1,α2,α3)
+λ
+∩Br(p)
+V s
+λ,−
+� 1
+s
+≤ C (λr)−1 + C (λr)
+1
+8− 7
+8s + C (λr)1− 3
+2s
+for all pairwise distinct elements α1, α2, α3 ∈ A. On the other hand, Σλ =
+�
+α1,α2,α3 G(α1,α2,α3)
+λ
+, where the union is taken over all pairwise distinct el-
+ements α1, α2, α3 ∈ A.
+Hence, the assertion follows by summation over
+α1, α2, α3. This completes the proof of Proposition 3.12.
+Corollary 3.13. Assume that the mean curvature of the hypersurface {uα =
+0} with respect to g is nonnegative at each point in Ω ∩ {uα = 0} and that
+the Matching Angle Hypothesis is satisfied. Let us fix an exponent s ∈ [1, 3
+2).
+Then
+sup
+p∈Rn sup
+r≤1
+�
+rs+1−n
+�
+Σλ∩Br(p)
+V s
+λ,−
+� 1
+s
+→ 0
+as λ → ∞.
+Proof. Let us consider an arbitrary sequence λl → ∞. By Proposition
+3.6, we can find a sequence of positive real numbers δl → 0 such that
+sup
+p∈Rn
+sup
+r≤(δlλl)−1
+�
+rs+1−n
+�
+Σλl∩Br(p)
+V s
+λl,−
+� 1
+s
+→ 0
+as l → ∞. On the other hand, Proposition 3.12 implies that
+sup
+p∈Rn
+sup
+(δlλl)−1≤r≤1
+�
+rs+1−n
+�
+Σλl∩Br(p)
+V s
+λl,−
+� 1
+s
+→ 0
+as l → ∞. Putting these facts together, the assertion follows.
+4. Proof of the Theorem 1.1
+Throughout this section, we assume that n ≥ 3 is an odd integer, and Ω is
+a compact polytope in Rn with non-empty interior. Let g be a Riemannian
+metric which is defined on an open set containing Ω and has nonnegative
+scalar curvature at each point in Ω. We assume that the mean curvature of
+the hypersurface {uα = 0} with respect to g is nonnegative at each point in
+Ω ∩ {uα = 0} and that the Matching Angle Hypothesis is satisfied.
+Let U denote a Euclidean ball such that the closure of U is contained in
+the interior of Ω. Consider a sequence λl → ∞. Note that U ⊂ Ωλl if l is
+sufficiently large. By Proposition 2.14 we can find an m-tuple of harmonic
+spinors s(l) = (s(l)
+1 , . . . , s(l)
+m ) such that s(l) is defined on Ωλl; s(l) does not
+vanish identically; Ds(l) = 0 in Ωλl; and χs(l) = s(l) on Σλl.
+Standard
+
+SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES
+19
+unique continuation arguments imply that
+�
+U
+�m
+α=1 |s(l)
+α |2 dvolg > 0 if l is
+sufficiently large. By scaling, we can arrange that
+�
+U
+m
+�
+α=1
+|s(l)
+α |2 dvolg = 1
+for each l. Using Proposition 2.10, we obtain
+�
+Ωλl
+m
+�
+l=1
+|∇s(l)
+α |2 dvolg + 1
+4
+�
+Ωλl
+m
+�
+α=1
+R |s(l)
+α |2 dvolg
+≤ −1
+2
+�
+Σλl
+(H − ∥dN∥tr)
+� m
+�
+α=1
+|s(l)
+α |2
+�
+dσg.
+Proposition 3.5 implies that H − ∥dN∥tr ≥ Vλl at each point on Σλl. Con-
+sequently,
+�
+Ωλl
+m
+�
+l=1
+|∇s(l)
+α |2 dvolg + 1
+4
+�
+Ωλl
+m
+�
+α=1
+R |s(l)
+α |2 dvolg
+≤ 1
+2
+�
+Σλl
+Vλl,−
+� m
+�
+α=1
+|s(l)
+α |2
+�
+dσg.
+Note that the hypersurface Σλl = ∂Ωλl can be written as a radial graph
+with bounded slope. From this, it is easy to see that Ωλl is bi-Lipschitz
+equivalent to the Euclidean unit ball, with constants that are independent
+of λl. Using Theorem A.7 and Proposition 3.13, we obtain
+�
+Σλl
+Vλl,− F 2 dσg ≤ εl
+�
+Ωλl
+|∇F|2 dvolg + εl
+� �
+Σλl
+F dσg
+�2
+for every smooth function F on Ωλl, where εl → 0 as l → ∞. Moreover, the
+Sobolev trace theorem implies
+� �
+Σλl
+F dσg
+�2
+≤ C
+�
+Ωl
+|∇F|2 dvolg + C
+�
+U
+F 2 dvolg
+for every smooth function F on Ωλl, where C is a uniform constant inde-
+pendent of l. Putting these facts together, we conclude that
+�
+Σλl
+Vλl,− F 2 dσg ≤ Cεl
+�
+Ωλl
+|∇F|2 dvolg + Cεl
+�
+U
+F 2 dvolg
+
+20
+SIMON BRENDLE
+for every smooth function F on Ωλl. In the next step, we apply this inequal-
+ity with F =
+�
+δ2 + �m
+α=1 |s(l)
+α |2� 1
+2 , and send δ → 0. This gives
+�
+Ωλl
+m
+�
+l=1
+|∇s(l)
+α |2 dvolg + 1
+4
+�
+Ωλl
+m
+�
+α=1
+R |s(l)
+α |2 dvolg
+≤ 1
+2
+�
+Σλl
+Vλl,−
+� m
+�
+α=1
+|s(l)
+α |2
+�
+dσg
+≤ Cεl
+�
+Ωλl
+m
+�
+α=1
+|∇s(l)
+α |2 dvolg + Cεl
+�
+U
+m
+�
+α=1
+|s(l)
+α |2 dvolg
+for each l. Since the scalar curvature is nonnegative, it follows that
+�
+Ωλl
+m
+�
+α=1
+|∇s(l)
+α |2 dvolg ≤ Cεl
+�
+U
+m
+�
+α=1
+|s(l)
+α |2 dvolg
+if l is sufficiently large. Passing to the limit as l → ∞, we obtain a non-
+vanishing m-tuple of parallel spinors defined on the interior of Ω. Conse-
+quently, the Ricci tensor of g vanishes at each point in Ω. This completes
+the proof of Theorem 1.1.
+Appendix A. A variant of a theorem of Fefferman and Phong
+In this section, we describe a variant of an estimate due to Fefferman and
+Phong [4], which plays a central role in our argument. We denote by Q the
+collection of all (n − 1)-dimensional cubes of the form
+[2mj1, 2m(j1 + 1)] × . . . × [2mjn−1, 2m(jn−1 + 1)] × {0},
+where m ∈ Z and j1, . . . , jn−1 ∈ Z. For each Q ∈ Q, we denote by |Q| the
+(n − 1)-dimensional volume of Q.
+Theorem A.1. Fix an exponent s ∈ (1, n − 1). Let V be a nonnegative
+continuous function on the hyperplane Rn−1 × {0} ⊂ Rn with the property
+that
+diam(Q)s+1−n
+�
+Q
+V s ≤ 1
+for each (n − 1)-dimensional cube Q ∈ Q.
+Suppose that F is a smooth
+function on the half-space Rn
++ = {x ∈ Rn : xn ≥ 0}, and let f denote the
+restriction of F to the boundary ∂Rn
++ = Rn−1 × {0}. Then
+�
+Q
+V f 2 ≤ C
+�
+Q×[0,diam(Q)]
+|∇F|2 + C diam(Q)−1 |Q|−1
+� �
+Q
+|f|
+�2
+.
+for each (n − 1)-dimensional cube Q ∈ Q.
+
+SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES
+21
+The proof of Theorem A.1 involves a straightforward adaptation of the
+arguments of Fefferman and Phong [4]. Let us fix an exponent t such that
+s > t > 1. We define a nonnegative function W : Rn−1 × {0} → R by
+W(x) =
+sup
+Q∈Q,x∈Q
+�
+|Q|−1
+�
+Q
+V s
+� 1
+s
+for each point x ∈ Rn−1 × {0}. In other words, W s is the maximal function
+associated with the function V s. Clearly, V ≤ W.
+We assume that F is a smooth function on the half-space Rn
++ = {x ∈
+Rn : xn ≥ 0}, and let f denote the restriction of F to the boundary ∂Rn
++ =
+Rn−1 × {0}.
+For each (n − 1)-dimensional cube Q ∈ Q, we denote by
+fQ = |Q|−1 �
+Q f the mean value of f over the cube Q.
+Lemma A.2. For each (n − 1)-dimensional cube Q0 ∈ Q, we have
+�
+|Q0|−1
+�
+Q0
+W t
+� 1
+t
+≤ C
+sup
+Q∈Q,Q0⊂Q
+�
+|Q|−1
+�
+Q
+V s
+� 1
+s
+.
+Proof. For abbreviation, let
+Λ =
+sup
+Q∈Q,Q0⊂Q
+�
+|Q|−1
+�
+Q
+V s
+� 1
+s
+.
+We define a nonnegative function W0 : Q0 → R by
+W0(x) =
+sup
+Q∈Q,x∈Q⊂Q0
+�
+|Q|−1
+�
+Q
+V s
+� 1
+s
+for each point x ∈ Q0. Clearly,
+W(x) = max{Λ, W0(x)}
+for each point x ∈ Q0. The Hardy-Littlewood maximal inequality implies
+|Q0|−1 |{x ∈ Q0 : W0(x)s > α}| ≤ Cα−1 |Q0|−1
+�
+Q0
+V s ≤ Cα−1 Λs
+for all α > 0.
+We multiply both sides by α
+t
+s−1 and integrate over α ∈
+[Λs, ∞). This gives
+|Q0|−1
+�
+Q0
+W t
+0 ≤ C Λt,
+hence
+|Q0|−1
+�
+Q0
+W t ≤ C Λt.
+This completes the proof of Lemma A.2.
+
+22
+SIMON BRENDLE
+Lemma A.3. Given a real number ε > 0, we can find a real number δ > 0
+with the following property. If Q0 is an (n − 1)-dimensional cube in Q and
+A ⊂ Q0 is a Borel set with |A| ≤ δ |Q0|, then
+�
+A
+W ≤ ε
+�
+Q0
+W.
+Proof. Using Lemma A.2, we obtain
+�
+|Q0|−1
+�
+Q0
+W t
+� 1
+t
+≤ C
+sup
+Q∈Q,Q0⊂Q
+�
+|Q|−1
+�
+Q
+V s
+� 1
+s
+.
+Moreover,
+sup
+Q∈Q,Q0⊂Q
+�
+|Q|−1
+�
+Q
+V s
+� 1
+s
+≤ inf
+Q0 W ≤ |Q0|−1
+�
+Q0
+W.
+Therefore,
+�
+|Q0|−1
+�
+Q0
+W t
+� 1
+t
+≤ C |Q0|−1
+�
+Q0
+W.
+Hence, if A ⊂ Q0 is a Borel set with |A| ≤ δ Q0, then
+�
+A
+W ≤ |A|
+t−1
+t
+� �
+Q0
+W t
+� 1
+t
+≤ δ
+t−1
+t |Q0|
+t−1
+t
+� �
+Q0
+W t
+� 1
+t
+≤ Cδ
+t−1
+t
+�
+Q0
+W.
+This completes the proof of Lemma A.3.
+Lemma A.4. For each (n − 1)-dimensional cube Q0 ∈ Q, we have
+|Q0|−1
+�
+Q0
+W ≤ C diam(Q0)−1.
+Proof. Using Lemma A.2, we obtain
+�
+|Q0|−1
+�
+Q0
+W t
+� 1
+t
+≤ C
+sup
+Q∈Q,Q0⊂Q
+�
+|Q|−1
+�
+Q
+V s
+� 1
+s
+.
+Moreover, our assumption implies that
+�
+|Q|−1
+�
+Q
+V s
+� 1
+s
+≤ C diam(Q)−1
+for each (n − 1)-dimensional cube Q ∈ Q. Putting these facts together, the
+assertion follows.
+Lemma A.5. For each (n − 1)-dimensional cube Q0 ∈ Q, we have
+�
+Q0
+V |f − fQ0|2 ≤ C
+�
+Q0
+Wg2,
+
+SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES
+23
+where the function g : Q0 → R is defined by
+g(x) =
+sup
+Q∈Q,x∈Q⊂Q0
+|Q|−1
+�
+Q
+|f − fQ|
+for x ∈ Q0.
+Proof. Fix an (n − 1)-dimensional cube Q0 ∈ Q. We define a function
+h : Q0 → R by
+h(x) =
+sup
+Q∈Q,x∈Q⊂Q0
+|Q|−1
+�
+Q
+|f − fQ0|
+for x ∈ Q0. Note that V ≤ W and |f −fQ0| ≤ h at each point in Q0. Hence,
+it suffices to prove that
+�
+Q0
+Wh2 ≤ C
+�
+Q0
+Wg2.
+To prove this inequality, let α0 = |Q0|−1 �
+Q0 |f − fQ0|. For each α > α0,
+we denote by Qα the set of all (n − 1)-dimensional cubes Q ∈ Q with the
+following properties:
+• Q ⊂ Q0.
+• |Q|−1 �
+Q |f − fQ0| > α.
+• If ˜Q is an (n − 1)-dimensional cube in Q with Q ⊊ ˜Q and ˜Q ⊂ Q0,
+then | ˜Q|−1 �
+˜Q |f − fQ0| ≤ α.
+It is easy to see that
+|Q|−1
+�
+Q
+|f − fQ0| ≤ 2n−1α
+for all Q ∈ Qα. In particular, |fQ − fQ0| ≤ 2n−1α for all Q ∈ Qα. Moreover,
+�
+Q∈Qα
+Q = {x ∈ Q0 : h(x) > α}.
+Finally, no point can be contained in the interior of more than one cube in
+Qα.
+Let K > 1 and δ ∈ (0, 1) be two real numbers that will be chosen later.
+For each (n−1)-dimensional cube Q ∈ Qα satisfying |Q|−1 �
+Q |f −fQ| ≤ δα,
+
+24
+SIMON BRENDLE
+we have
+(Kα − |fQ − fQ0|)
+�
+˜Q∈QKα, ˜Q⊂Q
+| ˜Q|
+≤
+�
+˜Q∈QKα, ˜Q⊂Q
+� �
+˜Q
+|f − fQ0| −
+�
+˜Q
+|fQ − fQ0|
+�
+≤
+�
+˜Q∈QKα, ˜Q⊂Q
+�
+˜Q
+|f − fQ|
+≤
+�
+Q
+|f − fQ|
+≤ δα |Q|.
+Recall that |fQ − fQ0| ≤ 2n−1α for all Q ∈ Qα. Hence, if we choose K > 2n,
+then we obtain
+�
+˜Q∈QKα, ˜Q⊂Q
+| ˜Q| ≤ 21−n δ |Q|
+for each (n − 1)-dimensional cube Q ∈ Qα satisfying |Q|−1 �
+Q |f − fQ| ≤ δα.
+We next apply Lemma A.3 with ε = 1
+2 K−2. Hence, we can choose δ ∈
+(0, 1) sufficiently small (depending on K) so that
+�
+˜Q∈QKα, ˜Q⊂Q
+�
+˜Q
+W ≤ 1
+2 K−2
+�
+Q
+W
+for each (n − 1)-dimensional cube Q ∈ Qα satisfying |Q|−1 �
+Q |f − fQ| ≤ δα.
+For each (n−1)-dimensional cube Q ∈ Qα, the set Q∩{h > Kα} is contained
+in the union �
+˜Q∈QKα, ˜Q⊂Q ˜Q. This implies
+�
+Q∩{h>Kα}
+W ≤ 1
+2 K−2
+�
+Q
+W
+for each (n − 1)-dimensional cube Q ∈ Qα satisfying |Q|−1 �
+Q |f − fQ| ≤ δα.
+On the other hand, if Q is an (n − 1)-dimensional cube in Qα satisfying
+|Q|−1 �
+Q |f − fQ| > δα, then g > δα at each point in Q. Therefore,
+�
+Q∩{h>Kα}
+W ≤
+�
+Q∩{g>δα}
+W
+for each (n − 1)-dimensional cube Q ∈ Qα satisfying |Q|−1 �
+Q |f − fQ| > δα.
+Putting these facts together, we conclude that
+�
+Q∩{h>Kα}
+W ≤ 1
+2 K−2
+�
+Q
+W +
+�
+Q∩{g>δα}
+W
+
+SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES
+25
+for each (n−1)-dimensional cube Q ∈ Qα. Summation over all cubes Q ∈ Qα
+gives
+�
+{h>Kα}
+W ≤ 1
+2 K−2
+�
+{h>α}
+W +
+�
+{g>δα}
+W.
+This inequality holds for each α > α0. Moreover, since g ≥ α0 at each point
+in Q0, the inequality is trivially true for α ≤ α0. Finally, we multiply the
+inequality by α
+2 and integrate over α ∈ (0, ∞). This gives
+K−2
+�
+Q0
+Wh2 ≤ 1
+2 K−2
+�
+Q0
+Wh2 + δ−2
+�
+Q0
+Wg2.
+This completes the proof of Lemma A.5.
+Lemma A.6. For each (n − 1)-dimensional cube Q0 ∈ Q, we have
+�
+Q0
+Wg2 ≤ C
+�
+Q0×[0,diam(Q)]
+|∇F|2,
+where the function g : Q0 → R is defined by
+g(x) =
+sup
+Q∈Q,x∈Q⊂Q0
+|Q|−1
+�
+Q
+|f − fQ|
+for x ∈ Q0.
+Proof. Fix an (n−1)-dimensional cube Q0 ∈ Q, and let α0 = |Q0|−1 �
+Q0 |f−
+fQ0|. For each α > α0, we denote by Qα the set of all (n − 1)-dimensional
+cubes Q ∈ Q with the following properties:
+• Q ⊂ Q0.
+• |Q|−1 �
+Q |f − fQ| > α.
+• If ˜Q is an (n − 1)-dimensional cube in Q with Q ⊊ ˜Q and ˜Q ⊂ Q0,
+then | ˜Q|−1 �
+˜Q |f − f ˜Q| ≤ α.
+It is easy to see that
+|Q|−1
+�
+Q
+|f − fQ| ≤ 2nα
+for all Q ∈ Qα. Moreover,
+�
+Q∈Qα
+Q = {x ∈ Q0 : g(x) > α}.
+Finally, no point can be contained in the interior of more than one cube in
+Qα.
+
+26
+SIMON BRENDLE
+Let K > 1 be a real number that will be chosen later. For each (n − 1)-
+dimensional cube Q ∈ Qα, we have
+Kα
+�
+˜Q∈QKα, ˜Q⊂Q
+| ˜Q| ≤
+�
+˜Q∈QKα, ˜Q⊂Q
+�
+˜Q
+|f − f ˜Q|
+≤ 2
+�
+˜Q∈QKα, ˜Q⊂Q
+�
+˜Q
+|f − fQ|
+≤ 2
+�
+Q
+|f − fQ|
+≤ 2n+1α |Q|.
+Hence, if we choose K > 2n+2, then
+�
+˜Q∈QKα, ˜Q⊂Q
+| ˜Q| ≤ 1
+2 |Q|
+for each cube Q ∈ Qα. For each (n − 1)-dimensional cube Q ∈ Qα, the set
+Q ∩ {g > Kα} is contained in the union �
+˜Q∈QKα, ˜Q⊂Q ˜Q. This implies
+|Q ∩ {g > Kα}| ≤
+�
+˜Q∈QKα, ˜Q⊂Q
+| ˜Q| ≤ 1
+2 |Q|,
+hence
+|Q ∩ {g ≤ Kα}| ≥ 1
+2 |Q|
+for each (n−1)-dimensional cube Q ∈ Qα. We define a nonnegative function
+ϕ : Rn−1 × {0} → R by
+ϕ(x1, . . . , xn−1, 0) =
+� � diam(Q0)
+0
+|∇F(x1, . . . , xn−1, xn)|2 dxn
+� 1
+2
+.
+Moreover, we define a nonnegative function ψ : Q0 → R by
+ψ(x) =
+sup
+Q∈Q,x∈Q⊂Q0
+|Q|−1
+�
+Q
+ϕ
+for each point x ∈ Q0. In other words, ψ is the maximal function associated
+with ϕ. Using the Sobolev trace theorem, we obtain
+α ≤ |Q|−1
+�
+Q
+|f − fQ|
+≤ 2 |Q|−1 inf
+a∈R
+�
+Q
+|f − a|
+≤ C |Q|−1 inf
+a∈R
+� �
+Q×[0,diam(Q)]
+|∇(F − a)|
++ diam(Q)−1
+�
+Q×[0,diam(Q)]
+|F − a|
+�
+
+SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES
+27
+for each (n − 1)-dimensional cube Q ∈ Qα. Using the Poincar´e inequality,
+we conclude that
+α ≤ C |Q|−1
+�
+Q×[0,diam(Q)]
+|∇F|
+≤ C diam(Q)
+1
+2 |Q|−1
+�
+Q
+ϕ
+≤ C diam(Q)
+1
+2 inf
+Q ψ
+for each (n − 1)-dimensional cube Q ∈ Qα. This implies
+α2 diam(Q)−1 |Q| ≤ C
+�
+Q∩{g≤Kα}
+ψ2
+for each (n − 1)-dimensional cube Q ∈ Qα. Combining this estimate with
+Lemma A.4, we obtain
+α2
+�
+Q
+W ≤ C
+�
+Q∩{g≤Kα}
+ψ2
+for each (n−1)-dimensional cube Q ∈ Qα. Summation over all cubes Q ∈ Qα
+gives
+α2
+�
+{g>α}
+W ≤
+�
+{α<g≤Kα}
+ψ2.
+This estimate holds for each α > α0. We now multiply both sides by α−1
+and integrate over α ∈ (2α0, ∞). This gives
+�
+{g>4α0}
+Wg2 ≤ C
+�
+Q0
+ψ2.
+In the next step, we bound the contribution from the set {g ≤ 4α0}. Using
+the Sobolev trace theorem, we obtain
+α0 = |Q0|−1
+�
+Q0
+|f − fQ0|
+≤ 2 |Q0|−1 inf
+a∈R
+�
+Q0
+|f − a|
+≤ C |Q0|−1 inf
+a∈R
+� �
+Q0×[0,diam(Q0)]
+|∇(F − a)|
++ diam(Q0)−1
+�
+Q0×[0,diam(Q)]
+|F − a|
+�
+.
+
+28
+SIMON BRENDLE
+Using the Poincar´e inequality, we conclude that
+α0 ≤ C |Q0|−1
+�
+Q0×[0,diam(Q0)]
+|∇F|
+≤ C diam(Q0)
+1
+2 |Q0|−1
+�
+Q0
+ϕ
+≤ C diam(Q0)
+1
+2 inf
+Q0 ψ.
+This implies
+α2
+0 diam(Q0)−1 |Q0| ≤ C
+�
+Q0
+ψ2.
+Combining this estimate with Lemma A.4, we obtain
+α2
+0
+�
+Q0
+W ≤ C
+�
+Q0
+ψ2,
+hence
+�
+{g≤4α0}
+Wg2 ≤ C
+�
+Q0
+ψ2.
+Putting these facts together, we conclude that
+�
+Q0
+Wg2 ≤ C
+�
+Q0
+ψ2.
+On the other hand, the Hardy-Littlewood maximal inequality implies
+�
+Q0
+ψ2 ≤ C
+�
+Q0
+ϕ2 = C
+�
+Q0×[0,diam(Q0)]
+|∇F|2.
+This completes the proof of Lemma A.6.
+After these preparations, we now complete the proof of Theorem A.1.
+Combining Lemma A.5 and Lemma A.6, we conclude that
+�
+Q0
+V |f − fQ0|2 ≤ C
+�
+Q0×[0,diam(Q0)]
+|∇F|2
+for each (n − 1)-dimensional cube Q0 ∈ Q. This implies
+�
+Q0
+V f 2 ≤ C
+�
+Q0×[0,diam(Q0)]
+|∇F|2 + C |Q0|−2
+� �
+Q0
+V
+� � �
+Q0
+|f|
+�2
+for each (n − 1)-dimensional cube Q0 ∈ Q. Using the estimate
+|Q0|−1
+�
+Q0
+V ≤
+�
+|Q0|−1
+�
+Q0
+V s
+� 1
+s
+≤ C diam(Q0)−1,
+we conclude that
+�
+Q0
+V f 2 ≤ C
+�
+Q0×[0,diam(Q0)]
+|∇F|2 + C diam(Q0)−1 |Q0|−1
+� �
+Q0
+|f|
+�2
+
+SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES
+29
+for each (n − 1)-dimensional cube Q0 ∈ Q.
+Corollary A.7. Fix an exponent s ∈ (1, n − 1). Let V be a nonnegative
+continuous function on the unit sphere Sn−1 ⊂ Rn with the property that
+rs+1−n
+�
+Sn−1∩Br(p)
+V s ≤ 1
+for all points p ∈ Rn and all r ≤ 1. Suppose that F is a smooth function on
+the unit ball Bn = {x ∈ Rn : |x| ≤ 1}, and let f denote the restriction of F
+to the boundary ∂Bn = Sn−1. Then
+�
+Sn−1 V f 2 ≤ C
+�
+Bn |∇F|2 + C
+� �
+Sn−1 |f|
+�2
+.
+References
+[1] C. B¨ar and W. Ballmann, Boundary value problems for elliptic differential opera-
+tors of first order, Surveys in Differential Geometry vol. 17, pp. 1–78, Intern. Press,
+Somerville, 2012
+[2] C. B¨ar and W. Ballmann, Guide to boundary value problems for Dirac-type operators,
+arXiv:1307.3021
+[3] C. B¨ar, B. Hanke, and T. Schick, Remarks on the paper ”On Gromov’s dihedral
+extremality and rigidity conjectures” by Jinmin Wang, Zhizhang Xie, and Guoliang
+Yu, arXiv:2202.05180
+[4] C. Fefferman and D. Phong, Lower bounds for Schr¨odinger equations, Conference on
+Partial Differential Equations (Saint Jean de Monts, 1982), Conf. No. 7, pp. 1–7, Soc.
+Math. France, Paris, 1982
+[5] W. Fulton and J. Harris, Representation Theory, Springer-Verlag, 1991
+[6] M. Gromov, Dirac and Plateau billiards in domains with corners, Central European
+Journal of Mathematics 12, 1109–1156 (2014)
+[7] M. Gromov, Four Lectures on Scalar Curvature, arXiv:1908.10612
+[8] M. Gromov, Convex Polytopes, dihedral angles, mean curvature, and scalar curvature,
+arXiv:2207.13346
+[9] C. Li, A polyhedron comparison theorem for 3-manifolds with positive scalar curvature,
+Invent. Math. 219, 1–37 (2020)
+[10] J. Wang, Z. Xie, and G. Yu, On Gromov’s dihedral extremality and rigidity conjec-
+tures, arXiv:2112.01510
+Columbia University, 2990 Broadway, New York NY 10027, USA
+

7NAyT4oBgHgl3EQfcvfE/content/tmp_files/2301.00290v1.pdf.txt

@@ -0,0 +1,1045 @@
+BARVINN: Arbitrary Precision DNN Accelerator Controlled by a
+RISC-V CPU
+Mohammadhossein Askarihemmat1, Sean Wagner2, Olexa Bilaniuk3,
+Yassine Hariri4, Yvon Savaria1, Jean-Pierre David1
+1Ecole Polytechnique Montreal, Canada, 2 IBM, Toronto, Canada, 3 Mila, Montreal, Canada,
+4 CMC Microsystems, Kingston, Canada
+{mohammad.hossein.askari-hemmat, yvon.savaria, jpdavid}@polymtl.ca, wagnerse@ca.ibm.com,
+olexa.bilaniuk@mila.quebec, hariri@cmc.ca
+ABSTRACT
+We present a DNN accelerator that allows inference at arbitrary
+precision with dedicated processing elements that are configurable
+at the bit level. Our DNN accelerator has 8 Processing Elements
+controlled by a RISC-V controller with a combined 8.2 TMACs of
+computational power when implemented with the recent Alveo
+U250 FPGA platform. We develop a code generator tool that ingests
+CNN models in ONNX format and generates an executable com-
+mand stream for the RISC-V controller. We demonstrate the scalable
+throughput of our accelerator by running different DNN kernels
+and models when different quantization levels are selected. Com-
+pared to other low precision accelerators, our accelerator provides
+run time programmability without hardware reconfiguration and
+can accelerate DNNs with multiple quantization levels, regardless
+of the target FPGA size. BARVINN is an open source project and it
+is available at https://github.com/hossein1387/BARVINN.
+KEYWORDS
+neural networks, hardware acceleration, FPGA, low-precision
+ACM Reference Format:
+Mohammadhossein Askarihemmat1, Sean Wagner2, Olexa Bilaniuk3,, Yas-
+sine Hariri4, Yvon Savaria1, Jean-Pierre David1. 2023. BARVINN: Arbi-
+trary Precision DNN Accelerator Controlled by a RISC-V CPU. In 28th
+Asia and South Pacific Design Automation Conference (ASPDAC ’23), Janu-
+ary 16–19, 2023, Tokyo, Japan. ACM, New York, NY, USA, 7 pages. https:
+//doi.org/10.1145/3566097.3567872
+1
+INTRODUCTION
+Deep neural networks (DNNs) traditionally rely on floating point
+computations. These operations are slow and costly in terms of
+power consumption and required silicon area compared to fixed-
+point/integer operations. One way to accelerate computation in
+a DNN is to use less precision for computation via quantization
+[12]. This also reduces memory consumption as well as energy
+consumption. For instance, in a 45 nm process, 8-bit integer multi-
+plication and addition take 0.2 pJ and 0.03 pJ, respectively, while the
+Permission to make digital or hard copies of all or part of this work for personal or
+classroom use is granted without fee provided that copies are not made or distributed
+for profit or commercial advantage and that copies bear this notice and the full citation
+on the first page. Copyrights for components of this work owned by others than ACM
+must be honored. Abstracting with credit is permitted. To copy otherwise, or republish,
+to post on servers or to redistribute to lists, requires prior specific permission and/or a
+fee. Request permissions from permissions@acm.org.
+ASPDAC ’23, January 16–19, 2023, Tokyo, Japan
+© 2023 Association for Computing Machinery.
+ACM ISBN 978-1-4503-9783-4/23/01...$15.00
+https://doi.org/10.1145/3566097.3567872
+Table 1: Effects of Quantization on Accuracy and Model Size.
+Task
+Dataset
+Model
+Precision
+A/W
+Acc/
+MAP
+Size
+(MB)
+Classification
+CIFAR
+100
+ResNet18
+LSQ(2/2)
+76.81
+2.889
+LSQ(4/4)
+76.92
+5.559
+LSQ(8/8)
+78.45
+10.87
+FP32
+76.82
+42.8
+Object
+Detection
+VOC-
+2007
+SSD300-
+ResNet18
+LSQ(2/2)
+0.61
+10.34
+LSQ(4/4)
+0.60
+11.81
+LSQ(8/8)
+0.68
+14.77
+FP32
+0.59
+32.49
+same operations with 32-bit floating-point values requires 3.7 pJ
+for multiplication and 0.9 pJ for addition [11]. On an Intel Core i7
+4770 running at 3.4GHz, multiplication is more than 3 times faster
+for fixed-point compared to floating-point [15]. With recent quanti-
+zation techniques, these benefits are available with little to no loss
+in model performance and accuracy. In [9, 13], their quantization
+schemes showed accuracy losses of 1-3% at 2-bit precision on most
+classification and object detection models. Table 1 illustrates the
+result of applying Learned Scale Quantization (LSQ) [9] with differ-
+ent bit precisions on different models and tasks. Quantized models
+offer accuracy similar to full precision models, while having smaller
+size.
+Mixed-precision quantization [7, 16, 21, 23, 24] further provides
+finer control to reach an optimal solution by learning different
+precisions for each layer of a network. In [23], the authors illustrate
+that using their mixed-precision framework, they reduced model
+latency and energy consumption by a factor of almost 2× with little
+drop in accuracy compared with an 8-bit quantized model.
+Fully benefiting from low-precision in a DNN requires hardware
+that natively supports low-precision computations. Commodity
+hardware can perform arbitrary precision arithmetic by transform-
+ing data-layout and computing with bit-wise instructions [8]. How-
+ever, this approach is extremely costly for general processors, be-
+cause of the overhead for shifting, masking and packing bits to
+the correct format. At the time of writing this paper, there are no
+commercially available general processors (CPU or GPU) that can
+efficiently process data in arbitrary precision.
+In this paper, we propose an arbitrary low-precision DNN hard-
+ware accelerator called BARVINN. Our accelerator is software pro-
+grammable and can be integrated in the RISC-V standard devel-
+opment flow. It is designed as a highly optimized computational
+pipeline for DNNs that introduces low hardware overhead and of-
+fers low-power operation. The contributions of our paper are as
+follows:
+arXiv:2301.00290v1  [cs.AR]  31 Dec 2022
+
+ASPDAC ’23, January 16–19, 2023, Tokyo, Japan
+AskariHemmat et al
+• Implementation of a DNN hardware accelerator with arbi-
+trary fixed-point low-precision for matrix-vector multiply
+operations at high-throughput and low power.
+• Implementation of a custom embedded RISC-V CPU to con-
+trol an array of DNN accelerators by software.
+• Data structures for efficiently storing and processing weights
+and activations for high-throughput serial computation.
+• Development of a software code generator for transforming
+DNNs into RISC-V code that executes on our accelerator.
+In section 2, we review relevant DNN accelerators from the liter-
+ature. Section 3 presents the architecture of BARVINN. In section
+4, a detailed performance analysis of BARVINN is provided and
+compared with other DNN accelerators.
+2
+RELATED WORKS
+Several DNN hardware accelerators supporting quantization and
+low-precision have been presented in recent years for both FPGA
+and ASIC targets. Here, we discuss the accelerator architectures
+most relevant to our work.
+Recent FPGA-based accelerators include FINN [6, 22], DNNBuilder
+[25], and FILM-QNN [20]. In FINN and DNNBuilder, a software
+toolchain is used to map a trained DNN to generated logic modules
+that are integrated together. An overall processing pipeline is gen-
+erated and then synthesized for the target device. The advantage
+of this approach is that the logic efficiently implements a specific
+DNN with minimal overhead on a device that can be reconfigured
+to different DNNs at different times. However, this approach re-
+quires that all DNN layers be implemented in the logic all at once,
+which limits the size of the DNN to the amount of logic resources
+available on a given FPGA. While FINN supports low-precision
+down to binary and DNNBuilder down to 4-bit, neither supports
+arbitrary and mixed-precision at different DNN layers. In contrast,
+FILM-QNN does support DNN models of arbitrary sizes and quan-
+tized DNNs with mixed precisions. However, it is limited to only 4-
+or 8-bit weights and 5-bit activations due to a bit-packing scheme
+used with the DSP blocks in the FPGA.
+Several ASIC accelerator designs support arbitrary precision. Bit
+Fusion [19] uses a large array of 2-bit processing elements that
+can be fused together to perform up to 8-bit operations. Loom
+[18], and BitBlade [17] employ bit serial computation schemes for
+added flexibility. While bit-serial computation of any single math
+computation (e.g. multiplication) inherently requires additional
+clock cycles and latency over bit-parallel circuits, these designs
+exploit the large number of computations in a DNN that can be
+done in parallel. This is done by implementing a large number
+of bit-serial computational units operating simultaneously, which
+compensates high latency with high throughput. For example, the
+Loom engine consists of 128 × 16 = 2048 Serial Inner-Product units
+(SIPs), each of which performs 16 1 × 1-bit products per cycle.
+3
+ARCHITECTURE
+BARVINN is designed to provide high-throughput and software
+programmability, while supporting DNNs of arbitrary size and type.
+The high-level architecture of BARVINN is illustrated in Figure 1.
+It consists of the following main components: 1) an array of Matrix
+Vector Units (MVU) [5], and 2) a RISC-V CPU called Pito [4] as a
+controller for the MVU array. The MVUs accelerate common DNN
+computations such as GEMV, GEMM, and convolutions along with
+other operations such as batch normalization, ReLU activation, and
+quantization. Pito coordinates the computations in the MVU array
+while also handling data transfers to and from the host system.
+As it is not possible to foresee all possible neural networks that
+may crop up in the literature in the future, high-level sequencing of
+tensor operations for BARVINN is done in software. To control the
+array of processing elements, unlike the aforementioned accelera-
+tors, BARVINN uses the standard RISC-V RV32I ISA. This allows
+us to leverage the pre-existing software ecosystem. Furthermore,
+by using a CPU that supports a well known ISA, BARVINN is more
+flexible and it can be adapted to support new DNN architectures.
+3.1
+Matrix Vector Units
+The base configuration of BARVINN is implemented with an array
+of 8 MVUs. Figure 1 shows each MVU is a 64-element vector pipeline
+with several modules: a) a Matrix Vector Product unit (MVP), b)
+RAMs for activations/weights/scalers/biases, c) a scaler unit, d)
+a pooling/activation unit, and e) a quantizer. MVUs compute 64
+output vector elements per clock cycle using a 64 element input data
+vector from the activation RAM and a 64×64 element matrix from
+the weight RAM. Activation and weight RAMs store data in low-
+precision. MVP units operate in low-precision, while subsequent
+units in the pipeline operate in high-precision fixed-point.
+To justify our design choice of operating on 64 element vectors,
+we analysed over 50 models available at the ONNX Model Zoo
+[2] to check the input channel size of convolution layers. Figure 2
+illustrates a distribution of input channel size of all layers among
+those models. We found that 79% of these models use convolution
+with input channel sizes that are multiples of 64.
+3.1.1
+Matrix-Vector Product Units. Matrix operations are carried
+out by the MVP units. They compute on fixed-point arbitrary pre-
+cision operands from 1- to 16-bit. Each MVP has 64 vector-vector
+product (VVP) pipelines. Each VVP has 64 input lanes with 1-bit
+multipliers, followed by an addition tree with 8-bit output, as shown
+in Figure 4. On every cycle, 64 bits from the activation RAM are
+broadcasted to each of the 64 VVPs, while a 64×64 matrix tile from
+the weight RAM is read with each row of the tile sent to sepa-
+rate VVPs. The VVPs compute a 64-element dot product on 1-bit
+operands in each pipeline. With 64 VVPs per MVP, the overall
+output is a 64-element vector.
+MVPs compute arbitrary bit precision dot-products using the bit-
+serial scheme of [5]. Weights and activations can be unsigned or 2’s-
+complement signed fixed-point. Bit-depth is set independently for
+both, thus allowing for mixed precision. The bit-serial dot-product,
+shown in Algorithm 1, is a multi-cycle sequence starting with the
+most significant bits (MSB) from 64 elements of the activation and
+weight tensors. Bits are multiplied in each lane and results are
+added together across lanes in an addition tree producing an 8-bit
+dot product. This is added to an accumulator/shifter (see Figure 4).
+The MSB×MSB result represents the highest order-of-magnitude
+partial sum of the overall dot product. The MVP then computes the
+next lower order-of-magnitude partial sum by drawing the needed
+bit combinations of the operands. When a change in the order-of-
+magnitude is made, the accumulator is shifted left by 1-bit to align
+
+BARVINN: Arbitrary Precision DNN Accelerator Controlled by a RISC-V CPU
+ASPDAC ’23, January 16–19, 2023, Tokyo, Japan
+Figure 1: BARVINN hardware architecture with MVU array and Pito RISC-V controller. Right side is MVU detail.
+to the order-of-magnitude prior to adding the addition tree output.
+MVPs are fully pipelined, allowing them to work on different bit
+combinations at different stages without stalling. The operation
+completes when the dot products of the least significant bits (LSB)
+of the operands are computed and accumulated. For 𝑏𝑤-bit weights
+and 𝑏𝑎-bit activations, the overall operation takes 𝑏𝑤𝑏𝑎 cycles to
+compute one tile of the output vector. The precision of the operands
+is configured separately for each MVU, thus each MVU can process
+different layers with different bit precisions.
+Algorithm 1 Bit-serial dot-product
+1: 𝑏𝑎, 𝑏𝑤: activation and weight bit precisions
+2: 𝑥,𝑤: activation and weight vectors of size 𝑛
+3: 𝑗,𝑘: bit position for activations and weights
+4: 𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑜𝑟 ← 0
+5: for 𝑖 ← 𝑏𝑤 + 𝑏𝑎 to 1 do
+6:
+for all (𝑗,𝑘) where 𝑗 + 𝑘 == 𝑖 do
+7:
+for 𝑙 ← 0 to 𝑛 − 1 do
+8:
+𝑜𝑛𝑒𝑏𝑖𝑡𝑝𝑟𝑜𝑑 = 𝑥𝑗 [𝑙] × 𝑤𝑘 [𝑙]
+9:
+𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑜𝑟 ← 𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑜𝑟 + 𝑜𝑛𝑒𝑏𝑖𝑡𝑝𝑟𝑜𝑑
+10:
+end for
+11:
+end for
+11:
+shift 𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑜𝑟 left 1-bit
+12: end for
+13: 𝑜𝑢𝑡𝑝𝑢𝑡 ← 𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑜𝑟
+Our bit-serial dot product scheme differs from other architec-
+tures. The computation scheme in BitFusion is based on computing
+the individual products of the overall dot-product, that are then
+summed. This requires a large number of shift-registers to align
+and sum partial products. BARVINN and BitBlade instead inter-
+change the ordering of the computation such that partial products
+of the same magnitude from all individual products are computed
+first and then summed. This reduces the number shifters needed.
+BARVINN additionally serializes the computation of partial prod-
+ucts of different magnitude, requiring only a single fixed shifter and
+a single adder tree, whereas BitBlade requires 16 variable shifters
+and 17 adder trees. BARVINN maintains throughput despite this
+serialized scheme by parallelizing across a wider number of input
+operands and producing a larger number of output products per
+clock cycle. BitFusion and BitBlade are further limited to operand
+sizes 2, 4, and 8-bit, whereas MVUs in BARVINN and SIPs in Loom
+support operands of any bit-depth down to 1-bit. However, Loom’s
+data loading scheme restricts the efficiency for general matrix mul-
+tiply operations when the weight bit depth is below 16, whereas
+BARVINNs is able to maintain full throughput down to 1-bit.
+3.1.2
+Memories and Data Layout. Activation and weight RAMs
+store data in a bit-transposed format shown in Figure 3 to exploit
+bit-serial computation. When precision is greater than 1 bit, tensor
+elements are organized in blocks where bits of the same order-of-
+magnitude are stored in the same memory word starting with the
+MSBs in the lowest address. A block of 𝑛 elements with precision 𝑏
+
+PITO RISC-V Core
+Matrix Vector Unit (MVU)
+Fetch
+Decode
+Execute
+Mem
+Commit
+Write Interconnect
+t  write Controller
+Read Controller Read Interco
+onnect
+Word
+Word
+Word
+Word
+imm :
+Instruction
+CSR WRITE
+ALU
+PRF1
+Memory
+D$
+8KB
+Decoder
+上
+CSR1
+Scaler
+PC+
+Activation Ram
+Data
+Ram
+select
+Memory
+Weight
+8KB
+Controller)
+Interface
+Ram
+pank3
+Bias
+APB][
+IRQ
+7
+Ram
+APB Bus
+1
+1
+Memory I
+64 bits
+64x64 bits
+MVU1
+MVU8
+MVP
+ Weight
+Input
+Bias
+Scaler
+Weight
+Bias
+Scaler
+Input
+32'bit 32'bit 32'bit
+32'bit
+Ram
+Ram
+Ram
+Ram
+Ram
+ Ram
+Ram
+Ram
+16bit
+4
+↑
+↑
+Scaler o
+ Scaler 1 
+Scaler 2
+Scaler 63
+32 bit
+32'bit 32'bit 32'bit
+32'bit
+Crossbar
+4
+Pol/ReLUPool/ReLu|Pool/ReLu
+Pool/ReLu
+Legend
+32'bit 32'bit 32'bit
+32'bit
+4
+[ Quantser
+PooIReLu
+> MVU to MVU Interface
+Quantser  Quantser
+<> Wire Interface
+1 bit
+1 bit
+1 bit
+1 bit
+> APB Bus CSR Config Interface
+AXlMemInterface
+L64 bit-ASPDAC ’23, January 16–19, 2023, Tokyo, Japan
+AskariHemmat et al
+Figure 2: Channel sizes in models from ONNX Model Zoo.
+requires 𝑏 memory words of width 𝑛. Activation vector elements
+are in blocks of 64 while weights matrix elements are in blocks of
+4096 bits in order to load a 64×64 matrix tile. A transposer module
+transforms input data from the host into the needed bit-transposed
+format. Transposition is only needed on the first layer of a DNN
+since MVUs write back to activation RAM in the bit-transposed
+format. Weights are pre-processed by a toolchain on the host and
+loaded into weights RAMs in the expected bit-transposed format.
+Figure 3: Bit-transposed data format for arbitrary precision.
+The layout of the tensors in the RAMs depends on the operation
+to be performed. For GEMV, activations are organized as vectors
+with blocks of 64 elements, while weight matrices are organized
+as a set of 64×64 tiles. For 2D convolutions, layout of activations
+is 𝑁𝐻𝑊𝐶, where the channel dimension 𝐶 is the innermost di-
+mension, followed by width 𝑊 , height 𝐻, and batch size 𝑁. The 𝐶
+dimension is the innermost since several common DNNs such as
+ResNet typically have hidden layer channel depths that are pow-
+ers of 2, and hence align to the 64 input lanes of the VVPs. When
+there are more than 64 channels, the first 64 channels are stored in
+the first block, the second 64 channels are stored into the second
+block and so on. As an example, an input tensor of [N=1, H=8, W=8,
+C=256] with 2-bit precision, will have 4 channel blocks, each block
+will have 64 rows of 2 by 64-bit elements.
+Our weight tensor memory layout for 2D convolutions is de-
+signed to support efficient execution by interleaving the input chan-
+nel dimension 𝐶𝑖 and output channel dimension 𝐶𝑜. Each weight
+memory word contains 64 subsets from the𝐶𝑜 dimension, with each
+subset containing 64 elements from the𝐶𝑖 dimension. A contiguous
+Figure 4: VVP unit with a shifter-accumulator. Bit 𝑗 from
+64 elements of the activation tensor 𝑥 and bit 𝑘 from 64 ele-
+ments of the weight tensor𝑤 are input in a bit-serial fashion.
+Note that some input bits and layers of the 5-deep adder tree
+are not shown.
+block of 𝑏𝑤 words that stores a complete set of bits for the needed
+weight precision is referred to as a channel block 𝐶𝑏. The layout
+for 2D convolution weights is 𝐶𝑜,𝑠𝐹𝐻 𝐹𝑊 𝐶𝑏, where 𝐶𝑜,𝑠 = 𝐶𝑜/64
+are output channel sets, and the kernel size is (𝐹𝑊 , 𝐹𝐻 ).
+3.1.3
+Job Configuration and Execution. MVUs are programmed to
+perform jobs such as GEMV or Conv2D operation. A controller sets
+configuration registers that orchestrate the sequence of calculations
+and memory reads to complete an operation in the MVUs. Once the
+job is finished, the MVU will generate an interrupt to the controller,
+indicating that the job is finished and results are ready to be sent
+back to the host or to trigger subsequent operations on the same
+MVU or other MVUs. While a MVU is busy, it can be programmed
+to prepare the next job to minimize idle time.
+Each MVU contains address generation units (AGU) that drive
+the memory access pattern across the activation and weight RAMs.
+The access pattern is managed by a set of up to five nested loops
+with parameters setting the number of iterations and the forward
+or backward address jumps to make on each iteration. The address
+jump scheme reduces the logic to a set of small accumulators to
+control the loops and small adders to compute addresses. Innermost
+loops are usually set to stride over the bit depth of the activations
+and weights. Outer loops are used to iterate over the bit combina-
+tions for the serial dot-product procedure and over the dimensions
+of the tensors. For GEMV, two nested loops are required for both
+activations and weights. Conv2D operations are programmed to
+compute one row of the output activation map per job, requiring
+four nested loops.
+3.1.4
+Pipeline Modules. Each MVU has modules downstream from
+the MVP to implement other DNN operations including a multi-
+plier/adder unit, a pooling/ReLU unit, and a quantizer/serializer
+unit. These modules operate at high-precision. Fixed-point mul-
+tiplier/adder units (Scaler in Figure 1), compute DNN operations
+such as batch normalization and quantization scaling as in LSQ [9].
+Scalers multiply the MVP output by a 16-bit operand sourced from
+the scaler RAM. In an FPGA, the multiplier is 27 × 16, which aligns
+with the port widths of on-chip fixed DSP units. An adder that
+follows adds 32-bit fixed-point bias terms from bias RAM. Scaler
+
+600
+Not Multiple of 64
+Multiple of 64
+500
+400
+ayers
+e-
+300
+#
+200
+100
+0
+128 256 512 1024
+1
+2
+4
+8
+16
+32
+64
+Input Channel Sizek elements
+address
+bit
+01
+k
+0
+n-1
+MSB
+1
+n-2
+block 0
+n-1
+0
+LSB
+MSB
+block 1
+LSBw,[0]
+x,[1] 
+32-bit shifter/accumulator
+w,[1]
+x,[2]
+W,[2]
+8-bit sum
+x[3]
+w,[3]
+x,[62]
+W,[62]
+x,[63]
+W.[63]BARVINN: Arbitrary Precision DNN Accelerator Controlled by a RISC-V CPU
+ASPDAC ’23, January 16–19, 2023, Tokyo, Japan
+and bias RAMs have independent AGUs. The module that follows
+combines max pooling and ReLU (Pool/ReLU in Figure 1), imple-
+mented as a comparator with an internal register. For ReLU, the
+incoming value is checked against the register initially set to 0. The
+combined MaxPool/ReLU is implemented by programming MVUs
+to produce data in the sequence needed for a MaxPool window.
+The pipeline ends at the quantization/serialization unit (QuantSer
+in Figure 1). It takes 32-bit fixed-point data from each of the 64 data
+paths and serializes them into 64 1-bit outputs. It is programmed to
+set the output bit-depth and the MSB position from the input word.
+Combined with scaler units, this is used to implement quantization
+schemes such as LSQ [9]. Serialized outputs of each datapath are
+grouped into a single 64-bit word that is sent either to the activation
+RAM of the same MVU, or to a different MVU via an interconnect.
+3.1.5
+Interconnect. MVUs can send data to each other via an inter-
+connect implemented as an 8-way crossbar switch with broadcast
+capability. A source MVU is programmed to send its output results
+in a serialized fashion to a given address in the activation memory
+of a destination MVU(s). At a destination MVU, a fixed-priority
+arbitration scheme to the write port of the target MVU activation
+RAM is used. The interconnect is given highest priority, followed
+by the controller, then lastly the MVU itself. When multiple MVUs
+attempt to write to the same destination MVU, a fixed priority
+scheme determines which MVU can write to its memory.
+3.1.6
+DNN Mapping. Each MVU can be assigned to different lay-
+ers of a DNN, such as convolutions and fully-connected layers.
+Alternatively, a single layer can be split between multiple MVUs
+with each MVUs processing a subset of the input activations and/or
+weights. Partial results are forwarded from one MVU to another
+via the interconnect to process subsequent layers of the network,
+thus creating an overall processing pipeline through the array. By
+sending partial results from one MVU to another, subsequent MVUs
+can begin processing as soon as sufficient data has been received
+from previous layers. For instance, a MVU processing a 3× 3 convo-
+lution requires only 3 rows of activations from the previous layer to
+produce one output row of the layer it is processing. This avoids the
+need to wait until all outputs from a layer are generated, which re-
+duces latency and idle time. Furthermore, the ability to immediately
+process partial layer outputs by subsequent MVUs keeps on-chip
+storage requirements low, since only the partial set of activations
+required to produce the next layer partial output needs to be stored.
+Depending on the performance goal, BARVINN can execute a
+DNN in either Pipelined mode or Distributed mode. In Pipelined
+mode (Figure 5.a), the MVU array can process up to 8 convolutions
+and fully-connected layers all at once. Each MVU can be configured
+to use different precisions. In cases where a DNN model contains
+more than 8 layers, the MVU array can be programmed to process
+the entire model by dividing it into subsets of up to 8 layers each.
+Each MVU can be loaded with weights from layers in each subset,
+either all from the start of processing if there is sufficient weight
+memory available in each MVU or on-the-fly from external memory
+if not. Output activations from the last MVU in the chain can also
+be stored temporarily in off-chip memory and fetched later in the
+case where the first MVU is still processing data from the current
+lap. In the Distributed mode, to minimize latency, the objective is
+to process single batch inputs as fast as possible. As can be seen
+in Figure 5.b, in this mode, the computation of a single layer is
+broken into 8 independent computation regions. All MVUs will be
+programmed to share the same set of weights. To make sure an MVU
+computation is independent of those performed on other MVUs, the
+user might need to copy the input regions that are shared between
+computation units. The programmability of BARVINN allows the
+user to mix and match these execution modes for different layers
+and models to achieve highest performance.
+a. Pipelined mode
+b. Distributed mode
+Figure 5: Execution flow of a DNN on the MVU array in
+Pipelined (a) and Distributed (b) modes. In Pipelined mode
+each MVU processes one layer at a time. In distributed mode,
+the computation of a single layer is distributed among mul-
+tiple MVUs.
+3.2
+Pito: RISC-V-based Controller
+To make use of MVUs for neural networks, a control unit is required.
+The controller is a barrel RISC-V processor designed to control
+the 8 MVUs using separate but communicating hardware threads
+(harts) that each manage their respective MVUs. DNN layers are ex-
+ecuted either in distributed or in a pipelined fashion, depending on
+whether the DNN is compiled to maximize throughput or minimize
+latency. This design allows MVUs to complete tensor operations
+independently of each other. The drawback is that it requires 8
+microprocessors to execute the 8 programs. We instead amortized
+the fixed costs of the processor by adopting barrel processing. With
+a 8-way threaded processor, we may assign one thread to control
+each of the MVUs. Because every thread comes up for execution
+only every 8 clock cycles, the five pipeline stages (fetch, decode,
+execute, data read & writes and commit) can be completely hidden.
+Branch prediction units are unnecessary. Since tensor operations
+can require hundreds of cycles to execute on a MVU, the barrel
+processor can fully turn over dozens of times in the interim, allow-
+ing each thread to issue the next command to its MVU in a few
+instructions.
+We adopted a Harvard architecture and divided the instruction
+and data RAM, 8KB each, and shared between all harts. This gives a
+
+MemoryInterface
+UART
+MvU Array
+MVU6
+MU5
+D$
+IS
+8KB
+8KB
+1
+CSR8 pe_irq
+pestart
+CSR2 pe irq
+Crossbar
+CSR1 pe_irq
+ant
+Pito RISC-V
+pe_start ld
+pe_command httus
+APB
+pe_quant
+pe_status
+Input
+Convo
+Conv4
+Conv1
+Conv5
+Conv2
+Conv6
+Input
+Weight
+Conv7
+Output
+Conv3
+[1x63x32x32]
+[64x64x3x3]
+[1x63x32x32]ASPDAC ’23, January 16–19, 2023, Tokyo, Japan
+AskariHemmat et al
+1K word space to store data and instructions to control each MVU.
+The processor executes instructions following compilation order
+and without any further scheduling. A hart scheduler provides
+access to the required resources for the hart at each stage. In the
+fetch stage, each hart loads instructions from the instruction RAM.
+The program counter (PC) and register file for each hart is different
+and the hart scheduler indicates which register should be accessed
+at a given time. The Decode stage decodes instruction and loads
+source registers or an immediate operand. Our RISC-V controller is
+compatible with RV32I RISC-V ISA with minimal support for privi-
+lege specification to make Control and Status Registers (CSRs) and
+Interrupts available to interface with the MVU array. In addition
+to the base CSRs, we have added 74 MVU-specific CSRs to allow
+software to control the processing element array. These CSRs con-
+trol different settings within an MVU such as weight and activation
+precision, AGU’s jump settings, input, weight and output memory
+address and pipeline module selection as described in 3.1.4.
+3.3
+Code Generator
+BARVINN performs GEMM/GEMV, Convolutions, Maxpooling and
+activation (ReLU). However, it is up to the user to sequence the
+operations within a DNN with software. To facilitate this, we de-
+veloped a code generator that takes a DNN described in ONNX
+[3] and configuration settings (weight/input/output precision), and
+generates RISC-V code for each operation. The code generator ex-
+ports weights to the bit-transposed format described in section 3.1.2.
+Since each MVU works on 64-bit words, the code generator tiles
+each weight tensor in blocks of 64×64. When this cannot be done
+(either tensor input channel or output channel is not a multiple of
+64), we pad the corresponding tile. Currently, our code generator
+does not apply graph optimization techniques. Also, for now, our
+code generator supports Pipelined mode execution. In the follow-
+ing section, we used our code generator to map PyTorch models to
+micro kernel codes which can then be directly used by BARVINN.
+4
+PERFORMANCE ANALYSIS AND RESULTS
+4.1
+Experimental Setup
+To illustrate the performance of BARVINN, we chose the ResNet9
+image classifier model for the CIFAR10 dataset. We trained and
+quantized a ResNet9 model on CIFAR10 using LSQ [9] and used the
+residual distillation [14] technique to remove shortcut connections
+(Plain CNN models). In many image classification DNN models such
+as ResNet [10], the input to the first layer typically consists of less
+than 64 channels. Furthermore, due to sensitivity of the first and
+last layer to information loss, most state-of-the-art compression
+and quantization methods do not apply optimization on input and
+output layers [9], hence keeping these layers untouched and in full
+precision. We have adopted the same technique to compute first
+and last layers on the host or on the RISC-V controller.
+Table 2 shows the performance of ResNet9 on CIFAR10 in the
+PyTorch framework. Once we were satisfied with the performance
+of our quantized model, we exported the trained model to ONNX
+and then used our code generator. Table 3 illustrates the per layer
+computation cost of running ResNet9 on BARVINN with 2-bit ac-
+tivations and weights. All convolutions use a padding of 1. As
+discussed before, we skipped running the first and last layer on
+Table 2: ResNet9 with different bit precision on CIFAR10
+ResNet9 Model
+Precision
+Accuracy
+Size (Bytes)
+Original
+Fp32
+90.8%
+19605141
+Plain-CNN
+Fp32
+91.1%
+18912487
+Quantized Plain-CNN
+Int2
+89.2%
+1181360
+Table 3: ResNet9 layers for CIFAR10 dataset and computa-
+tion cost. All layers are quantized to 2-bit for activation and
+weights, except for the first and last layers.
+Layer
+Input
+Kernel
+Output
+Cycles
+conv0
+[3, 32, 32]
+[64, 3, 3, 3]
+[64, 32, 32]
+N/A
+conv1
+[64, 32, 32]
+[64, 64, 3, 3]
+[64, 32, 32]
+34560
+conv2
+[64, 32, 32]
+[64, 64, 3, 3]
+[64, 32, 32]
+34560
+conv3
+[64, 32, 32]
+[128, 64, 3, 3]
+[128, 16, 16]
+17280
+conv4
+[128, 16, 16]
+[128, 128, 3, 3]
+[128, 8, 8]
+32256
+conv5
+[128, 8, 8]
+[256, 128, 3, 3]
+[128, 8, 8]
+16128
+conv6
+[128, 8, 8]
+[256, 256, 3, 3]
+[256, 4, 4]
+27648
+conv7
+[256, 4, 4]
+[512, 256, 3, 3]
+[256, 4, 4]
+13824
+conv8
+[256, 4, 4]
+[512, 512, 3, 3]
+[512, 4, 4]
+18432
+fc
+[512, 4, 4]
+[10, 512]
+[10]
+N/A
+Total:
+194688
+BARVINN and we kept them in their original format. The overall
+computation takes 194,688 cycles to complete.
+Our design was written in Verilog and synthesized using Xilinx
+Vivado 2021.1 for the Xilinx Alveo U250 accelerator card. Synthesis
+results for the RISC-V controller, the processing array, and the accel-
+erator are presented in Table 4. Power consumption was estimated
+using the software tools in Vivado.
+4.2
+Discussion
+We compared BARVINN with FINN [22], which is a templated Vi-
+vado HLS C++ library of common DNN layers. Like BARVINN,
+FINN can generate hardware for arbitrary precision, but is not
+software programmable. Hence, once the FINN hardware is gen-
+erated, the user cannot change the computation data stream. We
+attempted to compare the performance of BARVINN with FINN
+using the ResNet9 model we used earlier. However, at the time of
+writing, FINN supports simple linear topologies and we were not
+able to get performance metrics for our model. Instead, we used the
+available CIFAR10-CNV model from the FINN repository that was
+tuned for the FINN dataflow for our comparison. Table 5 shows the
+performance of BARVINN and FINN. For this experiment, we used
+different precisions for weights and activation. For both tools, we
+used the performance estimation numbers for frames per second
+(FPS). For FINN, we used the default folding configurations publicly
+available in FINN-example repository [1]. As illustrated in Table
+5, we provide 7-15 times better throughput albeit with higher LUT
+usage. On the other hand, for higher bit precisions, FINN provides
+a better FPS/LUT, suggesting a scalable solution for bigger models.
+We also compared the performance on a ResNet-50 model. Table
+6 shows our estimated FPS for BARVINN executing in Pipelined
+mode along with reported performance for FINN [1] synthesized for
+the Xilinx U250 and for FILM-QNN [20] synthesized for the Xilinx
+ZCU102 FPGA. While FINN has the highest FPS, BARVINN shows
+
+BARVINN: Arbitrary Precision DNN Accelerator Controlled by a RISC-V CPU
+ASPDAC ’23, January 16–19, 2023, Tokyo, Japan
+Table 4: Post-synthesis resource utilization of BARVINN.
+Resource
+Pito RISC-V
+MVU Array
+Overall
+LUT
+10454
+190625
+201079
+BRAM
+15
+1312
+1327
+DSP
+0
+512
+512
+Dynamic Power
+0.410 W
+21.066 W
+21.504 W
+Frequency
+250 MHz
+250 MHz
+250 MHz
+Table 5: Estimated performance of running CNV model on
+CIFAR10 on Alveo U250 when different bit precision is used.
+Bits
+(W/A)
+kLUT
+BRAM
+DSP
+FPS
+FPS/
+kLUT
+Ours
+1/1
+201.1 (15.0%)
+1327
+512
+61035
+303.5
+1/2
+201.1 (15.0%)
+1327
+512
+30517
+151.7
+2/2
+201.1 (15.0%)
+1327
+512
+15258
+75.8
+FINN
+1/1
+28.2 (2.1%)
+150
+0
+7716
+273.6
+1/2
+19.8(1.47%)
+103
+0
+2170
+109.6
+2/2
+24.3(1.81%)
+202
+0
+2170
+89.3
+Table 6: Performance for ResNet-50 model on ImageNet.
+Bits (W/A)
+Clock Freq.
+FPS
+FPS/Watt
+Ours
+1/2
+250 MHz
+2296
+106.8
+FINN-R [1][6]
+1/2
+178 MHz
+2873
+41.0
+FILM-QNN [20]
+4(8)/5
+150 MHz
+109
+8.4
+the best performance per Watt. According to the FINN-example
+repository [1], a fine-tuned ResNet50 model, requires more than
+87% of Alveo U250 accelerator’s resources. This shows the limits
+of FINN dealing with bigger models. BARVINN requires the same
+LUT usage regardless of the model size and bit-width.
+5
+CONCLUSION
+In this paper, we presented an FPGA-based DNN accelerator that
+supports arbitrary bit precision computations. We tested the perfor-
+mance of BARVINN over different DNN kernels and models with
+different bit precision. For model deployment, we developed a code
+generator tool that takes in a model in ONNX format and generates
+RISC-V assembly code for the controller. Compared to other low
+precision accelerators, we provide a programmable solution which
+makes BARVINN more flexible. With the programmable MVUs, the
+user can run different models regardless of their size. BARVINN
+allows trading off throughput and latency by running DNN lay-
+ers either in distributed or in pipeline modes. Unlike other low
+precision accelerators, our proposed solution offers implementing
+various trade-offs through software and the end user can control
+them for each individual layer without FPGA reconfiguration at
+run time. Compared to programmable accelerators, BARVINN was
+shown to provide a better throughput per Watt performance.
+ACKNOWLEDGMENTS
+The authors acknowledge support for this project from the IBM
+AI Horizons Network, CMC Microsystems, Fonds de Recherche du
+Quebec–Nature et Technologies (FRQNT), MITACS and from the
+NSERC COHESA Strategic Research Network.
+REFERENCES
+[1] 2021. FINN Dataflow Accelerator Examples. (2021). https://github.com/Xilinx/
+finn-examples
+[2] 2021. ONNX Model Zoo. (2021). https://github.com/onnx/models
+[3] 2021. Open Neural Network Exchange. (2021). https://onnx.ai/
+[4] MohammadHossein AskariHemmat, Olexa Bilaniuk, Sean Wagner, Yvon Savaria,
+and Jean-Pierre David. 2021. RISC-V Barrel Processor for Deep Neural Network
+Acceleration. In 2021 IEEE International Symposium on Circuits and Systems
+(ISCAS). 1–5. https://doi.org/10.1109/ISCAS51556.2021.9401617
+[5] Olexa Bilaniuk, Sean Wagner, Yvon Savaria, and Jean-Pierre David. 2019. Bit-
+Slicing FPGA Accelerator for Quantized Neural Networks. In 2019 IEEE Interna-
+tional Symposium on Circuits and Systems (ISCAS). 1–5.
+[6] Michaela Blott, Thomas B. Preußer, Nicholas J. Fraser, Giulio Gambardella, Ken-
+neth O’brien, Yaman Umuroglu, Miriam Leeser, and Kees Vissers. 2018. FINN-R:
+An End-to-End Deep-Learning Framework for Fast Exploration of Quantized
+Neural Networks. ACM Trans. Reconfigurable Technol. Syst. 11, 3, Article 16 (dec
+2018), 23 pages. https://doi.org/10.1145/3242897
+[7] Adrian Bulat and Georgios Tzimiropoulos. 2021. Bit-Mixer: Mixed-Precision
+Networks With Runtime Bit-Width Selection. In Proceedings of the IEEE/CVF
+International Conference on Computer Vision (ICCV). 5188–5197.
+[8] Meghan Cowan, Thierry Moreau, Tianqi Chen, James Bornholt, and Luis Ceze.
+2020. Automatic Generation of High-Performance Quantized Machine Learning
+Kernels. Association for Computing Machinery, New York, NY, USA, 305–316.
+https://doi.org/10.1145/3368826.3377912
+[9] Steven K. Esser, Jeffrey L. McKinstry, Deepika Bablani, Rathinakumar Ap-
+puswamy, and Dharmendra S. Modha. 2020. Learned Step Size Quantization. In
+International Conference on Learning Representations.
+[10] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. 2016. Deep Residual
+Learning for Image Recognition. In Proceedings of the IEEE Conference on Computer
+Vision and Pattern Recognition (CVPR).
+[11] Mark Horowitz. 2014. Computing’s Energy Problem (and what we can do about
+it). Interational Solid State Circuits Conference (2014).
+[12] Itay Hubara et al. 2018. Quantized Neural Networks: Training Neural Networks
+with Low Precision Weights and Activations. JMLR 18, 187 (2018), 1–30.
+[13] B. Ham J. Lee, D. Kim. 2021. Network Quantization with Element-wise Gradient
+Scaling. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern
+Recognition.
+[14] Guilin Li et al. 2020. Residual Distillation: Towards Portable Deep Neural Net-
+works without Shortcuts. In Advances in Neural Information Processing Systems,
+Vol. 33. 8935–8946.
+[15] Nicolas Limare. 2014. Floating-Point Math Speed vs Precision. (2014).
+http:
+//nicolas.limare.net/pro/notes/2014/12/16_math_speed/
+[16] Paulius Micikevicius et al. 2017.
+Mixed precision training.
+arXiv preprint
+arXiv:1710.03740 (2017).
+[17] Sungju Ryu, Hyungjun Kim, Wooseok Yi, and Jae-Joon Kim. 2019. Bitblade: Area
+and energy-efficient precision-scalable neural network accelerator with bitwise
+summation. In Proceedings of the 56th Annual Design Automation Conference 2019.
+1–6.
+[18] Sayeh Sharify, Alberto Delmas Lascorz, Kevin Siu, Patrick Judd, and Andreas
+Moshovos. 2018. Loom: Exploiting Weight and Activation Precisions to Accelerate
+Convolutional Neural Networks. In 2018 55th ACM/ESDA/IEEE Design Automation
+Conference (DAC). 1–6. https://doi.org/10.1109/DAC.2018.8465915
+[19] Hardik Sharma et al. 2018. Bit fusion: Bit-level dynamically composable archi-
+tecture for accelerating deep neural network. In 2018 ACM/IEEE 45th Annual
+International Symposium on Computer Architecture (ISCA). IEEE, 764–775.
+[20] Mengshu Sun et al. 2022. FILM-QNN: Efficient FPGA Acceleration of Deep Neural
+Networks with Intra-Layer, Mixed-Precision Quantization. In Proceedings of the
+2022 ACM/SIGDA International Symposium on Field-Programmable Gate Arrays.
+134–145.
+[21] Stefan Uhlich et al. 2020.
+Mixed Precision DNNs: All you need is a good
+parametrization. In 8th International Conference on Learning Representations,
+ICLR 2020, Addis Ababa, Ethiopia, April 26-30, 2020.
+[22] Yaman Umuroglu et al. 2017. FINN: A framework for fast, scalable binarized
+neural network inference. In Proceedings of the 2017 ACM/SIGDA International
+Symposium on Field-Programmable Gate Arrays. ACM, 65–74.
+[23] Kuan Wang, Zhijian Liu, Yujun Lin, Ji Lin, and Song Han. 2019. HAQ: Hardware-
+Aware Automated Quantization With Mixed Precision. In 2019 IEEE/CVF Con-
+ference on Computer Vision and Pattern Recognition (CVPR). 8604–8612. https:
+//doi.org/10.1109/CVPR.2019.00881
+[24] Haichao Yu, Haoxiang Li, Humphrey Shi, Thomas S. Huang, and Gang Hua.
+2021. Any-Precision Deep Neural Networks. In Thirty-Fifth AAAI Conference on
+Artificial Intelligence. AAAI Press, 10763–10771. https://ojs.aaai.org/index.php/
+AAAI/article/view/17286
+[25] Xiaofan Zhang et al. 2018. DNNBuilder: an Automated Tool for Building High-
+Performance DNN Hardware Accelerators for FPGAs. In (ICCAD).
+

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+ 
+ 
+1 
+Abstract— Objectives: Identifying suicidality including suicidal 
+ideation, attempts, and risk factors in electronic health record data 
+in clinical notes is difficult. A major difficulty is the lack of training 
+samples given the small number of true positive instances among 
+the increasingly large number of patients being screened.  This 
+paper describes a novel methodology that identifies suicidality in 
+clinical notes by addressing this data sparsity issue through zero-
+shot learning. Materials and Methods: U.S. Veterans Affairs 
+clinical notes served as data. The training dataset label was 
+determined using diagnostic codes of suicide attempt and self-
+harm. A base string associated with the target label of suicidality 
+was used to provide auxiliary information by narrowing the 
+positive training cases to those containing the base string. A deep 
+neural network was trained by mapping the training documents’ 
+contents to a semantic space.  For comparison, we trained another 
+deep neural network using the identical training dataset labels and 
+bag-of-words features. Results: The zero shot learning model 
+outperformed the baseline model in terms of AUC, sensitivity, 
+specificity, and positive predictive value at multiple probability 
+thresholds. In applying a 0.90 probability threshold, the 
+methodology identified notes not associated with a relevant ICD-
+10-CM code that documented suicidality, with 94% accuracy. 
+Conclusion: This new method can effectively identify suicidality 
+without requiring manual annotation.   
+ 
+Keywords— Suicide, Clinical Notes, NLP, Zero-Shot Learning 
+I. INTRODUCTION 
+uicide is a significant problem in the United States, 
+increasing 35.2% from 1999 to 2018, and from 10.5 to 14.2 
+suicides per every 100,000 individuals in that same time period 
+[1] In 2020, 45,979 people died from suicide, and 
+approximately 1.2 million attempted suicide in the United 
+States [2] Its estimated cost is over $70 billion annually in lost 
+productivity and medical care [3]; this calculation does not 
+include residual costs from the estimated 4-17 people closely 
+tied to the suicide decedent who are left bereaved [4]. Suicide, 
+however, is a complicated problem that includes a dynamic web 
+of individual-level risk factors (e.g., depression, substance use 
+behaviors, personality traits), interpersonal risk factors (e.g., 
+violence, victimization), and community-level factors (e.g., 
+unemployment, stigmatization of mental illness) [5, 6].   
+ 
+1Biomedical Informatics Center; The George Washington University; 
+Washington DC, USA; 
+2VA Medical Center, Washington, DC, USA; 
+3Department of Emergency Medicine, Yale School of Medicine, New Haven, 
+CT, USA; 4PRIME Center, VA Connecticut Healthcare System, West Haven, 
+CT, USA; 5Research, VA Connecticut Healthcare System, West Haven, CT, 
+Veterans are especially affected by suicide, with an age- and 
+sex-adjusted rate that is 1.5 times higher than nonveterans [7]. 
+The Department of Veterans Affairs (VA) operates the single 
+largest integrated health care system in the U.S., and has 
+devoted resources to suicide prevention, including the Suicide 
+Prevention Applications Network (SPAN), embedding suicide 
+prevention coordinators and special reporting measures in 
+facilities [8], increased mental health staffing, partnerships with 
+community care organizations, and enhanced surveillance and 
+monitoring through its electronic health record (EHR) system 
+[9, 10]. Additionally, the VA has continual efforts to develop 
+predictive analytics to identify patients at the highest risk of 
+suicide [8, 11] The data elements for these predictive analytic 
+algorithms rely on structured data (e.g., International 
+Classification of Disease [ICD] diagnosis codes, prescription 
+data, socio-demographic data, care utilization metrics) [12] 
+which often provide an incomplete record [13, 14]. Less is 
+known about how unstructured data, such as contained in 
+clinical notes, can contribute to suicidality (i.e., suicidal 
+ideation or attempt) identification and prevention. Given that a 
+suicide attempt is one of the greatest risk factors for subsequent 
+suicide death, a more thorough means of detecting such events 
+is warranted [15].  
+A. Background and Significance 
+Natural language processing (NLP) combined with machine 
+learning may add value to suicide documentation research.  
+Supervised machine learning methods use “supervised”, or pre-
+classified data.  However, naïve attempts at note retrieval using 
+keyword search alone quickly demonstrate the difficulty of this 
+problem, as words such as “suicide” occur in standard 
+questionnaires which are included in many notes, with few 
+actually documenting suicidality. For instance, in a prior 
+experiment we carried out, we randomly collected 1,000 VA 
+notes containing the term “suicidal” or “suicide” from 1,000 
+individual patients and performed manual chart review for 
+affirmed suicidality. Only 1.57% of these notes documented 
+actual suicidality. Patient reluctance to disclose suicidal 
+ideation provides a further complicating factor [16, 17]. As a 
+result, a patient’s negative response to a suicide ideation inquiry 
+may not reflect their real feelings or intentions.  Additionally, 
+USA; 6VA Connecticut Healthcare System, West Haven, CT, USA; 7Suzanne 
+Dworak-Peck School of Social Work, University of Southern California, Los 
+Angeles, CA, USA; 8Department of Internal Medicine, Yale School of 
+Medicine, West Haven, CT; 
+ 
+Leveraging Contextual Relatedness to Identify Suicide 
+Documentation in Clinical Notes through Zero Shot 
+Learning 
+T. Elizabeth Workman, Ph.D.1,2, Joseph L. Goulet, Ph.D.3,6, Cynthia A. Brandt, M.D.3,6, Allison R. 
+Warren, Ph.D.4, Jacob Eleazer, Ph.D.4, Melissa Skanderson, M.S.W.5, Luke Lindemann, Ph.D.6, John 
+R. Blosnich, Ph.D.7, John O’Leary, M.Ed.6,8, Qing Zeng-Treitler, Ph.D.1,2  
+S 
+
+ 
+ 
+2 
+relying on structured data alone will result in incomplete 
+identification of patients who have or are experiencing 
+suicidality, because relevant coding is prone to underuse [8].  
+However, not all clinical notes associated with relevant 
+structured data document suicidality.  For example, a note 
+documenting a secondary service such as group therapy, or a 
+note documenting fluid intake may not directly document 
+suicidality.  
+Prior attempts to apply NLP and machine learning are often 
+limited to mental health-oriented notes and may suffer if using 
+imbalanced data. Levis et al.[18] applied sentiment analysis and 
+various machine learning algorithms to classify suicide, using 
+VA psychotherapy notes, yielding area under the curve (AUC) 
+ratings comparable to chance. Fernandes et al.[19] obtained 
+excellent NLP performance in their study of clinical notes from 
+the Clinical Record Initiative Search (CRIS), but performance 
+was computed after removing neutral (non-suicide) results from 
+their machine learning output. Carson et al. enriched notes 
+associated with suicide attempt that were then used to train a 
+random forest model achieving 83% sensitivity, but only 22% 
+specificity [20].  Cook et al. [21] applied a bag-of-words 
+approach with machine learning to identify suicide ideation and 
+psychiatric symptoms using notes for patients identified as 
+having performed self-harm, achieving 61% PPV (positive 
+predictive value), 59% sensitivity, and 60% specificity, with 
+results varying  depending on the task. Zhang et al. sought to 
+identify psychological stressors using a pre-annotated dataset of 
+psychiatric evaluation records from the CEGS N-GRID 2016 
+challenge [22] as a gold standard, for a conditional random 
+fields machine learning model, [23] yielding final F scores of 
+73.91% and 89.01%, respectively, on exact and inexact stressor 
+matching, and 97.73% and 100% respectively, for exact and 
+inexact suicide recognition on instances of the positive 
+keywords with the stressors; however, their evaluation methods 
+for this are not detailed.   
+Zhong et al. applied structured data and NLP to identify 
+suicidal behavior in pregnant women, achieving PPV of 76% 
+and 30%, for women identified through relevant diagnostic 
+codes and through NLP for women not receiving a relevant 
+diagnostic code, respectively [24].  Obeid et al.[25] trained a 
+convolutional neural network that achieved an AUC of 0.882 
+and an F1 score of 0.769 in predicting relevant suicide ICD 
+codes in subsequent years.  Using notes from psychiatric 
+encounters, Cusick et al. [26] developed a rule-based NLP tool 
+to identify positive instances of suicide-oriented keywords that 
+leveraged NegEx. [27] They also developed different weakly-
+supervised machine learning models. A convolutional neural 
+network receiving Word2Vec [28] word embeddings as input 
+achieved precision, recall, F1 score, and AUC values of 0.81, 
+0.83, 0.82, and 0.946.  In a subsequent evaluation the 
+convolutional neural network correctly classified 87% of the 23 
+notes (of 5000 clinical notes) receiving a positive classification, 
+from notes for patients diagnosed with depression or prescribed 
+an antidepressant.  In a related task Tsui et al. [29] used prior 
+structured and unstructured data (clinical notes from history, 
+physical examination, progress notes and discharge summaries) 
+of inpatient and emergency room patients with a coded suicide 
+attempt, to identify first-time suicide attempts in a case-control 
+study.  An ensemble of extreme gradient boosting (EXGB) 
+yielded best performance, with an AUC ranging from 91.9% to 
+93.2%, according to time window between prior data and 
+suicide attempt diagnosis.  Recently, Rozova et al. obtained 
+promising results (87% AUC) using a gradient boosting model, 
+although the study was limited to emergency room triage notes 
+[30]. 
+Seeking suicidality in all types of clinical notes, among all 
+types of patients, or when hampered by imbalanced data, is 
+indeed a complex task. Some of the methods in the papers cited 
+above tend to suffer from low precision, specificity, and 
+possibly also low sensitivity (recall).  Identifying probability 
+thresholds addresses these problems, providing flexibility for a 
+given task. For example, a high probability threshold (e.g., the 
+top ten percent) can serve as a means for identifying 
+documentation indicating suicidality and its risk with high 
+precision. When the prevalence is very low, which is often the 
+case of true positive suicidality documentation, the optimal 
+threshold needs to balance metrics such as the true positive rate 
+(sensitivity, also known as recall), specificity, and the positive 
+predictive value (precision).  A strategic implementation of a 
+technique like Zero-Shot Learning may also provide accurate 
+identification of suicidality in clinical notes. 
+B.   Zero-Shot Learning 
+Zero-Shot Learning (ZSL) enables predictions on unseen 
+data using a model trained on data that has labels that are 
+different than those of the unseen data [31, 32].  It largely 
+operates by mapping select properties of the data (i.e., the 
+“feature space”) to a semantic representation (i.e., the “semantic 
+space”) that enables prediction of unseen classes [33]. In other 
+words, auxiliary information must be provided on the labels of 
+the unseen classes to make it possible for a trained model to 
+recognize them in the testing data.   
+ ZSL has been applied in several computer vision tasks [34, 
+35], as well as NLP tasks [36].  Accordingly, a feature space 
+can consist of data derived from images [37] or text [36].  The 
+semantic representation can be based on several different 
+approaches, including data attributes, semantic word vectors as 
+those provided by skip-gram or continuous-bag-of-word  
+architectures [33] or BERT output [38], or knowledge graphs 
+[33].  Examples in NLP applications include semantic utterance 
+classification [39] multilingual translation [40] and emotion 
+detection [41]. However, other than Sivarajkumar and Wang’s 
+work [38] there is little ZSL research in unstructured clinical 
+text data.  
+Naturally, different semantic representations affect the 
+accuracy of ZSL [42]. In this study, we leveraged word 
+embedding and usage context.    
+C. Objectives 
+We investigated a ZSL methodology applied to a binary 
+suicidality classification task. The training dataset was 
+constructed using diagnostic codes (ICD-10-CM codes) related 
+to suicide. Our target label is the broader concept of suicidality. 
+To enable ZSL, a base string representing suicidality was 
+
+ 
+ 
+3 
+selected. We then built the semantic space by identifying key 
+features associated with suicidality in the training dataset. A 
+DNN model was developed using the training data and tested 
+on two different sets of unseen data with the unseen label of 
+suicidality. Specifically, we sought to answer: 
+ Will ZSL effectively identify suicidality documentation 
+from among all types of clinical notes, using review by 
+clinicians as the reference standard? 
+ Will ZSL effectively identify suicidality or suicide risk 
+documentation from among clinical notes not associated with 
+a relevant ICD-10-CM code, by probability threshold, in terms 
+of precision, using the same reference standard? 
+We are unaware of previous descriptions of this methodology 
+and to our knowledge it has not been used prior to this study.  
+II. METHODS 
+A. Training Data 
+A training dataset was created using two corpora. The first 
+corpus consisted of 50,000 randomly selected VA clinical notes 
+from outpatient encounters recorded between 2016 and 2019 
+which contained the base string “suicid” (e.g. “suicide”, 
+“suicidal” ) and were associated with at least one ICD-CM-10 
+code identified by the National Health Statistics Report from 
+the Centers for Disease Control and Prevention (CDC) 
+indicating suicide attempt or intentional self-harm.[43]  This 
+corpus is referred to as stringAndDx (9170 unique patients). 
+The second corpus consisted of 50,000 randomly selected VA 
+clinical notes from outpatient encounters recorded between 
+2016 and 2019 that were associated with other ICD-CM-10 
+codes that were irrelevant to suicidality or self-harm. These 
+notes were extracted from patients matching the stringAndDx 
+patients in age (at the time of document retrieval), race, and 
+ethnicity.  This second corpus is referred to as noDx (8638 
+unique patients).  Each corpus was preprocessed by 
+transforming all letters to lower case, removing basic 
+formatting markup and punctuation, separating character 
+strings into tokens (words), separating relevant concatenated 
+tokens (e.g., “suicidalhomicidal” to “suicidal” ”homicidal”), 
+and removing all tokens that did not entirely consist of letters.   
+B.  Semantic Space Feature Extraction and Mapping 
+The task to build the semantic space was carried out in three 
+steps: First, we identified a list of features that are potentially 
+relevant for the positive training label. Second, we created word 
+embeddings using a skip-gram architecture. Third, we 
+identified context words of the selected features using the word 
+embeddings.  In a fourth step, a contextual weight is assigned 
+to each feature for each document in mapping the semantic 
+space to the feature space.  
+In the first step, inverse document frequency (TF-IDF) 
+analysis was used to identify the n most important terms in each 
+corpus.  For this investigation, n = 1000.  TF-IDF evaluates 
+term frequency using the count of documents containing a given 
+term.  In each document, the relative frequency of each term is 
+weighted by the log of the number of documents in the corpus 
+divided by the number of documents containing the term, as 
+shown in (1) 
+𝑡𝑖,𝑗 = 𝑡𝑓𝑖,𝑗 ∗ 𝑙𝑜𝑔( 𝑛
+𝑑𝑓𝑖
+) 
+ 
+where ti,j is term i in document j, tfi,j is the relative frequency of 
+term i in document j, n is the total number of documents, and 
+dfi is the number of documents containing term i.  Because TF-
+IDF is a document-based measurement, we used the mean TF-
+IDF value for each term in its respective corpus.   The words 
+with the top TF-IDF scores that are unique to the stringAndDx 
+corpus were treated as features.  Figure 1 illustrates this process.  
+Each circle represents terms from one of the corpora. Sets a and 
+b are the words with the top n TFIDF scores for stringAndDx 
+and noDx, respectively. Set c is the overlap between a and b. 
+The feature set F contains words that are in set a, but not in the 
+overlap set c or in set b (f  a and f  c and f  b).  
+ 
+ 
+Figure 1. Feature identification. Words that are deemed as features are in set a, 
+excluding words in c and b. 
+ 
+In the second step, we created a Word2Vec model using the 
+stringAndDx corpus. In this study, the model was a shallow 
+neural network with the hidden layer containing 300 nodes, 
+applying the skip-gram architecture, with an analytic window 
+size of 5, trained through 10 iterations. 
+In the third step, we identified the top m context words for 
+each feature word using the word embeddings from the 
+Word2Vec model. The m words most similar to each feature 
+word, according to cosine similarity values, served as its 
+context words. In this investigation, m = 50. 
+In the fourth step, we map the feature space, i.e. a document’s 
+preprocessed content, to the semantic space. A weight v is 
+assigned to each feature word for each document, based on its 
+occurrence with its context words in a window in the 
+document’s text.  This weight is the summed total of the cosine 
+similarity between the feature and a co-occurring context word 
+multiplied by the mean TF-IDF value of the feature word. The 
+formula is shown in (2) 
+ 
+𝑣 =
+∑
+𝑐𝑜𝑠𝑆𝑖𝑚(𝑥, 𝑦) ∗ 𝑡𝑓𝑖𝑑𝑓(𝑥)
+𝑥∈𝐹,𝑦∈𝐷
+ 
+ 
+where x is a feature in F, the set of features in the semantic 
+space, and y is a context word of set D, the context words for x 
+in the semantic space, which occurs in a five-word window 
+around x in the document’s text. This process is illustrated in 
+Figure 2, where “pattern” (highlighted in light gray) is a feature 
+word, and “internalizing” and “fitful” (highlighted in dark gray) 
+are among its set of context words and appear in a five-word 
+window. 
+(2) 
+(1) 
+
+stringAndDx
+noDx
+a
+c
+6 
+ 
+4 
+ 
+ 
+Figure 2.  Example of deriving a feature weight using (2) 
+ 
+If a feature word is not in the text, its value is zero for the 
+given document. 
+C. Model Development  
+20,000 documents were randomly selected from each corpus 
+(stringAndDx and noDx). We trained a DNN model (here 
+referred to as the ZSL DNN) consisting of five fully-connected 
+hidden layers of alternating sizes of 30 or 70 nodes, with each 
+layer implementing a dropout rate of 0.5.  We implemented the 
+Adam optimizer [44], with a learning rate of 0.0012, beta 1 
+value of 0.92, beta 2 value of 0.9992, and an epsilon value of 
+1e-08, with binary cross entropy as the loss function, and the 
+sigmoid function in the output layer, since it was a binary 
+classification task.  The architecture and hyperparameters were 
+chosen on empirical grounds, after experimentation.  Each 
+document from the stringAndDx corpus was classified as “1” (a 
+generic positive instance), and each document from the noDx 
+corpus was classified as “0” (a generic negative instance).  
+These labels do not indicate whether or not the given document 
+directly pertains, or not pertains, to suicidality or its risks, but 
+an association with a structured data element, and for those 
+labeled “1”, also containing a base string.  Balancing the 
+positive and negative approximated training datasets in this 
+manner (i.e., providing balanced training examples) addressed 
+the problematic issue of otherwise training a model with few 
+positive and many negative instances. We implemented a 60% 
+training, 20% validation, and 20% testing split in developing 
+the ZSL DNN.  Figure 3 illustrates the method. 
+ 
+   
+Figure 3. Method. The corpora stringAndDx (2016-19), noDx (2016-19), 
+testSet1 (2020), and testSet2 (2020) are unique and extracted from all clinical 
+notes based on associated ICD-10-CM codes, and in the case of 
+stringAndDx, where a base string is also present; corpora content is 
+preprocessed.  The stringAndDx and noDx corpora are used in the TF-IDF 
+analysis to identify feature words that are unique to stringAndDx (step 1). 
+stringAndDx is applied to a skip-gram model to produce word embeddings 
+(step 2). Feature words and their significant context words (determined 
+through the word embeddings) form the semantic space (step 3). The 
+contents of stringAndDx and noDx are mapped to the semantic space, using 
+a function to determine feature word weights (step 4). The mapped contents 
+of stringAndDx and noDx documents are used to train the ZSL DNN, using 
+generic labels 1 and 0, respectively.    The mapped contents of unseen 
+testSet1 and testSet2 notes were classified by the trained ZSL DNN, for the 
+classes (a) containing suicidality documentation, or (b) not containing 
+suicidality documentation.  Human annotation independently classified 
+random documents from testSet1 and testSet2 for the same classes (a) 
+containing suicidality documentation, or (b) not containing suicidality 
+documentation; human annotation also assessed documents from testSet2 
+containing the base string that received a probability of 0.90 or greater, for 
+these classes and suicidality risk factors. 
+ 
+D. Evaluation 
+The authors randomly retrieved 5,000 different clinical notes 
+recorded in 2020 that were associated with at least one of the 
+relevant IDC-10-CM codes. This corpus is subsequently 
+labeled as testSet1. The authors also randomly retrieved 5,000 
+different clinical notes recorded in 2020 that were associated 
+with other IDC-10-CM codes irrelevant to suicidality or self-
+harm.  This corpus is subsequently labeled testSet2.   
+The contents of each of the notes in testSet1 and testSet2 were 
+mapped to the semantic space, i.e., deriving a weight for each 
+feature word as described earlier in the fourth step.  Then, the 
+trained ZSL DNN was used to classify the notes in testSet1 and 
+testSet2 as (a) containing suicidality documentation, or (b) not 
+containing suicidality documentation.   
+In joint sessions, two clinical psychologists familiar with VA 
+clinical note documentation together identified suicidality (i.e., 
+current or past suicide ideation or attempt) in 200 notes 
+randomly selected from testSet1 and testSet2 (100 from each 
+test set), after being instructed to look for documentation for 
+these specific events. They addressed differences of opinion 
+through discussion and mutual consensus during the joint 
+sessions.  In a second evaluation, to explore how the 
+application’s output may serve to identify patients who had 
+experienced or were at risk for suicidality, but never formally 
+diagnosed as such, the clinicians examined the testSet2 notes 
+containing the base string “suicid" that received a probability 
+value of 0.90 or greater from the trained ZSL DNN, for 
+documentation of suicidality and/or its risk factors, according 
+to NIH guidelines.[45] This threshold was chosen in order to 
+explore how high-probability documents (i.e. the top 10% in 
+terms of probability) would be representative in identifying 
+documented suicidality or its risk factors with high precision, 
+thus addressing our second question.   
+1) Baseline Comparison 
+For comparative purposes, the 163 most frequent bigrams 
+unique to the stringAndDx corpus were identified and used in a 
+bag-of-words baseline model.  We trained a DNN (here referred 
+to as the Baseline DNN) using these 163 bigrams as features for 
+the 20,000 stringAndDx documents and the 20,000 noDx 
+documents. This baseline DNN was also used to classify the 
+Document Text: “The patient has a pattern of internalizing 
+criticism from his family.  This pattern sometimes results in fitful 
+outbursts.” 
+ 
+TF-IDF value of feature word “pattern”: 0.0062 
+Cosine similarity of “pattern” and “internalizing”: 0.4673 
+Cosine similarity of “pattern” and “fitful”: 0.3824 
+Feature weight for “pattern”:  
+(0.0062 * 0.4673) + (0.0062 * 0.3824) = 0.0053 
+
+All Clinical
+Notes
+ICDcodes
+ICDcodes
+Base String
+Human
+testSetl
+testSet2
+stringAndDx
+noDx
+Annotation
+(suicidality)
+TFIDF
+Analysis
+Word
+Embeddings
+Semantic
+Map content
+Space
+Mapcontent
+Testing
+DNN
+Training
+Output is a classification of (a) containing suicidality
+documentation or (b) not containing suicidality documentation 
+ 
+5 
+notes in testSet1 and testSet2, for (a) containing suicidality 
+documentation, or (b) not containing suicidality documentation, 
+using the 163 most frequent bigrams as features. 
+III. RESULTS 
+The first step of the new method (described in Methods) 
+identified 163 feature words associated with suicidality 
+diagnosis.  The top thirty feature words are listed in Table I. No 
+form of the base string “suicid” was found among the 163 final 
+feature words. Both “suicide” and “suicidal” were prominent 
+terms in both the noDx and stringAndDx corpora, along with 
+terms like “psychiatrist” and “psychosocial”; this is likely due 
+to the proliferation of objects like questionnaires, and mental 
+health care documentation in notes that are unrelated to 
+suicidality. 
+TABLE I 
+TOP 30 FEATURE WORDS 
+flag 
+overdose 
+coordinator 
+took 
+spc 
+observation 
+called 
+warning 
+pills 
+prf 
+unknown 
+interrupted 
+gun 
+placement 
+lcsw 
+lethal 
+outcome 
+reportedly 
+notified 
+sdv 
+occurred 
+police 
+protocol 
+od 
+supports 
+seeking 
+category 
+preparatory 
+cut 
+determined 
+ 
+A. ZSL DNN and Baseline DNN Performance 
+The classifications by the clinicians and the probabilities 
+assigned by the ZSL DNN and the Baseline DNN were first 
+assessed by AUC score. The results are in Table II and Figure 
+4. 
+ 
+TABLE II 
+AUC PERFORMANCE 
+ZSL DNN 
+Baseline DNN 
+0.946 
+0.47 
+ 
+ 
+Figure 4. ZSL DNN AUC results (left), Baseline DNN AUC results (right) 
+ 
+In terms of AUC, the ZSL DNN trained through mapping the 
+semantic space to the feature space outperformed the Baseline 
+DNN trained with the bigram bag-of-words features.  
+The sensitivity, specificity, and PPV results at 0.15, 0.5, and 
+0.85 probability thresholds for each DNN are in Tables III-V.  
+Probability refers to the probability the DNN assigned to each 
+note for positive suicidality documentation. We applied the 
+median probability (0.1499, rounded) assigned by the ZSL 
+DNN to the testSet2 documents (the test set containing random 
+notes associated with irrelevant ICD-10-CM codes) in forming 
+minimum and maximum thresholds; 0.5 is a standard midpoint 
+probability threshold. The combined scores in these tables were 
+computed with all true positives, true negatives, false positives, 
+and false negatives for both test sets, for the indicated metrics.  
+Values of NaN (not a number) occurred where there were no 
+true positives or false positives. 
+ 
+TABLE III 
+EVALUATION RESULTS AT 0.15 PROBABILITY THRESHOLD 
+ZSL DNN 
+Sensitivity/Recall  
+Specificity 
+Precision/PPV 
+testSet1 
+97% 
+100% 
+91% 
+testSet2 
+100% 
+64% 
+05% 
+Combined 
+97% 
+59% 
+67% 
+Baseline DNN 
+ 
+ 
+ 
+testSet1 
+99% 
+0% 
+90% 
+testSet2 
+50% 
+09% 
+01% 
+Combined 
+98% 
+08% 
+48% 
+ 
+TABLE IV 
+EVALUATION RESULTS AT 0.5 PROBABILITY THRESHOLD 
+ZSL DNN 
+Sensitivity/Recall  
+Specificity 
+Precision/PPV 
+testSet1 
+92% 
+40% 
+93% 
+testSet2 
+50% 
+97% 
+25% 
+Combined 
+91% 
+92% 
+90% 
+Baseline DNN 
+ 
+ 
+ 
+testSet1 
+92% 
+0% 
+89% 
+testSet2 
+50% 
+10% 
+1% 
+Combined 
+91% 
+9% 
+46% 
+ 
+TABLE V 
+EVALUATION RESULTS AT 0.85 PROBABILITY THRESHOLD 
+ZSL DNN 
+Sensitivity/Recall  
+Specificity 
+Precision/PPV 
+testSet1 
+77% 
+70% 
+96% 
+testSet2 
+50% 
+100% 
+100% 
+Combined 
+76% 
+97% 
+96% 
+Baseline DNN 
+ 
+ 
+ 
+testSet1 
+0% 
+100% 
+NaN/div by 0 
+testSet2 
+0% 
+100% 
+NaN/div by 0 
+Combined 
+0% 
+100% 
+NaN/div by 0 
+ 
+The ZSL DNN outperformed the Baseline DNN in most 
+metrics at all probability thresholds.   
+B. Second Evaluation 
+To explore how this new methodology can identify clinical 
+notes documenting suicidality that are not associated with a 
+relevant ICD-10-CM code with high precision, the clinicians 
+also reviewed the 16 notes from testSet2 containing the base 
+string “suicid’ that received a probability at or above 0.90 from 
+the trained ZSL DNN. The clinicians noted suicide ideation or 
+attempt, and the presence of the following suicide risk factors, 
+based on National Institute of Mental Health guidelines [45]: 
+ Depression and other mental health disorders 
+ Substance abuse disorder 
+ Family history of a mental health or substance abuse 
+disorder 
+ Family history of suicide 
+ Family violence, including physical or sexual abuse 
+ Having guns or other firearms in the home 
+ Being in prison or jail 
+ Being exposed to others’ suicidal behavior 
+Of these 16 clinical notes (associated with 16 different 
+patients), 7 documented current or past suicide ideation or 
+attempt.  Eight of the remaining notes included one or more 
+risk factors for suicide (nearly all included multiple risk 
+factors). In all, 15 of the 16 notes contained documentation of 
+current or past suicide ideation or attempt, and/or suicide risk 
+ 
+ 
+
+1.0
+model results
+0.0
+0.2
+0.4
+0.8
+1.Dno distinction
+model results
+0.8-
+2 0.4
+0.2 
+0.0 
+0.0
+Q2
+0.4
+False Positive Rate
+0.8
+10  
+ 
+6 
+factors, for patients who had never received a suicidality ICD-
+10-CM code diagnosis during the study period, achieving a 
+PPV of 93.8%. 
+IV. DISCUSSION 
+Regarding the study’s original questions, our ZSL approach 
+effectively identified suicidality in all types of clinical notes, 
+surpassing the performance of the bag-of-words baseline in 
+conjunction with deep learning. It also effectively identified 
+suicidality or suicide risk documentation from among clinical 
+notes not associated with a relevant ICD-10-CM code with high 
+precision, on probability threshold. 
+A. Semantic Space 
+In this work, the semantic space development is framed as  
+feature extraction where mapping is enhanced by attaching 
+weights to features found in the data, an approach also used in 
+computer vision ZSL [46]. The semantic space captures natural 
+data properties by identifying salient terms and relevant 
+contextual 
+terms 
+in 
+collective 
+clinical 
+suicidality 
+documentation (i.e., a corpus of notes associated with relevant 
+ICD codes). Table 1 lists 30 prominent feature words associated 
+with collective suicidality documentation after removing terms 
+associated with other kinds of documents. There is an intuitive 
+sense to these words; “flag” is found in the phrase “high risk for 
+suicide flag”; “overdose” and “cut” refer to suicide methods; 
+“pills” and “gun” refer to suicide instruments. Identifying terms 
+contextually similar to these provides patterns in relevant 
+documentation.  Again, this has an intuitive logic. The most 
+contextually similar terms to “flag” include “reactivate” and 
+“deactivate” (for a high suicide risk flag) and “high” (the level 
+of risk).  The most contextually similar terms to “pills” include 
+“handful”, “fistfuls”, and “bunch”, implying large quantities, 
+along with “overdosing” and “took”, the associated actions.  
+The feature word  “spc” indicates VA’s suicide prevention 
+coordinators, which is a structural change that VA implemented 
+for suicide prevention [10]. Concordantly, “police” and “lcsw” 
+(i.e., licensed clinical social worker) refer to other professions 
+highly associated with individuals at risk for suicide. For 
+example, police may be activated for a rescue, and a licensed 
+clinical social worker may be involved in treatment planning or 
+referral connections for suicidal individuals. The feature words 
+“prf” and “sdv” refer to “patient record flag” and “self-directed 
+violence”, respectively.  The semantic space provided an 
+efficient representation for effective mapping to the feature 
+space.   
+B. Data Retrieval and Model Training 
+Using associated structured data elements like ICD-10-CM 
+codes, and a base string provides a means to locate equally sized 
+corpora for training that could be generically labeled “0” or “1”.  
+These labels were primarily based on a structured data 
+association, since their individual unstructured content was 
+mostly unknown. This approach solves the issue of imbalanced 
+training data. The predominant clinical note types (Appendix) 
+also illustrate this.  Most of the frequent note types associated 
+with one of the relevant CDC ICD-10-CM codes and containing 
+the base string are relevant to suicidality.  Addendum is a 
+common note type [47] associated with many domains [48]. 
+The most frequent note types not associated with a relevant 
+code resemble frequencies of all note types in the VA [47].  
+C. Identifying Suicidality Documentation 
+To our knowledge, this method has not been applied in other 
+studies.  Unlike VA surveillance methods using structured data, 
+it also leverages information found in EHR notes. Also, unlike 
+other NLP methods [18, 20, 21, 23, 24, 26, 29, 30] it can be 
+applied to all patients and note types.  In other studies, a bag-
+of-words approach has been applied to suicidality identification 
+and other machine learning tasks [21, 49, 50].  However, the 
+results of this current study suggest that the complexity of 
+suicidality documentation demands a more targeted approach.   
+This method could complement existing measures like 
+SPAN, alerting suicide prevention coordinators of additional 
+patients at risk.  The results of the two clinical psychologists’ 
+evaluations demonstrate the method’s efficiency in identifying 
+suicidality documentation for documents where there is no 
+relevant ICD-10-CM code. The performance on both test sets 
+demonstrates the methodology’s effectiveness in classifying 
+notes that are mixed in terms of ICD-10-CM coding. 
+Tables III - V suggest that the probability threshold can be 
+adjusted to suit a specific task like finding suicidality and its 
+risk factors with high precision among notes not associated with 
+a relevant ICD-10-CM code. This is especially true considering 
+the small prevalence of suicidality documentation in clinical 
+notes.  The second evaluation (which yielded 93.8% PPV) 
+demonstrates this.  By applying a high probability threshold of 
+0.90 to all 5000 testSet2 documents and focusing on clinical 
+notes containing the base string, of the 16 documents (for 16 
+different patients), 94% contained suicidality and/or suicidality 
+risk factor documentation, based on clinician review. These 
+results exceed those of Cusick et al.’s [26] similar task, where 
+87% of notes were correctly classified, among notes for patients 
+diagnosed with depression or prescribed an antidepressant.  In 
+this current study’s second evaluation, none of the 16 patients 
+identified had ever received a suicide ICD-10-CM code during 
+the study’s time period.  It is impossible to know if the patients 
+in the 8 notes simply containing documented risk factors were 
+suicidal or not based solely on electronic health records. 
+Suicidal patients sometimes deny suicide ideation or attempt 
+[16, 51].  For example, in one note from the chart review 
+associated with a relevant ICD-10-CM code, the patient 
+reportedly denied suicide ideation, even after checking into the 
+hospital hours earlier for a self-reported suicide attempt. 
+D. Future Work 
+ This work is part of a larger study of patients at risk for 
+suicide.[52] The next step is to combine these findings with 
+prior work. We also plan an analysis of patients from first 
+suicide ideation or attempt documented in the VA system, to 
+understand their evolution of care. 
+E. Limitations 
+VHA data largely cover a population of older men. However, 
+the amount of women and younger patients is increasing, thus 
+
+ 
+ 
+7 
+also increasing the generalizability of these findings. The 
+corpora retrieval method we used to train the ZSL DNN is 
+dependent on clinicians’ use of the relevant ICD-10-CM codes 
+in documenting care, which may be prone to underuse [8]. 
+However, the results of this study indicate the method’s utility. 
+Due to environmental computational limitations, we randomly 
+selected 20,000 notes from the stringAndDx corpus, and 20,000 
+notes from the noDx corpus for training the ZSL DNN.  
+V. CONCLUSION 
+We developed a new methodology to identify suicidality in 
+clinical notes using zero-shot learning (ZSL). A trained ZSL 
+deep neural network (DNN) outperformed a DNN trained using 
+a baseline bag-of-words method in AUC scores and other 
+metrics assessed at various probability thresholds on unseen 
+data, according to expert review. This novel methodology 
+identifies suicidality and its risk factors with high precision, 
+when applying a 0.90 probability threshold, in VA clinical notes 
+not associated with a relevant ICD-10-CM code.  This 
+methodology 
+could 
+complement 
+existing 
+suicidality 
+identification measures. These findings hold promise for future 
+research. 
+APPENDIX 
+ 
+Most Frequent Note Types in Training Data by Corpus 
+stringAndDx 
+noDx 
+Note Type 
+Count 
+Note Type 
+Count 
+Addendum 
+2844 
+Addendum 
+5683 
+Suicide Behavior 
+and Report 
+843 
+Primary Care Secure 
+Messaging 
+291 
+Suicide Prevention 
+Telephone Note 
+811 
+Nursing Note 
+228 
+Suicide Behavior 
+and Overdose 
+Report 
+613 
+Administrative Note 
+207 
+Suicide Prevention 
+Note 
+452 
+State Prescription 
+Drug Monitoring 
+Program 
+110 
+Suicide Prevention 
+Safety Plan 
+448 
+Care Flow Sheet 
+88 
+Mental Health 
+Nursing 
+Assessment Note 
+374 
+Telephone Contact 
+75 
+Veterans Crisis 
+Line Note 
+222 
+Mental Health 
+Diagnostic Study 
+Note 
+71 
+Social Work Note 
+213 
+Non VA Care 
+Consult Result Note 
+69 
+Suicide Prevention 
+Contact 
+212 
+Operation Report 
+64 
+ 
+ACKNOWLEDGMENT 
+The views expressed are those of the authors and do not 
+necessarily reflect those of the Department of Veterans Affairs, 
+the United States Government, or the academic affiliate 
+institutions. This work was funded by Veterans Affairs Health 
+Services Research and Development Services grant IIR 18-035 
+Understanding Suicide Risks among LGBT Veterans in VA 
+Care, and NIH National Center for Advancing Translational 
+Sciences grant UL1TR001876. 
+REFERENCES 
+ 
+[1] 
+H. Hedegaard, S. C. Curtin, and M. Warner. Suicide mortality in the 
+United States, 1999–2019, 2021. 
+[2] 
+American Foundation for Prevention. "Suicide Statistics." American 
+Foundation for Suicide Prevention. https://afsp.org/suicide-statistics  
+[3] 
+G. Gonzales and C. Henning-Smith, "Disparities in health and disability 
+among older adults in same-sex cohabiting relationships," J Aging Health, 
+vol. 27, no. 3, pp. 432-53, Apr 2015, doi: 10.1177/0898264314551332. 
+[4] 
+A. L. Berman, "Estimating the population of survivors of suicide: seeking 
+an evidence base," Suicide Life Threat Behav, vol. 41, no. 1, pp. 110-6, 
+Feb 2011, doi: 10.1111/j.1943-278X.2010.00009.x. 
+[5] 
+S. Stack, "Suicide: a 15-year review of the sociological literature. Part I: 
+cultural and economic factors," Suicide Life Threat Behav, vol. 30, no. 2, 
+pp. 
+145-62, 
+Summer 
+2000. 
+[Online]. 
+Available: 
+https://www.ncbi.nlm.nih.gov/pubmed/10888055. 
+[6] 
+K. Hawton, I. C. C. Casanas, C. Haw, and K. Saunders, "Risk factors for 
+suicide in individuals with depression: a systematic review," J Affect 
+Disord, 
+vol. 
+147, 
+no. 
+1-3, 
+pp. 
+17-28, 
+May 
+2013, 
+doi: 
+10.1016/j.jad.2013.01.004. 
+[7] 
+"2019 National Veteran Suicide Prevention Annual Report," U.S. 
+Department of Veterans Affairs, Office of Mental Health and Suicide 
+Prevention, 
+September 
+2019 
+2019. 
+[Online]. 
+Available: 
+https://www.mentalhealth.va.gov/docs/data-
+sheets/2019/2019_National_Veteran_Suicide_Prevention_Annual_Repo
+rt_508.pdf 
+[8] 
+C. Hoffmire, B. Stephens, S. Morley, C. Thompson, J. Kemp, and R. M. 
+Bossarte, "VA Suicide Prevention Applications Network: A National 
+Health Care System-Based Suicide Event Tracking System," Public 
+Health Rep, vol. 131, no. 6, pp. 816-821, Nov 2016, doi: 
+10.1177/0033354916670133. 
+[9] 
+I. Katz, "Lessons learned from mental health enhancement and suicide 
+prevention activities in the Veterans Health Administration," Am J Public 
+Health, 
+vol. 
+102 
+Suppl 
+1, 
+pp. 
+S14-6, 
+Mar 
+2012, 
+doi: 
+10.2105/AJPH.2011.300582. 
+[10] D. Carroll, L. K. Kearney, and M. A. Miller, "Addressing Suicide in the 
+Veteran Population: Engaging a Public Health Approach," Front 
+Psychiatry, vol. 11, p. 569069, 2020, doi: 10.3389/fpsyt.2020.569069. 
+[11] J. F. McCarthy et al., "Predictive Modeling and Concentration of the Risk 
+of Suicide: Implications for Preventive Interventions in the US 
+Department of Veterans Affairs," Am J Public Health, vol. 105, no. 9, pp. 
+1935-42, Sep 2015, doi: 10.2105/AJPH.2015.302737. 
+
+ 
+ 
+8 
+[12] R. C. Kessler et al., "Developing a practical suicide risk prediction model 
+for targeting high-risk patients in the Veterans health Administration," Int 
+J Methods Psychiatr Res, vol. 26, no. 3, Sep 2017, doi: 10.1002/mpr.1575. 
+[13] J. Moss, M. Andison, and H. Sobko, "An analysis of narrative nursing 
+documentation in an otherwise structured intensive care clinical 
+information system," AMIA Annu Symp Proc, pp. 543-7, Oct 11 2007. 
+[Online]. Available: https://www.ncbi.nlm.nih.gov/pubmed/18693895. 
+[14] H. J. Kong, "Managing Unstructured Big Data in Healthcare System," 
+Healthc Inform Res, vol. 25, no. 1, pp. 1-2, Jan 2019, doi: 
+10.4258/hir.2019.25.1.1. 
+[15] J. M. Bostwick, C. Pabbati, J. R. Geske, and A. J. McKean, "Suicide 
+Attempt as a Risk Factor for Completed Suicide: Even More Lethal Than 
+We Knew," Am J Psychiatry, vol. 173, no. 11, pp. 1094-1100, Nov 1 
+2016, doi: 10.1176/appi.ajp.2016.15070854. 
+[16] B. Harmer, S. Lee, H. Duong Tv, and A. Saadabadi, "Suicidal Ideation," 
+in StatPearls. Treasure Island (FL), 2021. 
+[17] S. A. Louzon, R. Bossarte, J. F. McCarthy, and I. R. Katz, "Does Suicidal 
+Ideation as Measured by the PHQ-9 Predict Suicide Among VA 
+Patients?," Psychiatr Serv, vol. 67, no. 5, pp. 517-22, May 1 2016, doi: 
+10.1176/appi.ps.201500149. 
+[18] M. Levis, C. Leonard Westgate, J. Gui, B. V. Watts, and B. Shiner, 
+"Natural language processing of clinical mental health notes may add 
+predictive value to existing suicide risk models," Psychol Med, pp. 1-10, 
+Feb 17 2020, doi: 10.1017/S0033291720000173. 
+[19] A. C. Fernandes, R. Dutta, S. Velupillai, J. Sanyal, R. Stewart, and D. 
+Chandran, "Identifying Suicide Ideation and Suicidal Attempts in a 
+Psychiatric Clinical Research Database using Natural Language 
+Processing," Sci Rep, vol. 8, no. 1, p. 7426, May 9 2018, doi: 
+10.1038/s41598-018-25773-2. 
+[20] N. J. Carson et al., "Identification of suicidal behavior among 
+psychiatrically 
+hospitalized 
+adolescents 
+using 
+natural 
+language 
+processing and machine learning of electronic health records," PloS one, 
+vol. 14, no. 2, p. e0211116, 2019. 
+[21] B. L. Cook, A. M. Progovac, P. Chen, B. Mullin, S. Hou, and E. Baca-
+Garcia, "Novel Use of Natural Language Processing (NLP) to Predict 
+Suicidal Ideation and Psychiatric Symptoms in a Text-Based Mental 
+Health Intervention in Madrid," Comput Math Methods Med, vol. 2016, 
+p. 8708434, 2016, doi: 10.1155/2016/8708434. 
+[22] O. Uzuner, A. Stubbs, and M. Filannino, "A natural language processing 
+challenge for clinical records: Research Domains Criteria (RDoC) for 
+psychiatry," J Biomed Inform, vol. 75S, pp. S1-S3, Nov 2017, doi: 
+10.1016/j.jbi.2017.10.005. 
+[23] Y. Zhang et al., "Psychiatric stressor recognition from clinical notes to 
+reveal association with suicide," Health Informatics J, vol. 25, no. 4, pp. 
+1846-1862, Dec 2019, doi: 10.1177/1460458218796598. 
+[24] Q.-Y. Zhong et al., "Screening pregnant women for suicidal behavior in 
+electronic medical records: diagnostic codes vs. clinical notes processed 
+by natural language processing," BMC medical informatics and decision 
+making, vol. 18, no. 1, pp. 1-11, 2018. 
+[25] J. S. Obeid et al., "Identifying and Predicting intentional self-harm in 
+electronic health record clinical notes: Deep learning approach," JMIR 
+medical informatics, vol. 8, no. 7, p. e17784, 2020. 
+[26] M. Cusick et al., "Using weak supervision and deep learning to classify 
+clinical notes for identification of current suicidal ideation," J Psychiatr 
+Res, 
+vol. 
+136, 
+pp. 
+95-102, 
+Apr 
+2021, 
+doi: 
+10.1016/j.jpsychires.2021.01.052. 
+[27] W. W. Chapman, W. Bridewell, P. Hanbury, G. F. Cooper, and B. G. 
+Buchanan, "A simple algorithm for identifying negated findings and 
+diseases in discharge summaries," J Biomed Inform, vol. 34, no. 5, pp. 
+301-10, Oct 2001, doi: 10.1006/jbin.2001.1029. 
+[28] T. Mikolov, K. Chen, G. Corrado, and J. Dean, "Efficient estimation of 
+word representations in vector space," arXiv preprint arXiv:1301.3781, 
+2013. 
+[29] F. R. Tsui et al., "Natural language processing and machine learning of 
+electronic health records for prediction of first-time suicide attempts," 
+JAMIA Open, 
+vol. 4, no. 1, p. ooab011, Jan 2021, doi: 
+10.1093/jamiaopen/ooab011. 
+[30] V. Rozova, K. Witt, J. Robinson, Y. Li, and K. Verspoor, "Detection of 
+self-harm and suicidal ideation in emergency department triage notes," J 
+Am Med Inform Assoc, vol. 29, no. 3, pp. 472-480, Jan 29 2022, doi: 
+10.1093/jamia/ocab261. 
+[31]  H. Larochelle, D. Erhan, and Y. Bengio, "Zero-data learning of new 
+tasks," in AAAI, 2008, vol. 1, no. 2, p. 3.  
+[32]  C. H. Lampert, H. Nickisch, and S. Harmeling, "Learning to detect 
+unseen object classes by between-class attribute transfer," in 2009 IEEE 
+conference on computer vision and pattern recognition, 2009: IEEE, pp. 
+951-958.  
+[33] X. Sun, J. Gu, and H. Sun, "Research progress of zero-shot learning," 
+Applied Intelligence, vol. 51, no. 6, pp. 3600-3614, 2021. 
+[34]  B. Romera-Paredes and P. Torr, "An embarrassingly simple approach to 
+zero-shot learning," in International conference on machine learning, 
+2015: PMLR, pp. 2152-2161.  
+[35] R. L. Hu, C. Xiong, and R. Socher, "Zero-shot image classification guided 
+by natural language descriptions of classes: A meta-learning approach," 
+NeurIPS, 2018. 
+[36] G. Dinu, A. Lazaridou, and M. Baroni, "Improving zero-shot learning by 
+mitigating the hubness problem," arXiv preprint arXiv:1412.6568, 2014. 
+[37] F. Pourpanah et al., "A review of generalized zero-shot learning 
+methods," IEEE transactions on pattern analysis and machine 
+intelligence, 2022. 
+[38] S. Sivarajkumar and Y. Wang, "HealthPrompt: A Zero-shot Learning 
+Paradigm for Clinical Natural Language Processing," arXiv preprint 
+arXiv:2203.05061, 2022. 
+[39] Y. N. Dauphin, G. Tur, D. Hakkani-Tur, and L. Heck, "Zero-shot learning 
+for semantic utterance classification," arXiv preprint arXiv:1401.0509, 
+2013. 
+[40] M. Johnson et al., "Google’s multilingual neural machine translation 
+system: Enabling zero-shot translation," Transactions of the Association 
+for Computational Linguistics, vol. 5, pp. 339-351, 2017. 
+[41] S. G. Tesfagergish, J. Kapočiūtė-Dzikienė, and R. Damaševičius, "Zero-
+Shot Emotion Detection for Semi-Supervised Sentiment Analysis Using 
+Sentence Transformers and Ensemble Learning," Applied Sciences, vol. 
+12, no. 17, p. 8662, 2022. 
+[42] T. Hascoet, Y. Ariki, and T. Takiguchi, "Semantic embeddings of generic 
+objects for zero-shot learning," EURASIP Journal on Image and Video 
+Processing, vol. 2019, no. 1, pp. 1-14, 2019. 
+[43] H. Hedegaard, M. Schoenbaum, C. Claassen, A. Crosby, K. Holland, and 
+S. Proescholdbell, "Issues in Developing a Surveillance Case Definition 
+for Nonfatal Suicide Attempt and Intentional Self-harm Using 
+International Classification of Diseases, Tenth Revision, Clinical 
+Modification (ICD-10-CM) Coded Data," Natl Health Stat Report, no. 
+108, 
+pp. 
+1-19, 
+Feb 
+2018. 
+[Online]. 
+Available: 
+https://www.ncbi.nlm.nih.gov/pubmed/29616901. 
+[44] D. P. Kingma and J. Ba, "Adam: A method for stochastic optimization," 
+arXiv preprint arXiv:1412.6980, 2014. 
+[45] National Institute of Mental Health/NIH. "Suicide In America: Frequently 
+Asked 
+Questions: 
+Who 
+is 
+at 
+Risk 
+for 
+Suicide?" 
+https://www.nimh.nih.gov/health/publications/suicide-faq/#pub2 
+[46] C. A. Caceres et al., "Feature Selection Methods for Zero-Shot Learning 
+of Neural Activity," Front Neuroinform, vol. 11, p. 41, 2017, doi: 
+10.3389/fninf.2017.00041. 
+[47]  Y. Shao, G. Divita, T. E. Workman, D. Redd, J. H. Garvin, and Q. Zeng-
+Treitler, "Clinical Sublanguage Trend and Usage Analysis from a Large 
+Clinical Corpus," in 2020 IEEE International Conference on Big Data 
+(Big Data), 2020: IEEE, pp. 3837-3845.  
+[48]  T. E. Workman, G. Divita, and Q. Zeng-Treitler, "Discovering 
+Sublanguages in a Large Clinical Corpus through Unsupervised Machine 
+Learning and Information Gain," in 2019 IEEE International Conference 
+on Big Data (Big Data), 2019: IEEE, pp. 4889-4898.  
+[49] M. A. Clapp, E. Kim, K. E. James, R. H. Perlis, A. J. Kaimal, and T. H. 
+McCoy, Jr., "Natural language processing of admission notes to predict 
+severe maternal morbidity during the delivery encounter," Am J Obstet 
+Gynecol, Apr 14 2022, doi: 10.1016/j.ajog.2022.04.008. 
+[50] R. L. Figueroa and C. A. Flores, "Extracting Information from Electronic 
+Medical Records to Identify the Obesity Status of a Patient Based on 
+Comorbidities and Bodyweight Measures," J Med Syst, vol. 40, no. 8, p. 
+191, Aug 2016, doi: 10.1007/s10916-016-0548-8. 
+[51] S. Merelle, E. Foppen, R. Gilissen, J. Mokkenstorm, R. Cluitmans, and 
+W. Van Ballegooijen, "Characteristics Associated with Non-Disclosure 
+of Suicidal Ideation in Adults," Int J Environ Res Public Health, vol. 15, 
+no. 5, May 9 2018, doi: 10.3390/ijerph15050943. 
+[52]  T. E. Workman et al., "A Prototype Application to Identify LGBT 
+Patients in Clinical Notes," in 2020 IEEE International Conference on Big 
+Data (Big Data), 2020: IEEE, pp. 4270-4275.  
+ 
+

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+A Bayesian latent position approach for community detection in
+single- and multi-layer networks with continuous attributes
+Zhumengmeng Jina, Juan Sosab and Brenda Betancourtc
+a University of Florida
+b Universidad Nacional de Colombia
+c NORC at the University of Chicago
+December 2022
+Abstract
+The increasing prevalence of multiplex networks has spurred a critical need to take into account po-
+tential dependencies across different layers, especially when the goal is community detection, which is a
+fundamental learning task in network analysis. We propose a full Bayesian mixture model for community
+detection in both single-layer and multi-layer networks. A key feature of our model is the joint modeling
+of the nodal attributes that often come with the network data as a spatial process over the latent space.
+In addition, our model for multi-layer networks allows layers to have different strengths of dependency
+in the unique latent position structure and assumes that the probability of a relation between two actors
+(in a layer) depends on the distances between their latent positions (multiplied by a layer-specific factor)
+and the difference between their nodal attributes. Under our prior specifications, the actors’ positions
+in the latent space arise from a finite mixture of Gaussian distributions, each corresponding to a cluster.
+Simulated examples show that our model performs favorably compared to the existing ones. The model
+is also applied to a real three-layer network of employees in a law firm.
+1
+Introduction
+Network data conveniently describes the relationships between actors in complex systems and is ubiquitous
+in many statistical applications, including finance, social science, criminology, biology, epidemiology, and
+computer science, among others. Understanding the relationships between actors can aid domain experts.
+Key words and phrases. multiplex network, community detection, latent position model, mixture model, spatial process, visu-
+alization
+1
+arXiv:2301.00055v1  [stat.AP]  30 Dec 2022
+
+For instance, in epidemiology, people in a certain area can be portrayed in a contact network that can be
+studied to detect infectious disease outbreaks. In criminology, communications between terrorists form a
+terrorist network, helping intelligence agencies to better counter terrorism.
+Many models have been developed for the inference of networks over the past decades (e.g., Erdös and
+Rényi, 1959, Frank and Strauss, 1986), among which the broad class of latent space models is one of the
+most widely used (see, e.g., Sosa, 2021 for an exhaustive review). Suppose the network under study has
+N actors, then under latent space models, there are N independent and identically distributed (i.i.d.) latent
+variables z1, . . . , zN, one for each actor. Under a mild exchangeability assumption in Hoff [2007], results
+in Aldous [1985] and Hoover [1982] show that edge variables yi,j depend on latent variables through a
+symmetric function γ(zi, zj) that is meant to capture any pattern in the network beyond any known covariate
+information.
+Many well-known models fall into the category of latent space models, which can be distinguished between
+two cases depending on whether latent variables are discrete or continuous [Matias and Robin, 2014]. For in-
+stance, stochastic block models [Nowicki and Snijders, 2001, Wang and Wong, 1987] – hereafter SBM – are
+special cases of latent space models with discrete latent variables zi ∈ {1, 2, . . . , K}. When latent variables
+are assumed to be continuous, another approach using latent variables is the class of latent position models
+(LPM) proposed by Hoff et al. [2002] which our model in the paper is built upon. In its basic formulation,
+LPMs model the edge variables yi,j as conditionally independent given the distance between latent variables
+γ(zi, zj) = −∥zi − zj∥, which naturally accounts for transitivity effects through the latent space (typically
+a Euclidean K-dimensional space for a predetermined K) where zi lives. Later on, Handcock et al. [2007]
+proposed an extension on Hoff et al.’s LPM, namely the latent position cluster model (LPCM), by imposing
+a Gaussian mixture prior on the latent positions to perform clustering tasks. Krivitsky et al. [2009] further
+extended Handcock et al.’s model by adding the random sender and receiver effects proposed by Hoff [2005].
+Other formulations of γ(·, ·) can be found in Schweinberger and Snijders [2003], Hoff [2005, 2009], Athreya
+et al. [2017], Minhas et al. [2019], among others.
+Besides edge information of a network, extra information like node and edge attributes and different types
+of edges are often available, and should ideally be leveraged for inference. Typical ways to incorporate
+attributes in a network model include: (1) modeling the network as a function of the attributes (see, e.g.,
+Hoff et al., 2002, Hoff, 2005); (2) modeling the attributes as a function of the network [Guha and Rodriguez,
+2021]; (3) jointly modeling the network and attributes (Linkletter, 2007, Kim and Leskovec, 2012, Fosdick
+and Hoff, 2015, Ciminelli et al., 2019). The first approach is arguably the most common approach to incor-
+porate covariates into the model, but we consider an approach of joint modeling proposed by Ciminelli et al.
+2
+
+[2019], namely the social network spatial model (SNSM), where the authors modeled edges yi,j as condi-
+tionally independent given ∥zi − zj∥ and the distance of the continuous node attributes ∥xi − xj∥, and node
+attributes are further modeled as a spatial process over the latent space. Note that joint modeling does not
+require the network or the attributes to be fully observed as the first two approaches, hence one could predict
+missing network and attribute data (if there is any). In addition, it improves model fitting by capturing the
+dependence structure between latent variables and the attributes (when such dependency exists), as we will
+see in Section 3.
+We propose a full hierarchical Bayesian model that builds on Ciminelli et al.’s SNSM. Instead of using a
+Gaussian distribution as the prior for latent positions as in Ciminelli et al. [2019], we impose a Gaussian
+mixture prior as in Handcock et al. [2007], so that our model could also capture the group structure in the
+network. Detecting communities or clusters among actors in the network is an important task in network
+analysis and has spurred the development of many models and algorithms, among which the SBM has
+motivated an active line of research that deals with community detection (see, e.g., Lee and Wilkinson
+[2019] for a review). However, SBM may not fit well when many actors fall between clusters [Hoff et al.,
+2002]. We will compare our model with an SBM that incorporates covariates as fixed effects (i.e., model
+the edge variables as a function of latent classes and covariates [Leger, 2016]), and we call this model a
+covariate-assisted stochastic block model (CSBM). We will show that our model presents improved model
+fitting while producing similar clustering results as CSBM.
+We also propose an extension of our model to multi-layer network settings. Multi-layer networks can gen-
+erally be categorized into two cases: cross-sectional networks that have different types of connections (e.g.,
+social networks of friendship, coworker-ship, etc.) and time-varying networks where the same type of con-
+nections are measured over time (e.g., a trade network that changes over time). We consider a type of
+cross-sectional multi-layer network where each layer has a common set of actors. Substantial work has been
+done on latent space models for cross-sectional multi-layer networks that take a Bayesian approach (see, e.g.,
+Gollini and Murphy, 2016, Salter-Townshend and McCormick, 2017, D’Angelo et al., 2019, Sosa and Betan-
+court, 2022, Durante and Dunson, 2018, Wang et al., 2019, MacDonald et al., 2020). In extending our model
+to the multiple networks setting, we adopt the approach in Sosa and Betancourt [2022] in a parsimonious
+way, where latent positions are assumed to be the same for all layers, but the strength of borrowing such
+latent structure information is allowed to be different across different layers. Note that, the original model
+in Sosa and Betancourt [2022] assumed different latent positions for different layers and had an additional
+hierarchy on the hyperparameters. The specification of our model is given in the next section.
+The remainder of the paper is organized as follows. Section 2 contains general background on the spatial
+3
+
+process and introduces the proposed model (for single- and multi-layer network settings) which we call
+the latent position joint mixture model (LPJMM) in the rest of the paper. In addition, prior specification,
+identifiable problem, and inference will also be discussed in this section. Several simulation studies are
+conducted in section 3, where LPJMM is compared with Handcock et al.’s LPCM, Ciminelli et al.’s SNSM
+and CSBM in single-layer settings and the model is also evaluated in multi-layer settings. In section 4, we
+apply LPJMM to a real-world multi-layer network data set. Finally, we conclude with some discussion in
+section 5.
+2
+Models
+We first review the LPM introduced in Hoff et al. [2002], and then build upon it with a spatial process to allow
+for joint modeling of the network and the nodal attributes, and with a finite Gaussian mixture distribution for
+latent positions to allow for clustering.
+Consider a binary single-layer network with N actors. Denote its adjacency matrix as Y = (yi,j) ∈
+{0, 1}N×N, where yi,j = 1 if actors i and j are connected, and yi,j = 0 if they are not connected. Suppose
+the network data comes with a one-dimensional nodal attribute xi for each actor, and denote the covariate as
+x = (xi) ∈ RN. The LPM assumes that each actor i has an observed latent position zi in a K-dimensional
+Euclidean latent space, the so-called latent space, for some K ∈ N. Let z = (zi) ∈ RN×K, then LPM
+models edge yi,j as conditionally independent given distances between nodal attributes as well as distances
+between latent positions via logistic regression. But instead of the logistic link, we use the probit link in our
+model. The analysis of probit regression models can often be facilitated by a Gibbs sampler constructed using
+the data augmentation approach that introduces latent variables with truncated normal distributions [Albert
+and Chib, 1993]. (See also Sosa and Betancourt (2022) for a discussion on the choice of link functions.)
+Specifically, for i, j ∈ {1, . . . , N} and i ̸= j,
+yi,j | z, x, a, b, θ ind
+∼ Ber
+�
+Φ(a + b|xi − xj| − θ∥zi − zj∥)
+�
+,
+(1)
+where a, b ∈ R and θ ∈ R+, Ber(p) is a Bernoulli distribution that takes value 1 with some probability p,
+∥ · ∥ is the Euclidean norm on RK and Φ(·) is the cumulative distribution function of the standard normal
+distribution. Note that we impose a factor θ for the distance between latent positions, which is different from
+Hoff et al. [2002] and Krivitsky et al. [2009]. Although θ is unidentifiable in single-layer networks, it plays
+a non-trivial role in multi-layer network settings (introduced in Section 2.1). We defer a detailed discussion
+of θ to Section 2.4.
+4
+
+To allow for joint modeling of the network and nodal attributes, we model the nodal attributes as a spatial
+process over the latent space RK. Hence, nodal attributes are treated as random variables indexed by their
+latent positions, and the distance between these random variables is found by the distance between their
+corresponding positions. As in Ciminelli et al. [2019], we specify the spatial process as a Gaussian process
+that is stationary with mean β and isotropic (see Banerjee et al., 2015 for definitions). In this case, the
+process is completely defined by its covariance function Cov(d), where d is the distance between two random
+variables in the Gaussian process. In particular, we specify Cov(d) with an exponential kernel, that is,
+Cov(d) =
+�
+�
+�
+�
+�
+τ 2 + σ2,
+if d = 0;
+σ2 exp(−φd),
+if d > 0,
+where τ ≥ 0, σ > 0 and φ > 0. It is well-known that such a covariance structure is valid, i.e., the covariance
+matrix for any finite collection of random variables in the process is positive definite [Banerjee et al., 2015].
+Let Mz = (mij) ∈ RN×N where mij = exp(−φ∥zi − zj∥) and denote IN as the N-dimensional identity
+matrix, then the Gaussian process of the nodal attributes is constructed as follows,
+x | z, β, σ, τ, φ ∼ NN(β111N, σ2M(z, φ) + τ 2IN),
+(2)
+where Nd is a d-dimensional multivariate normal distribution for some dimension d ∈ {2, 3, . . . }, and 111N is
+an N-dimensional vector with all 1s.
+As in Krivitsky et al. [2009], we impose a Gaussian mixture distribution on latent positions, which allows us
+to cluster actors into different groups. Suppose there are H < ∞ predetermined number of components in
+the Gaussian mixture distribution, then
+zi | ωωω,µµµ,κκκ ind
+∼
+H
+�
+h=1
+ωhNK(µh, κ2
+hIK) ,
+(3)
+where ωωω = {ω1, . . . , ωH}, µµµ = {µ1, . . . , µH}, κκκ = {κ1, . . . , κH}. Note that µh is a K-dimensional mean
+vector where h ∈ {1, . . . , H}, and ωh is the probability that an actor belongs to the h-th group such that
+ωh ∈ (0, 1) and �H
+h=1 ωh = 1.
+In single-layer network settings, the model is given by Eqs. (1) to (3). Under our model, nodal attributes
+of two actors whose latent positions are close are more likely to be similar according to the exponential
+covariance structure. If b < 0 (b > 0), actors with similar attributes are more (less) likely to be connected.
+When b = 0, nodal attributes do not affect the distribution of the network directly (but it still has an indirect
+5
+
+Figure 1: DAG representation of the LPJMM in multi-layer settings.
+impact on the network through latent positions by Eq. (2)).
+2.1
+An extension to multi-layer networks.
+Our model can also be extended to multi-layer network settings in the following way. Suppose we have L
+layers Y1, . . . , YL in the network, where all layers are defined over the same set of actors. We assume the
+same latent positions z for all layers but allow the strength of borrowing such latent structure information to
+be different by imposing layer-specific factors θℓ for ℓ ∈ {1, . . . , L}. Our model in multi-layer settings is
+then presented as follows
+yi,j,ℓ | z, x, aℓ, bℓ, θℓ
+ind
+∼ Ber
+�
+Φ(aℓ + bℓ|xi − xj| − θℓ∥zi − zj∥)
+�
+,
+(4)
+x | z, β, σ, τ, φ ∼ NN(β111N, σ2M(z, φ) + τ 2IN) ,
+(5)
+zi | ωωω,µµµ,κκκ i.i.d.
+∼
+H
+�
+h=1
+ωhNK(µh, κ2
+hIK) ,
+(6)
+where yi,j,ℓ is the edge variable between actors i and j in layer ℓ ∈ {1, . . . , L}, aℓ, bℓ and θℓ are layer-specific
+parameters. Note that Eqs. (5) and (6) are the same as Eqs. (2) and (3). Fig. 1 shows a directed acyclic graph
+(DAG) representation of the model given by Eqs. (4) to (6).
+6
+
+2.2
+Prior specification
+We take a Bayesian approach to estimate the model parameters. Without loss of generality, a Bayesian ver-
+sion of the model given by Eqs. (4) to (6) is formed by placing prior distributions on the unknown parameters
+aℓ, bℓ, θℓ, β, σ, τ, φ, ωωω, µµµh, κh, for ℓ = {1, . . . , L} and h = {1, . . . , H}. In the model we consider, these
+parameters are assumed a priori independent. For parameters in the probit regression tier as specified by
+Eq. (4), their priors are specified as follows:
+aℓ
+i.i.d.
+∼ N(ma, ν2
+a) ,
+bℓ
+i.i.d.
+∼ N(mb, ν2
+b ) ,
+θℓ
+i.i.d.
+∼ Gamma(λ1, λ2) .
+The priors for the parameters in the spatial process tier as given in Eq. (5) are given as follows:
+β ∼ N(0, ν2
+β) ,
+σ2 ∼ InvG(η1, η2) ,
+τ 2 ∼ InvG(ξ1, ξ2) ,
+φ ∼ U(u1, u2) .
+Finally, we put the following priors on the rest of the parameters:
+ωωω ∼ Dir(α) ,
+µh
+i.i.d.
+∼ NK(mµ, ν2
+µIK) ,
+κ2
+h
+i.i.d.
+∼ InvG(γ1, γ2) .
+Note that, ma, νa, mb, νb, λ1, λ2, νβ, η1, η2, ξ1, ξ2, u1, u2, α, mµ, νµ, γ1 and γ2 are user-specified
+hyperparameters, and Gamma(·, ·), InvG(·, ·), U(·, ·), Dir(·) represents Gamma, Inverse-Gamma, uniform,
+and Dirichlet distributions respectively.
+2.3
+Posterior distribution and model estimation
+As is standard in Bayesian estimation of mixture models (see, e.g., Diebolt and Robert [1994]), we define a
+new variable gi that serves as the missing data of group membership of actor i whose distribution depends
+on ωωω. In particular, gi = h if actor i belongs to the h-th group. The joint density of (zi, gi) given ωωω, µµµ and κκκ
+is then given by
+H
+�
+h=1
+�
+ωh
+1
+�
+2πκ2
+h
+exp
+�
+−
+1
+2κ2
+h
+∥zi − µh∥2��I{gi=h}
+,
+where the indicator function I{gi=h} = 1 if gi = h, and I{gi=h} = 0 otherwise. Let g = (gi)N
+i=1 be the group
+membership for all actors and L(·) be the law of a random variable. Then the posterior distribution of z, g
+7
+
+and the parameters upon which priors are specified in Section 2.2 is given by
+Π(z, g, a1, . . . , aL, b1, . . . , bL, θ1, . . . , θL, β, τ 2, σ2, φ,ωωω,µµµ,κκκ | Y1, . . . , YL, x)
+∝
+� L
+�
+ℓ=1
+L(Yℓ | z, x, aℓ, bℓ, θℓ)
+�
+L(x | z, σ, τ, φ)L(z, g | ωωω,µµµ,κκκ)
+� L
+�
+ℓ=1
+L(aℓ)L(bℓ)L(θℓ)
+�
+× L(β)L(σ2)L(τ 2)L(φ)L(ωωω)L(µµµ)L(κκκ) .
+Note that the dimension of the posterior distribution has dimension NK + N + 3L + 3H + 4 and the
+corresponding posterior density is presented as follows,
+π(z, g, a1, . . . , aL, b1, . . . , bL, θ1, . . . , θL, β, τ 2, σ2, φ,ωωω,µµµ,κκκ | Y1, . . . , YL, x)
+∝
+N
+�
+i,j=1
+i̸=j
+L
+�
+ℓ=1
+�
+Φ(aℓ + bℓ|xi − xj| − θℓ∥zi − zj∥)
+�yi,j,ℓ�
+1 − Φ(aℓ + bℓ|xi − xj| − θℓ∥zi − zj∥)
+�1−yi,j,ℓ
+× |σ2M(z, φ) + τ 2IN|− 1
+2 exp
+�
+− 1
+2(x − β1)⊺�
+σ2M(z, φ) + τ 2IN
+�−1(x − β1)
+�
+×
+N
+�
+i=1
+H
+�
+h=1
+� ωh
+�
+κ2
+h
+exp
+�
+−
+1
+2κ2
+h
+∥zi − µh∥2��I{gi=h}
+× exp
+� 1
+2ν2a
+L
+�
+ℓ=1
+(aℓ − ma)2 +
+1
+2ν2
+b
+L
+�
+ℓ=1
+(bℓ − mb)2� L
+�
+ℓ=1
+θλ1−1
+ℓ
+exp(−λ2θℓ)
+× exp
+� β2
+2ν2
+β
+�
+(σ2)−η1−1(τ 2)−ξ1−1 exp
+�
+− η2
+σ2 − ξ2
+τ 2
+�
+I{φ∈[u1,u2]}
+×
+H
+�
+h=1
+�
+ωαh−1
+h
+I{�H
+h=1 ωh=1} exp
+�
+−
+1
+2ν2µ
+∥µh − mµ∥2�
+(κ2
+h)−γ1−1 exp
+�
+− γ2
+κ2
+h
+��
+.
+2.4
+Inference and identifiability of parameters
+Note that the posterior distribution is highly intractable, hence we must resort to Markov chain Monte Carlo
+(MCMC) methods for inferences on model parameters. A Markov chain of the parameters is generated via
+the program “Just Another Gibbs Sampler” (JAGS) which is implemented in R [R Core Team, 2021] using
+the rjags package [Plummer, 2022].
+Several parameters are not identifiable in our model. Firstly, due to factors θℓ and φ, and the fact that latent
+positions are incorporated in the posterior only through their distances, the posterior is, therefore, invariant
+to θℓs and φ, and is invariant to scaling, reflection, rotation, and translation of the latent positions z. (Note
+that, Hoff et al., 2002 and Krivitsky et al., 2009 did not have θℓs, hence their posterior is not invariant to the
+8
+
+scaling of latent positions.) Although θℓs are not identifiable and do not affect the model fitting, in multi-
+layer settings, their ratios θℓ1/θℓ2 still provide valid information on layer’s relative strength of borrowing
+information from the latent space.
+Despite being unidentifiable, one can still make inferences on the latent positions and find a reasonable
+estimate for z through a post-process which we now describe. Similar to the definition in [Hoff et al., 2002],
+we define the equivalence class of z ∈ RN×K, denoted as [z], to be the set of positions that are equivalent
+to z under scaling, reflection, rotation, and translation. Given a fixed reference position zref, a position
+z∗ is found in [z] such that z∗ = arg minz′∈[z] tr(zref − z′)⊺(zref − z′), which is the so-called Procrustes
+transformation. In simulation studies, zref is naturally chosen to be the true latent position, while in practical
+applications, we could use the last iteration of the Markov chain of latent positions as the reference. The
+Procrustes transformation is performed for each iteration of the Markov chain of the latent positions {zn},
+and an estimate for z is taken as the mean of the Procrustes transformations of {zn}.
+As occurs in Bayesian mixture models, the label-switching problem for the group membership g is an-
+other source of non-identifiability. That is, the posterior is invariant under permutations of clustering labels.
+Many algorithms have been proposed to obtain a single clustering estimate based on the MCMC sample of
+the group membership {gn}, including an optimization method (which we call “MaxPEAR” hereafter) in
+Fritsch and Ickstadt [2009] that finds a clustering that maximizes posterior expected adjusted rand index, an
+optimization method (“MinBinder”) in Lau and Green [2007] that minimizes Binder’s loss function, and a
+greedy algorithm (“GreedyEPL”) in Rastelli and Friel [2018] that aims to minimize the variation of informa-
+tion, among others. These approaches may generate different clustering estimates, and to get a better under-
+standing of the model performance, all aforementioned algorithms (MaxPEAR, MinBinder and GreedyEPL)
+are used to assess the model. Estimates based on these approaches are found using the packages GreedyEPL
+[Rastelli, 2021] and mcclust [Fritsch, 2022].
+3
+Simulation
+Two simulation studies are carried out in this section to evaluate our model. A single-layer network is
+considered in the first simulation where we compare LPJMM with three other models designed only for
+single-layer networks, namely LPCM in Handcock et al. [2007], SNSM in Ciminelli et al. [2019], and CSBM
+in Leger [2016], where SNSM is also implemented using the rjags package, and LPCM and CSBM are
+implemented using the latentnet [Krivitsky and Handcock, 2022] and sbm [Chiquet et al., 2022] packages
+respectively. The model specifications for these models can be found in Appendix A. Models assessments
+include how well a model could recover the group membership and the latent position configuration, and a
+9
+
+ 
+1
+Figure 2: Left: A visualization of the network based on the true latent position and color indicates group
+membership g. Right: Heatmap of the adjacency matrix (where actors are reordered according to g).
+goodness-of-fit test using summaries of networks including density, transitivity, and assortative coefficient
+with respect to the group membership g (see Kolaczyk and Csárdi, 2020 for definitions). We also evaluate
+our model by how accurately it could estimate certain parameters in the model. The second simulation is
+conducted in two-layer network settings, where the performance of our model could be further evaluated by
+how well the ratio θ1/θ2 can be recovered that reflects differences in each layer’s dependency on the latent
+position structure.
+3.1
+Simulation 1: a single-layer network
+Consider a single-layer network (i.e., L = 1) with N = 100 actors generated as follows. Firstly, generate
+latent positions z from a mixture of H = 5 multivariate normal distributions, and then generate attributes x
+jointly from a multivariate normal distribution with mean β1N = 0 and covariance matrix given by Cov(d)
+in Section 2 where φ = 0.5, τ 2 = 0.3, σ2 = 1. Finally, the network data is generated according to Eq. (1)
+with a = 5, b = −2, and θ = 2.72. See Fig. 2 for a visualization of the simulated network. The network is
+fairly sparse with a density equal to 0.1531, and shows fairly strong transitivity and assortative mixing with
+coefficients 0.5049 and 0.5512 respectively.
+As for the prior specifications, we set ma = mb = 0, and ν2
+a = ν2
+b = 9 to allow a wide range of values for a
+and b. Let θ ∼ Gamma(1, 1) so that θ has mean 1. An almost flat prior is imposed on β by setting νβ = 104.
+The same uniform prior U(0, 1) as in Ciminelli et al. [2019] is specified for φ. We suggest the sum of the prior
+means of τ 2 and σ2 to be on the same scale as the sample variance of x, and here we use σ2 ∼ InvG(2, 1)
+and τ 2 ∼ InvG(2, 1). Let α = 1 so that the prior on ωωω is a flat Dirichlet distribution. Following the
+heuristics in Sosa and Betancourt [2022], we specify µh
+i.i.d.
+∼ NK(0, 2/3IK) and κ2
+h
+i.i.d.
+∼ InvG(3, 2/3) so
+that var(zij|gi) = 1.
+10
+
+our model name
+LPCM
+CSBM
+MaxPEAR
+0.737 (5)
+0.707 (4)
+–
+MinBinder
+0.712 (11)
+0.748 (10)
+–
+GreedyEPL
+0.664 (4)
+0.688 (4)
+–
+Variational-EM
+–
+–
+0.707 (6)
+Table 1: Adjusted Rand indices corresponding to different estimation methods for group membership. Num-
+bers in the parentheses represent numbers of estimated groups.
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+21
+22
+23
+24
+25
+26
+27
+28
+29
+30
+31
+32
+33
+34
+35
+36
+37
+38
+39
+40
+41
+42
+43
+44
+45
+46
+47
+48
+49
+50
+51
+52
+53
+54
+55
+56
+57
+58
+59
+60
+61
+62
+63
+64
+65
+66
+67
+68
+69
+70
+71
+72
+73
+74
+75
+76
+77
+78
+79
+80
+81
+82
+83
+84
+85
+86
+87
+88
+89
+90
+91
+92
+93
+94
+95
+96
+97
+98
+99
+100
+(a) Truth
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+21
+22
+23
+24
+25
+26
+27
+28
+29
+30
+31
+32
+33
+34
+35
+36
+37
+38
+39
+40
+41
+42
+43
+44
+45
+46
+47
+48
+49
+50
+51
+52
+53
+54
+55
+56
+57
+58
+59
+60
+61
+62
+63
+64
+65
+66
+67
+68
+69
+70
+71
+72
+73
+74
+75
+76
+77
+78
+79
+80
+81
+82
+83
+84
+85
+86
+87
+88
+89
+90
+91
+92
+93
+94
+95
+96
+97
+98
+99
+100
+(b) LPJMM
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+21
+22
+23
+24
+25
+26
+27
+28
+29
+30
+31
+32
+33
+34
+35
+36
+37
+38
+39
+40
+41
+42
+43
+44
+45
+46
+47
+48
+49
+50
+51
+52
+53
+54
+55
+56
+57
+58
+59
+60
+61
+62
+63
+64
+65
+66
+67
+68
+69
+70
+71
+72
+73
+74
+75
+76
+77
+78
+79
+80
+81
+82
+83
+84
+85
+86
+87
+88
+89
+90
+91
+92
+93
+94
+95
+96
+97
+98
+99
+100
+(c) LPCM
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+21
+22
+23
+24
+25
+26
+27
+28
+29
+30
+31
+32
+33
+34
+35
+36
+37
+38
+39
+40
+41
+42
+43
+44
+45
+46
+47
+48
+49
+50
+51
+52
+53
+54
+55
+56
+57
+58
+59
+60
+61
+62
+63
+64
+65
+66
+67
+68
+69
+70
+71
+72
+73
+74
+75
+76
+77
+78
+79
+80
+81
+82
+83
+84
+85
+86
+87
+88
+89
+90
+91
+92
+93
+94
+95
+96
+97
+98
+99
+100
+(d) CSBM
+Figure 3: (A): Color indicates the true group membership g. (B)-(D): Color indicates the estimated group
+memberships ˆg of LPJMM, LPCM and CSBM respectively. Positions of the points in all plots are true latent
+positions z.
+Note that the latent space dimension K and the number of clusters H in the model need to be prespecified
+along with the priors. We take K to be the true dimensions of the latent space (i.e., K = 2) since this
+facilitates model assessment by allowing visualizations of the estimated latent positions. One could also
+use the Watanabe-Akaike Information Criterion (WAIC) to select a K with the smallest WAIC as in Sosa
+and Betancourt [2022]. However, WAIC and other information criteria like Deviance Information Criterion
+(DIC) are not helpful in choosing the number of clusters H. We noticed that the model assessment is
+significantly worse when H is chosen to be smaller than the truth. However, model assessments are similar
+among models whose H is at least as large as the truth. A comparison of the model assessment for different
+specified H is given in Appendix B. From the comparison, we could also see that when H is specified to be
+larger, the number of clusters in the estimated group membership ˆg also tends to be larger. Therefore, we
+suggest choosing H to be the largest number of groups that one is willing to accept, and in this example, we
+choose H to be 5.
+We then fit LPJMM using MCMC sampling with 20 000 burn-in iterations and a further 10 000 iterations
+which are kept for posterior analysis. The Markov chain mixes reasonably well and shows no signs of lack
+of convergence (see Appendix C for the traceplot of the log-likelihood chain).
+11
+
+LPCM
+LPJMM
+(simulation 1)
+LPJMM
+(simulation 2)
+Sum of Euclidean distances
+23.08
+28.06
+12.20
+Table 2: Sum of distances between the estimated and true latent positions.
+To evaluate a model’s ability to recover the group membership, we first find estimates of clustering using
+the optimization algorithms (i.e., MaxPEAR, MinBinder and GreedyEPL) mentioned in Section 2.4. The
+adjusted Rand index is then calculated for each clustering estimate. Note that SNSM does not define clusters,
+therefore we only compare the adjusted Rand index between LPJMM, LPCM, and CSBM. Since the sbm
+package takes a non-Bayesian approach that uses a Variational-EM algorithm to find a point estimator for
+the group membership g, optimization methods like MaxPEAR are not necessary to analyze results from
+CSBM. The results shown in Table 1 suggest that these three models have a similar ability in recovering
+group membership, with rand indices of LPJMM using the MaxPEAR and MinBinder algorithms being
+higher than the rand index (0.707) under the CSBM model. A visualization of the estimated clusters based
+on the true latent positions is given in Fig. 3. Also, notice that the MinBinder algorithm tends to overestimate
+the number of clusters in the network.
+To further compare the ability to recover latent position configuration between LPJMM and LPCM, we find
+an estimate of the latent positions as follows. Firstly, we perform the Procrustes transformation on zn for each
+iteration n, and then take the estimate ˆz of z to be the average of zn. We then calculate the Euclidean distance
+between the estimated latent position ˆzi (which is the i-th row in ˆz) and the true latent position zi (i.e., the
+i-th row in z) for each actor i and use the sum of distances of all actors to quantify the similarity between
+the estimated and the true latent position configurations. The results are shown in Table 2 which suggests
+that these two models have similar recovery of the latent positions. Plots of the estimated latent positions of
+LPJMM and LPCM can be found in Appendix D, which also suggest similar estimated configurations of z
+as Table 2.
+Following Sosa and Betancourt [2022], we assess if models have a good fit in the sense of good reproduction
+of a variety of summary statistics, which are calculated based on a collection of simulated networks generated
+as follows. For LPJMM and SNSM, a network is simulated for every 10-th iteration using the parameters in
+that iteration. For LPCM and CSBM, 1000 networks are simulated using their respective packages. Then
+for each model, we calculate the density, transitivity, and assortative coefficient (if applicable) with respect
+to the true group membership for each simulated network. Boxplots of these summary statistics are given in
+Fig. 4 and the averages of these summary statistics for each model are given in Table 3. Note that our model
+12
+
+density
+transitivity
+assortativity
+0.14
+0.15
+0.16
+0.35 0.40 0.45 0.50 0.55
+0.45
+0.50
+0.55
+0.60
+LPJMM
+LPCM
+SNSM
+CSBM
+Figure 4: Boxplots of summary statistics for each model. Red dotted lines indicate the true values for
+network characteristics respectively.
+true value
+LPJMM
+LPCM
+SNSM
+CSBM
+density
+0.1531
+0.1539
+0.1530
+0.1504
+0.1499
+transitivity
+0.5049
+0.5144
+0.5467
+0.4027
+0.3776
+assortativity
+0.5512
+0.5468
+0.5475
+0.4811
+0.4954
+Table 3: Means of the summary statistics of the simulated networks for each model in simulation 1.
+appropriately captures these structural features of the network data, while LPCM tends to overestimate tran-
+sitivity in the network, and both SNSM and CSBM tend to underestimate both transitivity and assortativity
+in the network.
+3.2
+Simulation 2: a two-layer network
+Continue using the parameter setup in simulation 1 and its generated network as the first layer (i.e., a1 = 5,
+b1 = −2, θ1 = 2.72), we generate a second layer of the network with a2 = 3, b2 = 1, θ2 = 4. As in
+simulation 1, we fit LPJMM with K = 2 and H = 5 and evaluate the model’s ability to recover the group
+membership using the adjusted Rand indices based on four clustering summaries. The results are given in
+Table 4, which shows similar clustering estimates as in simulation 1 where only one layer is considered.
+However, the sum of Euclidean distances between the estimated and true latent positions of all actors (see
+Table 2) in simulation 2 is 12.20, which is a significant improvement compared to 28.06 in simulation 1.
+The plot of the estimated latent position configurations is given in Fig. 5 (B), which visualizes the model’s
+recovery of latent positions and group membership.
+We also carry out the goodness-of-fit test as in simulation 1 and the result is given in Table 5, which shows
+that LPJMM captures these structural features accurately, and the result for layer 1 is similar to the result in
+simulation 1.
+13
+
+MaxPEAR
+MinBinder
+GreedyEPL
+adjusted Rand index
+0.748 (6)
+0.753 (12)
+0.662 (4)
+Table 4: Adjusted Rand indices corresponding to different estimation methods for group membership in
+simulation 2. Numbers in the parentheses represent numbers of estimated groups.
+1
+2
+3
+4
+5
+6
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+(a) Truth
+1
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+6
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+10
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+14
+15
+16
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+88
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+90
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+96
+97
+98
+99
+100
+(b) Estimated z and g
+Figure 5: (A): Points are plotted based on true latent position z and true group membership g. (B): Points
+are plotted using the estimated latent positions in simulation 2, and color represents the estimated group
+membership using the MaxPEAR method.
+Recall that θ1 and θ2 are of no direct interest since they are not identifiable. However, we are still interested
+in the ratio θ1/θ2 since it reflects the relative strength of borrowing information from the latent space of each
+layer. Although aℓ and bℓ are of no direct interest, we pay attention to their signs, especially that of bℓ because
+different signs of bℓ have different interpretations of the effect of attributes as discussed in Section 2. We also
+assess the model’s ability to estimate parameters β, τ 2, and σ2 using posterior means and 95% credible inter-
+vals. The results are given in Table 6. Overall, the performance of LPJMM in recovering the true values of
+these model parameters is pretty well, except for τ 2 and σ2. Both LPJMM and SNSM tend to underestimate
+σ2 and overestimate τ 2. That is, the covariance of the attributes tends to be underestimated, and although τ 2
+is slightly overestimated, the variance of the attributes (τ 2 + σ2) still tends to be underestimated.
+4
+Real data analysis
+In this section, we consider a three-layer network data set collected by [Lazega, 2001] from a corporate
+law firm from 1988-1991 in New England. This network describes three types of relationships (namely,
+networks of advice, friendship, and coworker contacts) between 71 lawyers in the law firm. Several actor
+attributes are also collected: age, gender, seniority (years with the firm), office (located in Boston, Hartford,
+or Providence), practice (litigation or corporate law), law school the lawyers attended (Harvard or Yale,
+University of Connecticut, or other universities) and status (partner or associate). A principal component
+14
+
+true value
+mean
+density
+layer 1
+0.1531
+0.1535
+layer 2
+0.1024
+0.1023
+transitivity
+layer 1
+0.5049
+0.5088
+layer 2
+0.5477
+0.5546
+assortativity
+layer 1
+0.5512
+0.5466
+layer 2
+0.6923
+0.6890
+Table 5: Means of the summary statistics in simulation 2.
+true value
+posterior mean
+95% credible interval
+θ1/θ2
+0.680
+0.653
+(0.579, 0.721)
+a1
+5
+5.01
+(4.719, 5.262)
+a2
+3
+3.25
+(2.976, 3.572)
+b1
+-2
+-1.919
+(-2.053, -1.766)
+b2
+1
+1.058
+(0.901, 1.252)
+β
+0
+-0.047
+(-1.027, 1.01 )
+τ 2
+0.3
+0.409
+(0.261, 0.592)
+σ2
+1
+0.642
+(0.230, 1.684)
+Table 6: Posterior means and 95% credible intervals.
+analysis (PCA) is performed on age and seniority attributes, and the first principal component explains
+89% of the variance which is of no surprise since age and seniority are highly correlated with a correlation
+coefficient being 0.78. We chose the first principal component to be the attribute x and let H = 8 since it
+is the largest number of clusters we expect in the network. Then the model is fitted to the network using the
+same prior and Markov chain setup as in Section 3.
+The study of the Lazega network in this paper is meant to find out how the three types of relations can be
+explained by the findings deduced from the model fitting. We first visualize the estimated latent positions
+z colored by different categorical attributes (gender, office, practice, law school, and status) in Fig. 6. As
+we can see from these plots, the estimated positions z are well separated by the office (especially offices
+in Boston and Hartford) and practice. Compare these plots with z colored by MaxPEAR and GreedyEPL
+estimated clustering g in Fig. 7, we can see that both estimated g roughly clusters lawyers into three groups:
+lawyers in Hartford office, litigation lawyers in Boston or Providence offices, and corporate lawyers in
+Boston or Providence offices.
+Plots of adjacency matrices of the three layers (where lawyers are grouped by the MaxPEAR estimate of g)
+and their corresponding networks are given in Fig. 8, where we could see that the coworker network shows
+15
+
+1
+2
+3
+4
+5
+7
+8
+9
+10
+11
+12
+13
+14
+15
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+22
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+28
+29
+30
+32
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+34
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+40
+44
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+46
+47
+49
+50
+52
+53
+54
+59
+60
+62
+63
+65
+67
+68
+70
+male
+female
+gender
+1
+2
+3
+4
+56
+7
+8
+9
+10
+11
+12
+13
+14
+15
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+56 57
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+59
+60
+61
+62
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+64
+65
+66
+67
+68
+69
+70
+71
+Boston
+Hartford
+Providence
+office
+1
+2
+3
+4
+56
+7
+8
+9
+10
+11
+12
+13
+14
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+68
+69
+70
+71
+litigation
+corporate
+practice
+1
+2
+3
+4
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+7
+8
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+10
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+65
+66
+67
+68
+69
+70
+71
+Harvard/Yale
+Ucon
+other
+law school
+1
+2
+3
+4
+56
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
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+56 57
+58
+59
+60
+61
+62
+63
+64
+65
+66
+67
+68
+69
+70
+71
+partner
+associate
+status
+Figure 6: Points in all plots are drawn based on the estimated latent positions z, and are colored based on
+their categories in gender, office, practice, law school, and status.
+the most estimated clustering pattern, while the advice network presents the least of such pattern, which
+could also be seen from the relative ratios of θℓs in Table 7. This means that lawyers from the same office
+and doing the same practice are more likely to become coworkers and friends, but who they seek advice
+from does not depend much on office and practice. Furthermore, we can deduce from the posteriors of bℓ in
+Table 7 that these lawyers tend to seek advice from people of similar age (or seniority) since the posterior
+estimate of b1 is negative, while lawyers of different ages (or seniority) are more likely to become friends and
+coworkers. This conclusion is in line with the assortativity coefficients with respect to the nodal attributes
+(lawyer’s age) given in Table 8.
+5
+Discussion
+This paper presents a latent position model that extends LPCM of Handcock et al. [2007] and SNSM of
+Ciminelli et al. [2019] to jointly model network data and the nodal attributes and perform model-based
+clustering. By jointly modeling the network and the attributes, we are able to describe how the attributes
+16
+
+1
+2
+3
+4
+56
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+21
+22
+23
+24
+25
+26
+27
+28
+29
+30
+31
+32
+33
+34
+35
+36
+37
+38
+39
+40
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+46
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+54
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+56 57
+58
+59
+60
+61
+62
+63
+64
+65
+66
+67
+68
+69
+70
+71
+(a) MaxPEAR
+1
+2
+3
+4
+56
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+21
+22
+23
+24
+25
+26
+27
+28
+29
+30
+31
+32
+33
+34
+35
+36
+37
+38
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+41
+42
+43
+44
+45
+46
+47
+48
+49
+50
+51
+52
+53
+54
+55
+56 57
+58
+59
+60
+61
+62
+63
+64
+65
+66
+67
+68
+69
+70
+71
+(b) GreedyEPL
+Figure 7: Points are plotted using the estimated latent positions and color indicates the estimated group
+membership using MaxPEAR and GreedyEPL methods respectively.
+posterior mean
+95% credible interval
+θ1/θ2
+0.3229
+(0.2352, 0.4152)
+θ1/θ3
+0.2035
+(0.1479, 0.2606)
+θ2/θ3
+0.6319
+(0.5536, 0.7198)
+b1
+-0.0986
+(-0.1401, -0.0579)
+b2
+0.0708
+(0.0263, 0.1137)
+b3
+0.133
+(0.0854, 0.186)
+Table 7: Posterior means and 95% credible intervals.
+change over the network and explain how relations could be influenced by attributes. LPJMM also provides
+an extension to multi-layer network settings on the assumption that all layers share the same latent position
+structure but with different strengths of borrowing such latent structure information. We applied our method
+to two simulated networks, one with a single layer and another with two layers, and found our model to
+give satisfactory fits to these two data sets and is competitive in terms of goodness-of-fit and group detection
+compared with SNSM, LPCM, and CSBM. The model is also applied to a three-layer real network data set
+and we are able to draw reasonable conclusions from the modeling results.
+We have suggested choosing the number of groups H to be the largest number of groups that one is willing to
+accept in the network because we have found that varying the number of groups has almost no impact on the
+model fit and prediction outcome as long as it is in a reasonable range. One could also fit the CSBM to the
+network first, and choose H based on its estimated number of groups. One problem we have not addressed
+in the paper is of choosing the dimension of the latent space. This can be done by using Bayesian model
+selection like WAIC as in Sosa and Betancourt [2022].
+Our model could be extended in several ways. Firstly, other extensions of our model to multi-layer settings
+could be considered. For example, Sosa and Betancourt [2022] assumed conditionally independent layer-
+17
+
+advice
+friendship
+coworker
+ 
+ 
+ 
+Figure 8: Upper: Heatmaps of the adjacency matrices (where lawyers are reordered according to the Max-
+PEAR estimate of g). Lower: A visualization of the three layers based on the estimated z and color indicates
+the MaxPEAR estimate of g.
+advice
+friendship
+coworker
+assortativity
+0.2536
+-0.1107
+-0.1224
+Table 8: Assortativity coefficients with respect to lawyer’s age.
+specific latent positions, whereas MacDonald et al. [2022] assumed that the latent position of an actor in all
+layers is (d0 + d1)-dimensional, where the first d0 components of the latent position are the same across all
+layers, and only the last d1 components are layer-specific. Secondly, instead of assigning a user-specified
+number of groups H to the model, we could learn the number of groups by using a Bayesian nonpara-
+metric approach with a Dirichlet Process prior to model community memberships (see, e.g., Amini et al.,
+2019).
+LPJMM could also be extended to leverage multivariate covariates. So far, we have limited ourselves to mod-
+eling univariate nodal attributes that are approximately Gaussian. For continuous nodal attributes with more
+than one dimension, we have used the first principal component from the principal component analysis. To
+take full advantage of high-dimensional nodal attributes, one could use multivariate spatial process modeling
+to replace Eq. (2). Other extensions of more sophisticated spatial modeling include spatiotemporal modeling
+of attributes for time-varying networks, which would help to describe changes in actors over time.
+18
+
+Appendix
+A
+Model Specifications for SNSM, LPCM and CSBM
+Note that the original SNSM in Ciminelli et al. [2019] uses the logit link. In order to make a fair comparison,
+we also use the probit link in SNSM as in LPJMM. The model specification for SNSM used in this paper is
+given as follows:
+yi,j | z, x, aℓ, bℓ, θℓ
+ind
+∼ Ber
+�
+Φ(a + b|xi − xj| − ∥zi − zj∥)
+�
+,
+x | z, β, σ, τ, φ ∼ NN(β111N, σ2M(z, φ) + τ 2IN) ,
+and the priors are set to be the same as the priors in LPJMM (if possible). To be specific,
+zi
+i.i.d.
+∼ N2(000, I2) ,
+β ∼ N(0, 104) ,
+σ2 ∼ InvG(2, 1) ,
+τ 2 ∼ InvG(2, 1) ,
+φ ∼ U(0, 1) ,
+and the priors on the parameters in the probit regression tier are given by:
+a i.i.d.
+∼ N(0, 9) ,
+b i.i.d.
+∼ N(0, 9) .
+SNSM in this paper is implemented using JAGS.
+The model specification for LPCM (see Handcock et al., 2007) is given as the follows,
+yi,j | z, x, β0, β1
+ind
+∼ Ber
+�
+logit(β⊺
+0xi,j − β1∥zi − zj∥)
+�
+,
+zi | ωωω,µµµ,κκκ i.i.d.
+∼
+5
+�
+h=1
+ωhN5(µh, κ2
+hIK) ,
+and we use the default priors given in the latentnet package for prior specifications.
+We first introduce several notations before presenting CSBM in Leger [2016]. Suppose there are Q groups
+in the network. Denote the N × Q group membership matrix as ZZZ = {Ziq}, and Ziq = 1 if actor i belongs
+to group q, Ziq = 0 if otherwise. It is assumed that an actor can only belong to one group. The model
+specification for CSBM is given as follows,
+yi,j | Zi, Zj, x, β ind
+∼ Ber
+�
+logit(mqi,qj + β⊺xi,j)
+�
+,
+where Zi is the i-th row of ZZZ, qi is the group membership for actor i and the group effect mqi,qj ∈ R.
+19
+
+B
+Comparing model performances for different number of groups
+We conduct a comparison of LPJMM with different H ∈ {3, 4, . . . , 9} using the data set in simulation 1.
+Table 9 presents the adjusted rand indices, and the results are similar for models that assume H to be equal to
+or larger than the true number of groups (which is 5 in this example). However, the adjusted rand indices for
+all three estimates are significantly smaller when the model assumes H to be smaller than 5. Also, notice that
+the estimated number of groups increases with H. Visualizations of how adjusted rand indices and estimated
+number of groups changes over H are given in Fig. 9.
+H
+MaxPEAR
+MinBinder
+GreedyEPL
+3
+0.4067 (3)
+0.4008 (5)
+0.4321 (3)
+4
+0.4882 (3)
+0.4977 (6 )
+0.6521 (4)
+5
+0.7374 (5)
+0.7115 (11)
+0.6635 (4)
+6
+0.7237 (6)
+0.7442 (20)
+0.7134 (4)
+7
+0.7449 (7)
+0.6624 (25)
+0.7313 (4)
+8
+0.7422 (8)
+0.6674 (25)
+0.7293 (8)
+9
+0.7056 (12)
+0.7041 (25)
+0.7043 (11)
+Table 9: Adjusted Rand indices of different estimates under LPJMM with different H. Numbers in the
+parentheses denote the numbers of estimated groups.
+3
+4
+5
+6
+7
+8
+9
+H
+Adjusted rand index
+0.4
+0.6
+0.8
+3
+4
+5
+6
+7
+8
+9
+H
+number of groups
+5
+10
+15
+20
+25
+MaxPEAR
+MinBinder
+GreedyEPL
+Figure 9: Left: Adjusted rand indices of the clustering estimates found by using the MaxPear, MinBinder,
+and GreedyEPL methods. Right: Estimated number of groups using the three methods.
+The goodness-of-fit test outlined in Section 3 is also carried out here to compare the means of several sum-
+mary statistics, which are plotted in Fig. 10. As we can see from the plots, the model’s fit is not affected by
+the choice of H even for H smaller than the actual number of clusters in the network.
+20
+
+3
+4
+5
+6
+7
+8
+9
+H
+0.1536
+0.1540
+0.1544
+(a) density
+3
+4
+5
+6
+7
+8
+9
+H
+0.513
+0.514
+0.515
+(b) transitivity
+3
+4
+5
+6
+7
+8
+9
+H
+0.538
+0.546
+0.554
+(c) assortativity
+Figure 10: The means of summary statistics for different H.
+C
+Traceplots of log-likelihood
+The traceplots of the log-likelihood (after thinning the Markov chain every 10 iterations) in simulation stud-
+ies and real applications in Sections 3 and 4 are given in Fig. 11.
+0
+2000
+6000
+10000
+−1100
+−1080
+−1060
+−1040
+(a) simulation 1
+0
+2000
+6000
+10000
+−1920
+−1900
+−1880
+−1860
+(b) simulation 2
+0
+2000
+6000
+10000
+−5230
+−5214
+−5198
+−5182
+(c) Lazega network
+Figure 11: Traceplots of the log-likelihood.
+D
+Visualizations of results from LPJMM and LPCM
+Visualizations of the estimated latent positions and estimated group membership using the MaxPEAR, Min-
+Binder, and GreedyEPL methods under LPJMM and LPCM are shown in Figs. 12 and 13 respectively.
+21
+
+1
+2
+3
+4
+5
+6
+7
+8
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+10
+11
+12
+13
+14
+15
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+17
+18
+19
+20
+21
+22
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+24
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+28
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+66
+67
+68
+69
+70
+71
+72
+73
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+76
+77
+78
+79
+80
+81
+82
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+84
+85
+86
+87
+88
+89
+90
+91
+92
+93
+94
+95
+96
+97
+98
+99
+100
+(a) MaxPEAR
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
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+22
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+24
+25
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+64
+65
+66
+67
+68
+69
+70
+71
+72
+73
+74
+75
+76
+77
+78
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+(b) MinBinder
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+(c) GreedyEPL
+Figure 12: Points are plotted based on the estimated latent position z and three estimated group memberships
+ˆg of LPJMM.
+1
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+(a) MaxPEAR (LPCM)
+1
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+(b) MinBinder (LPCM)
+1
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+100
+(c) GreedyEPL (LPCM)
+Figure 13: Points are plotted based on estimated z and three estimated ˆg of LPCM.
+References
+J. H. Albert and S. Chib.
+Bayesian analysis of binary and polychotomous response data. Journal of the
+American Statistical Association, 88:669–679, 1993.
+D. J. Aldous. Exchangeability and related topics. Springer, Berlin, Heidelberg, 1985.
+A. A. Amini, M. S. Paez, and L. Lin. Hierarchical stochastic block model for community detection in
+multiplex networks. arXiv:1904.05330, 2019.
+A. Athreya, D. E. Fishkind, M. Tang, C. E. Priebe, Y. Park, J. T. Vogelstein, K. Levin, V. Lyzinski, Y. Qin,
+and D. L. Sussman. Statistical inference on random dot product graphs: a survey. Journal of Machine
+Learning Research, 18:1–92, 2017.
+S. Banerjee, B. Carlin, and A. Gelf. Hierarchical Modeling and Analysis for Spatial Data. CRC Press, 2nd
+edition, 2015.
+22
+
+J. Chiquet, S. Donnet, and P. Barbillon. sbm: Stochastic Blockmodels, 2022. URL https://CRAN.
+R-project.org/package=sbm. R package version 0.4.4.
+J. T. Ciminelli, T. Love, and T. T. Wu. Social network spatial model. Spatial Statistics, 29:129–144, 2019.
+J. Diebolt and C. P. Robert. Estimation of finite mixture distributions through Bayesian sampling. Journal
+of the Royal Statistical Society. Series B, 56:363–375, 1994.
+D. Durante and D. Dunson. Bayesian inference and testing of group differences in brain networks. Bayesian
+Analysis, 13:29–58, 2018.
+S. D’Angelo, T. B. Murphy, and M. Alfò. Latent space modelling of multidimensional networks with appli-
+cation to the exchange of votes in Eurovision song contest. The Annals of Applied Statistics, 13:900–930,
+2019.
+P. Erdös and A. Rényi. On random graphs. I. Publicationes Mathematicae (Debrecen), 6:290–297, 1959.
+K. Fosdick and P. D. Hoff. Testing and modeling dependencies between a network and nodal attributes,.
+Journal of the American Statistical Association, 110:1047–1056, 2015.
+O. Frank and D. Strauss. Markov graphs. Journal of the American Statistical Association, 81:832–842,
+1986.
+A. Fritsch.
+mcclust:
+Process an MCMC Sample of Clusterings, 2022.
+URL https://CRAN.
+R-project.org/package=mcclust. R package version 1.0.1.
+A. Fritsch and K. Ickstadt. Improved criteria for clustering based on the posterior similarity matrix. Bayesian
+Analysis, 4:367–391, 2009.
+I. Gollini and T. B. Murphy. Joint modeling of multiple network views. Journal of Computational and
+Graphical Statistics, 25:246–265, 2016.
+S. Guha and A. Rodriguez. Bayesian regression with undirected network predictors with an application to
+brain connectome data. Journal of the American Statistical Association, 116(534):581–593, 2021.
+M. S. Handcock, A. E. Raftery, and J. M. Tantrum. Model-based clustering for social networks. Journal of
+the Royal Statistical Society: Series A, 170:301–354, 2007.
+P. Hoff. Modeling homophily and stochastic equivalence in symmetric relational data. In Advances in Neural
+Information Processing Systems, pages 657–664, 2007.
+23
+
+P. D. Hoff. Bilinear mixed-effects models for dyadic data. Journal of the American Statistical Association,
+100:286–295, 2005.
+P. D. Hoff. Multiplicative latent factor models for description and prediction of social networks. Computa-
+tional and Mathematical Organization Theory, 15:261–272, 2009.
+P. D. Hoff, A. E. Raftery, and M. S. Handcock. Latent space approaches to social network analysis. Journal
+of the American Statistical Association, 97:1090–1098, 2002.
+D. N. Hoover. Row-column exchangeability and a generalized model for probability. Exchangeability in
+probability and statistics (Rome, 1981), pages 281–291, 1982.
+M. Kim and J. Leskovec. Multiplicative attribute graph model of real-world networks. Internet Mathematics,
+8:113–160, 2012.
+E. D. Kolaczyk and G. Csárdi. Statistical Analysis of Network Data with R. Springer, 2nd edition, 2020.
+P. N. Krivitsky and M. S. Handcock. latentnet: Latent Position and Cluster Models for Statistical Net-
+works. The Statnet Project (https://statnet.org), 2022. URL https://CRAN.R-project.
+org/package=latentnet. R package version 2.10.6.
+P. N. Krivitsky, M. S. Handcock, A. E. Raftery, and P. D. Hoff. Representing degree distributions, clustering,
+and homophily in social networks with latent cluster random effects models. Social Networks, 31:204–
+213, 2009.
+J. W. Lau and P. J. Green. Bayesian model based clustering procedures. Journal of Computational and
+Graphical Statistics, 16:526–558, 2007.
+E. Lazega. The Collegial Phenomenon: The Social Mechanisms of Cooperation Among Peers in a Corporate
+Law Partnership. Oxford University Press, 2001.
+C. Lee and D. J. Wilkinson. A review of stochastic block models and extensions for graph clustering. Applied
+Network Science, 4:1–50, 2019.
+J.-B. Leger. Blockmodels: A R-package for estimating in Latent Block Model and Stochastic Block Model,
+with various probability functions, with or without covariates. arXiv:1602.07587, 2016.
+Crystal D Linkletter. Spatial process models for social network analysis. PhD thesis, Citeseer, 2007.
+P. W. MacDonald, E. Levina, and J. Zhu. Latent space models for multiplex networks with shared structure.
+arXiv:2012.14409, 2020.
+24
+
+P. W. MacDonald, E. Levina, and J. Zhu. Latent space models for multiplex networks with shared structure.
+Biometrika, 109:683–706, 2022.
+C. Matias and S. Robin. Modeling heterogeneity in random graphs through latent space models: a selective
+review. ESAIM: Proceedings and Surveys, 47:55–74, 2014.
+S. Minhas, P. D. Hoff, and M. D. Ward. Inferential approaches for network analysis: AMEN for latent factor
+models. Political Analysis, 27:208–222, 2019.
+K. Nowicki and T. A. B. Snijders. Estimation and prediction for stochastic block structures. Journal of the
+American Statistical Association, 96:1077–1087, 2001.
+M. Plummer.
+rjags:
+Bayesian Graphical Models using MCMC, 2022.
+URL https://CRAN.
+R-project.org/package=rjags. R package version 4-13.
+R Core Team. R: A Language and Environment for Statistical Computing. R Foundation for Statistical
+Computing, Vienna, Austria, 2021. URL https://www.R-project.org/.
+R. Rastelli. GreedyEPL: Greedy Expected Posterior Loss, 2021. URL https://CRAN.R-project.
+org/package=GreedyEPL. R package version 1.2.
+R. Rastelli and N. Friel. Optimal Bayesian estimators for latent variable cluster models. Statistics and
+Computing, 28:1169–1186, 2018.
+M. Salter-Townshend and T. H. McCormick. Latent space models for multiview network data. The Annals
+of Applied Statistics, 11:1217–1244, 2017.
+M. Schweinberger and A. B. Snijders. Settings in social networks: a measurement model. Sociological
+Methodology, 33:307–341, 2003.
+J. Sosa. A review on latent space models for social networks. Revista Colombiana de Estadistica, 44:
+171–200, 2021.
+J. Sosa and B. Betancourt. A latent space model for multilayer network data. Computational Statistics &
+Data Analysis, 169:107432, 2022.
+L. Wang, Z. Zhang, and D. Dunson. Common and individual structure of brain networks. Annals of Applied
+Statistics, 13:85–112, 2019.
+Y. J. Wang and G. Y. Wong. Stochastic blockmodels for directed graphs. Journal of the American Statistical
+Association, 82:8–19, 1987.
+25
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+Perspective: How to overcome dynamical density functional theory
+Daniel de las Heras,1 Toni Zimmermann,1 Florian Samm¨uller,1 Sophie Hermann,1 and Matthias Schmidt1
+1Theoretische Physik II, Physikalisches Institut, Universit¨at Bayreuth, D-95447 Bayreuth, Germany
+(Dated: 28 January 2023)
+We argue in favour of developing a comprehensive dynamical theory for rationalizing, predicting,
+and machine learning nonequilibrium phenomena that occur in soft matter. To give guidance for
+navigating the theoretical and practical challenges that lie ahead, we discuss and exemplify the
+limitations of dynamical density functional theory. Instead of the implied adiabatic sequence of
+equilibrium states that this approach provides as a makeshift for the true time evolution, we posit
+that the pending theoretical tasks lie in developing a systematic understanding of the dynamical
+functional relationships that govern the genuine nonequilibrium physics. While static density func-
+tional theory gives a comprehensive account of the equilibrium properties of many-body systems,
+we argue that power functional theory is the only present contender to shed similar insights into
+nonequilibrium dynamics, including the recognition and implementation of exact sum rules that
+result from the Noether theorem. As a demonstration of the power functional point of view, we
+consider an idealized steady sedimentation flow of the three-dimensional Lennard-Jones fluid and
+machine-learn the kinematic map from the mean motion to the internal force field. This proof of con-
+cept demonstrates the significant potential of machine learning the inherent functional relationships
+that govern nonequilibrium many-body physics.
+I.
+INTRODUCTION
+The coupled dynamics of the microscopic degrees of
+freedom in typical soft matter systems generates a wide
+array of relevant and also often unsolved nonequilibrium
+phenomena [1, 2].
+One central quantity for the char-
+acterization of self-assembly and structure formation in
+complex systems is the microscopically resolved one-body
+density distribution ρ(r, t), where r indicates position
+and t denotes time. The “density profile” ρ(r, t) acts as a
+central order parameter both due to its intuitive physical
+interpretation and clearcut mathematical definition [3].
+According to the dynamical density functional theory
+(DDFT), as originally proposed by Evans in 1979 [4],
+the time evolution of the microscopic density profile is
+assumed to be determined by the following partial differ-
+ential equation:
+∂ρ(r, t)
+∂t
+= γ−1∇ · ρ(r, t)∇
+� δF[ρ]
+δρ(r, t) + Vext(r, t)
+�
+.
+(1)
+Here γ is a friction constant, F[ρ] is an intrinsic free
+energy functional that depends functionally on the den-
+sity profile, and the external potential Vext(r, t) repre-
+sents interactions of the system with the environment.
+The system is set into motion by a temporal variation
+of Vext(r, t), such as e.g. step-like switching at an initial
+time.
+The time evolution according to Eq. (1) conserves the
+particle number locally and hence it constitutes dynam-
+ics of model B type [5].
+In standard applications one
+starts with an equilibrium state of the system and then
+the dynamics are monitored on the basis of numerical
+time integration of Eq. (1). In order to provide reference
+data and to allow for the generation of benchmark results
+to assess the quality of the theory, resorting to many-
+body computer simulations is common, with overdamped
+Brownian dynamics (BD) being a popular choice (Ref. [6]
+describes a modern and stable algorithm). Comparison
+of DDFT data with experimental results are more scarce,
+but notable exceptions include non-equilibrium sedimen-
+tation of colloids [7], the self-diffusion of particles in com-
+plex fluids [8], and the bulk dynamics of Brownian hard
+disks [9].
+The DDFT time evolution reaches a stationary state
+if the gradient on the right hand side of Eq. (1) vanishes,
+i.e. provided that the expression inside of the parentheses
+is constant:
+δF[ρ]
+δρ(r) + Vext(r) = µ.
+(2)
+Here we have dropped the dependence on time in the
+notation, as the situation is now static. The constant µ
+can be identified with the chemical potential, which in a
+grand canonical statistical mechanical setting is the con-
+jugate control parameter of the mean particle number.
+Equation (2) is exact in equilibrium, as was shown by
+Evans [4]. He proved the equilibrium intrinsic free en-
+ergy functional F[ρ] to exist, to be unique, and to form
+the starting point for a modern equilibrium theory of
+spatially inhomogeneous liquids and crystals [10, 11].
+In practice one needs to rely on approximations
+for F[ρ], given a microscopic fluid model under consid-
+eration.
+Once one has solved Eq. (2) for given values
+of µ and temperature T (the dependence of F[ρ] on T
+is suppressed in the notation), then in principle com-
+plete knowledge of the thermal system is available. The
+value of the density functional F[ρ] is the true intrinsic
+free energy, and higher-order correlation functions are
+determined via higher-order derivatives of the free en-
+ergy functional or via test-particle procedures. In par-
+ticular two-body correlations functions, such as the bulk
+pair correlation function g(r) as well as its generalization
+to inhomogeneous systems are accessible. These exhibit
+defining characteristics of liquids and more general soft
+arXiv:2301.12156v1  [cond-mat.soft]  28 Jan 2023
+
+2
+matter systems and they are formally fully contained in
+the static density functional theory framework.
+Together with a number of available reliable approxi-
+mate free energy functionals, density functional theory is
+a powerful theoretical framework that has been used to
+elucidate much intricate and complex behaviour in soft
+matter.
+Recent representative highlights include trac-
+ing hydrophobicity to critical drying at substrates [12–
+14], resolving three-dimensional structures of electrolyte
+aqueous solutions near surfaces [15, 16], and addressing
+the magnitude of the decay lengths in electrolytes [17].
+Rosenfeld’s celebrated hard sphere fundamental measure
+free energy functional [18–21] is at the core of much of
+this research activity.
+In the following we wish to address whether or not
+the DDFT has the prowess to play a similar role in
+nonequilibrium, as is often at least implicitly assumed.
+We demonstrate on the basis of an explicit and generic
+example, i.e., that of uniaxial compressional flow of the
+three-dimensional Lennard-Jones fluid, that the DDFT is
+fundamentally flawed and that in reality, as represented
+by many-body simulations, recognizing the flow field as
+a further relevant degree of freedom is required to rep-
+resent true nonequilibrium. These conclusions are based
+on analytical power functional approximations, adaptive
+BD simulation data, and explicit machine learning of the
+power functional map from motion to the interparticle
+one-body force field.
+This Perspective is organized as follows. We first make
+some key aspects of DDFT explicit in Sec. II and describe
+several prominent shortcomings of this theory. We then
+give an account of how to go towards the formally exact
+one-body dynamics in Sec. III and provide in Sec. IV a
+description of key aspects of the power functional frame-
+work, which as we wish to argue overcomes the funda-
+mental defects of DDFT. We describe the exemplary sta-
+tionary compressional flow situation in Sec. V and lay
+put the application of Noether’s theorem in this statis-
+tical mechanical setting in Sec. VI. We present machine
+learning results for the kinematic functional relationships
+of the streaming Lennard-Jones fluid in Sec. VII. We give
+conclusion and an outlook in Sec. VIII.
+II.
+LIMITS AND LIMITATIONS OF
+ADIABATIC DYNAMICS
+We go into some detail and describe why the DDFT
+represents adiabatic dynamics in the sense of a temporal
+sequence of spatially inhomogeneous equilibrium states.
+The equilibrium intrinsic free energy functional splits into
+ideal and excess (over ideal gas) contributions according
+to F[ρ] = Fid[ρ] + Fexc[ρ]. Here the excess free energy
+functional Fexc[ρ] accounts for the effects of the inter-
+particle interactions on the equilibrium properties of the
+system and it is in general unknown and requires approx-
+imations to be made. The ideal gas free energy functional
+however is exactly given by
+Fid[ρ] = kBT
+�
+drρ(r) ln(ρ(r)Λ3) − 1],
+(3)
+where kB denotes the Boltzmann constant, Λ is the
+thermal de Broglie wavelength, and we consider three-
+dimensional systems. The functional derivative, as it is
+relevant for Eq. (1), is δFid[ρ]/δρ(r) = kBT ln(ρ(r)Λ3).
+When disregarding the excess contribution and in-
+serting this result alone into the DDFT equation
+of motion (1), its right hand side becomes γ−1∇ ·
+ρ(r, t)∇[kBT ln(ρ(r, t)Λ3) + Vext(r, t)]. This can be re-
+written further such that for the case of the ideal gas,
+where Fexc[ρ] = 0 and F[ρ] = Fid[ρ], the equation of
+motion (1) attains the following form:
+∂ρ(r, t)
+∂t
+= D0∇2ρ(r, t) − ∇ · ρ(r, t)fext(r, t)/γ.
+(4)
+Here D0 = kBT/γ is the diffusion constant, ∇2 is the
+Laplace operator and the external force field is given
+(here) as fext(r, t) = −∇Vext(r, t). Equation (4) is the
+exact drift-diffusion equation for overdamped motion of
+a mutually noninteracting system, i.e., the ideal gas.
+Besides Evans’ original proposal [4] based on the con-
+tinuity equation and undoubtedly his physical intuition,
+derivations of the DDFT (1) were founded much more
+recently on Dean’s equation of motion for the density op-
+erator [22], the Smoluchowski equation [23], a stationary
+action principle for the density [24], the projection op-
+erator formalism [25], a phase-space approach [26], the
+mean-field approximation [27], a local equilibrium as-
+sumption [28], and a non-equilibrium free energy [29].
+The question of the well-posedness of the DDFT was ad-
+dressed [30] and several extensions beyond overdamped
+Brownian dynamics were formulated, such as e.g. for dy-
+namics including inertia [31–34] and for particles that ex-
+perience hydrodynamic interactions [34, 35] or undergo
+chemical reactions [36, 37].
+The DDFT was also used beyond the description of flu-
+ids, such as e.g. for opinion dynamics [38] and epidemic
+spreading [39].
+Recent reviews of DDFT are given in
+Refs. [40, 41]. The theory is put into a wider perspective,
+together with much background pedagogical material in
+Ref. [42]. A modern and well-accessible account of the
+general strategy of dynamical coarse-graining in statisti-
+cal physics, of which the DDFT can be viewed as being a
+representative, has recently been given by Schilling [43].
+The fact that both the static limit for the fully in-
+teracting system (2) as well as the full dynamics of the
+noninteracting system (4) are exact, taken together with
+the heft of the DDFT literature, appears to give much
+credibility to the equation of motion (1). However, de-
+spite the range of theoretical techniques employed [22–29]
+neither of these approaches has provided us with a con-
+crete way of going beyond Eq. (1). Apart from several
+case-by-case and rather ad hoc modifications, no system-
+atic or even only practical identification of what is miss-
+ing has been formulated. (We turn to power functional
+
+3
+theory in Sec. IV.) This is a problematic situation as
+two defects of Eq. (1) are immediately obvious upon in-
+spection: i) the description is local in time and there is
+no natural mechanism for the inclusion of memory while
+time-locality is not sufficient for general nonequilibrium
+situations; ii) only flow that leads to direct changes in
+the density profile is captured and hence effects of rota-
+tional flow, such as shearing, as well as of nonequilibrium
+effects in compression and expansion are lost (see below).
+Here we argue that these defects are indicative of a
+broader failure of Eq. (1) to describe nonequilibrium
+physics. We show that the DDFT is only fit to describe
+situations in which the dynamics follow an adiabatic path
+through a sequence of equilibrium states. The description
+of genuine nonequilibrium dynamics in a functional set-
+ting on the one-body level rather requires recognition of
+the local velocity field as a further relevant physical vari-
+able besides the density profile, and this is provided by
+power functional theory [42]. Before laying out key prin-
+ciples of this approach in Sec. IV, we first describe the mi-
+croscopically sharp coarse-graining on the one-body level
+of correlation functions.
+III.
+TOWARDS EXACT ONE-BODY
+DYNAMICS
+Evans based his original derivation [4] of Eq. (1) on the
+continuity equation,
+∂ρ(r, t)
+∂t
+= −∇ · J(r, t),
+(5)
+where J(r, t) is the microscopically resolved one-body
+current distribution. Equation (5) is exact in a variety of
+contexts, including overdamped Brownian dynamics, as
+described either on the Fokker-Planck level by the Smolu-
+chowski equation or by the corresponding overdamped
+Langevin equation that governs the trajectories, as they
+are realized in simulation work [6]. For BD the one-body
+current distribution is given exactly by [42]:
+γJ(r, t) = −kBT∇ρ(r, t) + Fint(r, t) + ρ(r, t)fext(r, t).
+(6)
+This identity expresses the force density balance of the
+negative friction force density (left hand side) with the
+force densities due to ideal thermal diffusion, interparti-
+cle interactions, and external influence (three contribu-
+tions on the right hand side). Here the interparticle force
+density distribution is given by the statistical average
+Fint(r, t) = −
+� �
+i
+δ(r − ri)∇iu(rN)
+����
+t,
+(7)
+where the angular brackets indicate an average at fixed
+time t over the nonequilibrium many-body distribu-
+tion, u(rN) is the interparticle interaction potential
+that depends on all particle position coordinates rN ≡
+r1, . . . , rN and ∇i indicates the derivative with respect to
+the position ri of particle i. The formulation of Eq. (7) is
+based on the concept of static operators and a dynami-
+cally evolving probability distribution. This is analogous
+to the Schr¨odinger picture of quantum mechanics. The
+Heisenberg picture is more closely related to simulation
+work. Here the probability distribution is that of the ini-
+tial microstates and the operators move forward in time,
+i.e., the position ri(t) of particle i changes over the course
+of time. Then the Dirac distribution in Eq. (7) becomes
+δ(r − ri(t)), with the generic position variable r however
+remaining static. The forces are those that act in the
+given microstate rN(t) at time t, i.e., the interparticle
+force on particle i at time t is −∇iu(rN(t)).
+In practice, using BD simulations, carrying out the
+average in Eq. (7) requires to build the mean over suf-
+ficiently many separate realizations of the microscopic
+evolution of the many-body system that differ in the ini-
+tial state and in the realization of the thermal noise. As
+Eq. (7) measures both the probability to find particle i at
+position r (via the delta function) and the interparticle
+force that acts via the negative gradient −∇iu(rN), we
+refer to Fint(r, t) as a force density. The corresponding
+force field fint(r, t) is obtained by simple normalization
+with the density profile, i.e. fint(r, t) = Fint(r, t)/ρ(r, t).
+Building this ratio scales out the probability effect and
+the force field then carries physical units of force, i.e.
+energy per length.
+In equilibrium the definition (7) remains intact. Com-
+plementing the statistical average, static density func-
+tional theory allows to express the equilibrium force den-
+sity as being functionally dependent on the density pro-
+file via the functional derivative of the excess free energy
+functional according to:
+Fint(r)
+��
+eq = −ρ(r)∇δFexc[ρ]
+δρ(r) .
+(8)
+Crucially, and in contrast to Eq. (7), here the internal
+force density is directly expressed as a density functional.
+This dependence has superseded the original dependence
+on the external potential, as is manifest in the probability
+distribution for building the average (7) in equilibrium.
+As a self-consistency check we insert the force density
+functional (8) into the equilibrium limit of the force den-
+sity balance (6). The current vanishes in the equilibrium
+case, J(r, t) ≡ 0, and we obtain
+−kBT∇ρ(r) + Fint(r)|eq + ρ(r)fext(r) = 0.
+(9)
+This result is independent of time and it consti-
+tutes the gradient of the static Euler-Lagrange equa-
+tion (2) when divided by the density profile.
+(Insert
+Eq. (8), identify the ideal gas contribution −kBT∇ρ(r) =
+−ρ(r)δFid[ρ]/δρ(r), and divide by ρ(r).)
+The classical
+force density balance result (9) by Yvon, Born and Green
+[3] has recently been derived from systematically address-
+ing thermal Noether invariance [44, 45] against locally
+resolved spatial deformations of the statistical ensemble
+[46–48], as also valid quantum mechanically [48] and at
+
+4
+second order in the displacement field [49, 50]; we give a
+brief account of this theory in Sec. VI below.
+A naive transfer of Eq. (8) to nonequilibrium lets
+one simply evaluate the equilibrium excess free energy
+functional at the instantaneous nonequilibrium density
+ρ(r, t). In order to separate this contribution from true
+static equilibrium, we refer to this force density as being
+adiabatic (subscript “ad”) and to be defined as
+Fad(r, t) = −ρ(r, t)∇δFexc[ρ]
+δρ(r, t) .
+(10)
+We recall that the right hand side offers a concrete com-
+putational structure that is of practical usefulness in ac-
+tual applications, as considerable knowledge about ap-
+proximative forms of the excess free energy density func-
+tional Fexc[ρ] is available. Using the adiabatic force den-
+sity as a proxy for the true nonequilibrium intrinsic force
+density distribution (7), i.e. setting Fint(r, t) = Fad(r, t)
+in the force density balance (6) together with the conti-
+nuity equation (5) leads to the DDFT equation of mo-
+tion (1). The adiabatic force density approximation is
+uncontrolled though and the theory inherently yields the
+dynamics as an adiabatic sequence of equilibrium states.
+Surely, more than 40 years after the conception of the
+DDFT [4], we have to be able to do better!
+IV.
+POWER FUNCTIONAL TECHNIQUES
+Power functional theory [42] offers a concrete math-
+ematical structure to go forward.
+We describe the es-
+sential steps that enable one to go beyond the DDFT
+and to hence address a significantly expanded realm of
+nonequilibrium physics which Eq. (1) is oblivious of.
+The interparticle force density profile (7) is identified
+to consist of two contributions according to:
+Fint(r, t) = Fad(r, t) + Fsup(r, t).
+(11)
+Here Fad(r, t) is the adiabatic force density profile, as
+given formally via the explicit equilibrium free energy
+derivative (10) and directly accessible in simulations via
+the custom flow method [51, 52]. The custom flow al-
+gorithm allows to systematically construct a hypotheti-
+cal adiabatic (equilibrium) system that shares its density
+profile with the nonequilibrium system at the given time.
+Then sampling the internal force density in the adiabatic
+system yields results for Fad(r, t).
+The second, superadiabatic contribution in Eq. (11),
+Fsup(r, t), contains all effects that are not expressible
+as an instantaneous density functional.
+This includes
+forces that lead to viscous and to nonequilibrium struc-
+ture forming phenomena, as we exemplify below in a con-
+crete model compressional flow situation. Formally, the
+superadiabatic force density is generated from the su-
+peradiabatic excess free power functional P exc
+t
+[ρ, J] upon
+functional differentiation with respect to the one-body
+current via [42, 53]:
+Fsup(r, t) = −ρ(r, t)δP exc
+t
+[ρ, J]
+δJ(r, t)
+.
+(12)
+The functional dependence of P exc
+t
+[ρ, J] on the density
+and current is causal, i.e. on the values of these fields
+at prior times to t; density and current need to satisfy
+the continuity equation. Upon using Eqs. (11) the force
+density balance (6) attains the following form:
+γJ(r, t) = −kBT∇ρ(r, t) + Fad(r, t)
++ Fsup(r, t) + ρ(r, t)fext(r, t).
+(13)
+This
+relationship
+holds
+beyond
+gradient
+forms
+of
+fext(r, t), i.e. for external force fields that contain non-
+conservative contributions.
+Crucially Fsup(r, t) will in
+general also acquire nonconservative contributions, such
+as e.g. damping effects that represent viscous behaviour.
+Moreover, nonequilibrium structure-forming effects will
+also arise in general. These affect directly the shape of
+the density profile, whether this evolves in time or per-
+sists in a nonequilibrium steady state.
+If one wishes to eliminate the explicit occurrence of the
+current from the dynamics, then inputting the force den-
+sity balance (13) into the continuity equation (5) leads
+to the following formally exact form of the equation of
+motion for the density profile:
+∂ρ(r, t)
+∂t
+= D0∇2ρ(r, t) + ∇ · ρ(r, t)
+γ
+∇δFexc[ρ]
+δρ(r, t)
+− ∇ · ρ(r, t)
+γ
+[fsup(r, t) + fext(r, t)].
+(14)
+Here it is apparent that the superadiabatic force field
+fsup(r, t) = Fsup(r, t)/ρ(r, t) has a direct effect on the
+system dynamics. The effect is similar to that of the ex-
+ternal force field. Crucially though, both force fields are
+independent of each other: the external force field rep-
+resents a prescribed and inert influence on the system.
+In contrast, the superadiabatic force field is an emer-
+gent phenomenon that arises due to interparticle inter-
+actions and, from the functional point of view, depends
+non-locally in position and causally in time on the one-
+body density and on the current profile.
+Although setting fsup(r, t) = 0 yields the DDFT (1),
+the superadiabatic force field fsup(r, t) was demonstrated
+to exist [54–60] and in general to play a major role in
+the dynamics on the one-body level and, based on test-
+particle concepts [61–66] also for two-body correlation
+functions [67–69] and for active matter [70–74]. Both the
+flow properties as well as the spatial structure formation
+in the system are affected.
+To reveal additional physics, it is useful to split into
+“structural” and “flow” contributions. This was estab-
+lished e.g. for complex flow patterns that occur in driven
+BD [55, 59], for active Brownian particles which form
+a self-sustained interface at motility-induced phase co-
+existence [70–74], as well as very recently for a sheared
+
+5
+FIG. 1. Illustration of unidirectional compressional flow of a liquid. The three-dimensional system is set into motion (red
+arrows) by the action of an external force profile fext(x) (blue arrows) which acts along the x-axis. The system retains planar
+geometry such that spatial inhomogeneities only occur as a function of x. The density profile ρ(x) (orange curve) and the
+velocity profile v(x) (red curve) are both stationary in time but inhomogeneous in position.
+The local one-body current
+J(x) = ρ(x)v(x) = const and as a result the system is in a nonequilibrium steady state. The corresponding adiabatic system
+is in equilibrium (it has no mean flow) and it has by construction an unchanged density profile ρ(x). In the adiabatic system
+the spatial variation of ρ(x) is stabilized by the action of an external force field −∇Vad(x) (olive arrows), which acts solely in
+the adiabatic system.
+three-body colloidal gel former [60]. Before we demon-
+strate these concepts for an example of steady nonequi-
+librium below, we first describe two simple model power
+functionals that respectively generate structure and vis-
+cously dampen the motion and that, as we will see, give
+a good account of the nonequilibrium flow considered be-
+low.
+We concentrate on the low-order terms that are rel-
+evant for compressional/extensional flow, i.e., for situa-
+tions where ∇ · v(r, t) ̸= 0.
+We focus on cases where
+there is no rotational motion (such as shearing) and hence
+∇ × v(r, t) = 0.
+The velocity gradient superadiabatic
+power functional consists of a sum,
+P exc
+t
+[ρ, v] = P flow
+t
+[ρ, v] + P str
+t
+[ρ, v].
+(15)
+Here the flow and structural [55, 59] contributions are
+approximated, respectively, by the following time-local
+(Markovian) and space-semilocal (i.e. involving ∇) forms
+P flow
+t
+[ρ, v] = η
+2
+�
+dr[ρ(r, t)∇ · v(r, t)]2,
+(16)
+P str
+t
+[ρ, v] = −χ
+3
+�
+dr[ρ(r, t)∇ · v(r, t)]3,
+(17)
+where the overall prefactors η and χ control the respec-
+tive magnitude.
+The flow functional (16) is quadratic
+both in density and in the velocity field; the structural
+functional (17) is of cubic order in each of these variables.
+Explicit higher-order functionals exist [59] and they be-
+come relevant when driving the system strongly. We will
+return to the consequences of Eqs. (16) and (17) after
+laying out in Sec. V the actual flow situation that we use
+as a model to exemplify the implications for the physics.
+Before doing so, we briefly describe several further key
+aspects of the power functional framework.
+Power functional theory provides a formal framework
+for the inclusion of time- and space-nonlocal dynamics
+[56, 68, 79]. While Eq. (12) applies to overdamped dy-
+namics, the acceleration field becomes a further relevant
+degree of freedom if inertia are relevant [78–81] whether
+classically in molecular dynamics [78, 79] or in quantum
+dynamics [80, 81].
+Here the memory functions act as
+convolution kernels on specific kinematic fields and rota-
+tional and compressional contributions to the dynamics
+are genuinely built in. As laid out above, the framework
+is based on an exact variational concept [42, 53], and the
+resulting functional mapping was shown to be explicitly
+accessible in many-body simulation via the custom flow
+computer simulation method [51, 52].
+Even
+simple
+mathematical
+model
+forms
+for
+the
+nonequilibrium contribution to the power functional,
+such as Eqs. (16) and (17), already capture essential
+physics (as we demonstrate below) and dynamical two-
+body correlation functions are accessible via test particle
+dynamics [8, 9, 61–69]. The power functional is thereby
+not to be confused with the often vague concept of a
+
+v(C
+p(α)
+noneguilibrium
+fext(α)
+(α) PeA△
+equilibrium6
+“nonequilibrium free energy”.
+The proper equilibrium
+free energy functional does play a central role in power
+functional theory though, via providing the description
+of the adiabatic reference state [42], see the generation
+of the force density distribution via functional differenti-
+ation (10), as is relevant for the interparticle force split-
+ting (11), and the full density equation of motion (14).
+The relevance of superadiabatic contributions to the
+dynamics, i.e. of those effects that lie beyond Eq. (1), has
+been amply demonstrated in the literature [54–59, 67–
+69]. Both adiabatic and superadiabatic effects arise from
+integrating out the dynamical degrees of freedom of the
+many-body problem.
+Ensemble differences between canonical dynamics and
+grand canonical equilibrium have been systematically ad-
+dressed [75–77] and these do not account for the observed
+differences between adiabatic and superadiabatic dynam-
+ics.
+The kinematic dependence on the motion of the
+system arises formally [42], it can be explicitly traced
+in many-body computer simulation work [59], and it
+is amenable to machine learning, as we demonstrate in
+Sec. VII. Before doing so, we first formulate the represen-
+tative flow problem that we will use to apply the above
+concepts.
+V.
+NONEQUILIBRIUM STEADY STATES
+We restrict ourselves to flow situations with one-body
+fields that are inhomogeneous in position but indepen-
+dent of time, i.e. ρ(r) and v(r). Then trivially ∂ρ(r)/∂t =
+0 and the continuity equation (5) constrains both fields
+to satisfy ∇ · [ρ(r)v(r)] = 0. As a representative case
+we illustrate in Fig. 1 a nonequilibrium steady state of a
+three-dimensional liquid undergoing unidirectional com-
+pressional flow. Flow along a single given direction occurs
+e.g. under the influence of gravity, where sedimentation
+of colloids leads to both compression in the lower parts of
+the sample and expansion in the upper parts of the sam-
+ple. Here we disregard transient phenomena and investi-
+gate an idealized periodic system, where flowing steady
+states can form.
+In order to elucidate the physics in such setups, we fol-
+low the splitting (15) of the superadiabatic power func-
+tional into structural and flow contributions and hence
+decompose the superadiabatic force field accordingly as
+fsup(r) = fstr(r) + fflow(r),
+(18)
+where the right hand side consists of the nonequilib-
+rium structural force field fstr(r) and the flow force
+field fflow(r).
+Both of these force contributions arise
+from the microscopic interparticle interactions, as coarse-
+grained in a microscopically sharp way to the one-body
+level.
+We lay out in the following the benefits of the
+structure-flow splitting (18) and its definition via flow
+reversal symmetry.
+First, on the more practical level, Eq. (18) allows to
+carry out a corresponding splitting of the force density
+balance (13) [we divide by ρ(r) to obtain force fields].
+The result is a set of two coupled equations of motion,
+with one of them depending explicitly on the velocity
+profile and the second one depending explicitly on the
+density profile:
+γv(r) = fflow(r) + fext,f(r),
+(19)
+0 = fstr(r) − kBT∇ ln ρ(r) + fad(r) + fext,s(r). (20)
+Building the sum of Eqs. (19) and (20) and multiplying
+by the density profile restores the full force density bal-
+ance (13). The external force field is split according to
+fext(r) = fext,f(r) + fext,s(r), where the two terms couple
+to the flow via fext,f(r) in Eq. (19) and to the structure
+via fext,s(r) in Eq. (20).
+On the superficial level the two equations (19) and
+(20) appear to be independent of each other, as no sin-
+gle field appears explicitly in both equations. However,
+the two equations are indeed intimately coupled to each
+other by the interparticle interactions, as represented by
+both the adiabatic and the two superadiabatic (flow and
+structural) force fields. These three intrinsic force con-
+tributions provide the physical representation of the true
+nonequilibrium steady state dynamics.
+The flow-structure splitting (18) is uniquely deter-
+mined by the symmetry properties of the forces upon
+motion reversal of the system [59].
+Motion reversal is
+a discrete symmetry operation, and hence different from
+continuous invariances where Noether’s theorem applies
+[44–50].
+One considers a “reversed” system, which is
+also in steady state and possesses an unchanged den-
+sity profile ρ(r). The flow, however, is directed against
+the velocity orientation in the original “forward” system.
+Hence the velocity profile in the reversed system is sim-
+ply −v(r). As a result the current also acquires a mi-
+nus sign, −ρ(r)v(r), which however does not affect the
+(vanishing) divergence, ∇ · [−ρ(r)v(r)] = 0. Thus the re-
+versed state indeed is stationary. The two superadiabatic
+contributions are then defined to be unchanged [fstr(r)]
+and inverted [−fflow(r)] in the reversed system. Conse-
+quentially, the superadiabatic force field in the reversed
+system is the difference fstr(r) − fflow(r).
+Analyzing the symmetry properties of the adiabatic
+force field is straightforward.
+We recall that fad(r) is
+a density functional via Eq. (10). The density profiles
+in the forward and in the reversed systems are identical
+though. Hence fad(r) is invariant under motion reversal.
+Motion reversal is a useful device in order to i) rationalize
+the nonequilibrium behaviour according to the split force
+balance (19) and (20), and to ii) classify the dependence
+of superadiabatic forces on the velocity field into even
+powers, which constitute fstr(r), and odd powers, which
+form fflow(r).
+We can demonstrate this mechanism explicitly on the
+basis of the above flow and structural power functionals
+(16) and (17). Superadiabatic force fields are generated
+via the functional derivative (12) with respect to the cur-
+rent or, analogously, by functionally deriving by v(r, t)
+
+7
+and dividing the result by ρ(r, t). The resulting supera-
+diabatic one-body force field consists of two components.
+The viscous flow force and [55, 58] and the structural
+force follow respectively as
+fflow(r) =
+η
+ρ(r)∇[ρ(r)2∇ · v(r)],
+(21)
+fstr(r) = − χ
+ρ(r)∇{ρ(r)3[∇ · v(r)]2},
+(22)
+where Eq. (21) is odd (linear) and Eq. (22) is even
+(quadratic) in the derivatives of the velocity field, as de-
+sired.
+One might wonder where all this genuine nonequilib-
+rium physics leaves the DDFT! Some readers will find the
+instantaneous dynamics, as generated from an adiabatic
+free energy according to (1), to be more appealing and in-
+tuitive than the thinking in terms of the above described
+apparently intricate functional relationships.
+Why not
+live with Eq. (1), use it, and simply accept its defects?
+In order to address this question and to demonstrate why
+this path is severely restricted from the outset, we turn
+in Sec. VII to an explicit demonstration of the functional
+relationship that governs the nonequilibrium physics, i.e.
+the kinematic functional map from the one-body mean
+motion to the internal force field. Before doing so, we
+demonstrate that Noether’s theorem of invariant varia-
+tions has much to say about our present setup.
+VI.
+NOETHER FORCE SUM RULES
+We discuss one of the arguably simplest cases of ex-
+ploitation of the inherent symmetries of a thermal many-
+body system, that of global translational invariance of its
+statistical mechanics [44, 45]. We consider a “shifting”
+transformation, where all particle coordinates change ac-
+cording to the map ri → ri + ϵ, where ϵ = const. This
+uniform shifting operation leaves all interparticle dis-
+tance unchanged, ri−rj → (ri+ϵ)−(rj+ϵ) ≡ ri−rj. As
+a consequence the interparticle potential is invariant un-
+der the transformation, which we can express as the iden-
+tity u(r1, . . . , rN) = u(r1 + ϵ, . . . , rN + ϵ). Here equality
+holds irrespectively of the magnitude and the direction
+of the shifting vector ϵ.
+The Noether argument proceeds with a twist.
+De-
+spite the absence of dependence on ϵ, we can neverthe-
+less differentiate both sides of the equation with respect
+to ϵ and the result will be a valid identity. We obtain
+0 = ∂u(ri + ϵ, . . . , rN + ϵ)/∂ϵ = �
+i ∇iu(r1, . . . , rN),
+where we have set ϵ = 0 after taking the derivative. We
+multiply by −1 and insert 1 =
+�
+drδ(r−ri), which yields
+−
+�
+dr
+�
+i
+δ(r − ri)∇iu(rN) = 0.
+(23)
+The expression on the left hand side allows to identify
+the locally resolved interparticle force operator ˆFint(r) =
+− �
+i δ(r − ri)∇iu(rN), such that Eq. (23) attains the
+form
+�
+drˆFint(r) = 0. This identity holds for each mi-
+crostate rN and hence it remains trivially valid upon av-
+eraging over the many-body distribution function, irre-
+spective of whether this is in- or out-of-equilibrium. We
+can hence conclude the vanishing of the global interpar-
+ticle force, expressed as the integral over the mean force
+density Fint(r) = ⟨ˆFint(r)⟩ as
+�
+drFint(r, t) = 0.
+(24)
+Equation (24) holds at all times t and it can be viewed as
+a consequence of Newton’s third law, see the discussion in
+Ref. [44]. Using the adiabatic-superadiabatic force split-
+ting (11) one can further conclude that the both global
+contributions need to vanish individually,
+�
+drFad(r, t) = 0,
+(25)
+�
+drFsup(r, t) = 0.
+(26)
+The proof can either be based on the fact that Eq. (25)
+is merely Eq. (24) for the special case of an equilibrium
+system, from which then Eq. (26) follows from the force
+splitting (11).
+Alternatively and starting from a very
+fundamental point of view, the global translational in-
+variance of the excess free energy functional Fexc[ρ] and
+of the superadiabatic free power functional P exc
+t
+[ρ, v],
+here considered instantaneously at time t, lead directly
+to Eqs. (25) and (26), see Refs. [44, 45] for the detailed
+account.
+It is interesting to apply the Noether concept to the
+flow-structure splitting Eq. (18) of the superadiabatic
+force field. One can see straightforwardly, from the sym-
+metry upon motion reversal, that both the global struc-
+tural force and the global flow force need to vanish indi-
+vidually:
+�
+drρ(r)fflow(r) = 0,
+(27)
+�
+drρ(r)fstr(r) = 0.
+(28)
+We prove by contradiction and assume that it is not
+the case, i.e. that each integral gives the same global
+force, but with opposite sign, such that the sum vanishes
+and Eq. (26) remains valid.
+Per construction, fflow(r)
+changes sign in the motion reversed system, but fstr(r)
+does not.
+Hence Eq. (26) can only be satisfied in the
+motion-reversed system provided that both the flow and
+structural contribution vanish separately.
+We
+can
+explicitly
+test
+the
+validity
+of
+the
+sum
+rules (27) and (28) for the above analytical force ap-
+proximations (21) and (22).
+The respective integrals
+are η
+�
+dr∇[ρ(r)2∇ · v(r)] = 0 and χ
+�
+dr∇{ρ(r)3[∇ ·
+v(r)]2} = 0, which follows from the divergence theorem,
+as boundary terms vanish. Hence the simple non-local
+velocity gradient power functional approximations (16)
+
+8
+density
+current
+external force field
+interparticle 
+force field
+Mermin 
+Evans 
+map 
+(DFT)
+kinematic fields
+kinematic 
+map
+adiabatic-superadiabatic 
+splitting
+structure-flow splitting
+superadiabatic 
+force field
+adiabatic 
+force field
+flow 
+force
+structural 
+force
+adaptive 
+BD
+super- 
+adiabatic 
+map 
+(PFT)
+ 0.3
+ 0.4
+ 0.5
+ 0.6
+ 0.7
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+��3
+x/�
+ 0
+ 1
+ 2
+ 3
+ 4
+ 5
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+J�2�
+x/�
+-1.5
+-1
+-0.5
+ 0
+ 0.5
+ 1
+ 1.5
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+fint�/�
+x/�
+-1.5
+-1
+-0.5
+ 0
+ 0.5
+ 1
+ 1.5
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+fad�/�
+x/�
+ 0
+ 4
+ 8
+ 12
+ 16
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+fext�/�
+x/�
+-0.6
+-0.4
+-0.2
+ 0
+ 0.2
+ 0.4
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+fsup�/�
+x/�
+-0.6
+-0.4
+-0.2
+ 0
+ 0.2
+ 0.4
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+f�ow�/�
+x/�
+-0.2
+-0.1
+ 0
+ 0.1
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+fstr�/�
+x/�
+FIG. 2. Kinematic profiles and force fields for uniaxial compressional flow of the LJ fluid. Results are shown from machine
+learning (lines) and from direct adaptive BD simulations (symbols). Functional relationships are represented by vertical arrows.
+Shown are the density profile ρ(x), the one-body current J(x) and the external force field fext(x) (top row) as a function of the
+scaled distance x/σ, where σ is the LJ length scale. The density and the current functionally determine both the interparticle
+force field fint(x) via the kinematic map and the superadiabatic force field fsup(x) via the superadiabatic kinematic map (middle
+row). The internal force field fint(x) splits into superadiabatic and adiabatic force contributions. The adiabatic force field fad(x)
+is a density functional via the Mermin-Evans map of density functional theory. The structural and flow force fields are split
+according to their symmetry upon motion reversal. The colour code represents different values of the current J0 = 0, 1, 2, 3, 4, 5
+(from violet to yellow, see the center panel in the top row); the two insets show the predictions from the analytical velocity
+gradient functionals (21) and (22). The system with J0 = 0 is at rest in equilibrium and it doubles as the adiabatic state as its
+density profile is identical to that of the flowing systems (first panel).
+and (17) have passed the global Noether validation test.
+This is nontrivial, as the proof rests on the specific struc-
+ture of the integrands being gradients, which for more
+general analytical forms will not be the case. This exem-
+plifies the merits of Noether sum rules for assessing and
+by extension also constructing theoretical nonequilibrium
+force approximations.
+The Noether concept carries much further. Reference
+[44] presents memory sum rules for so-called time di-
+rect correlation functions. These are defined via func-
+tional derivatives of the superadiabatic power functional,
+in generalization of the superadiabatic force density as
+generated via the derivative (12) with respect to the cur-
+rent distribution. We expect the corresponding identities
+to be helpful in the study of temporal nonlocality. Fur-
+ther work was addressed at the variance of global fluctu-
+ations, which were shown to be constrained by Noether
+invariance at the second order global level [49]. Noether’s
+theorem also yields the locally resolved force balance re-
+lationship in quantum mechanical many-body systems
+[48].
+Very recently, striking two-body force-force and
+force-gradient correlation functions for the precise and
+novel characterization of disordered (liquid and gel) sys-
+tems [50] were revealed. Exploiting Noether’s concept in
+a stastical mechanical setting is robust against changes of
+ensemble, Ref. [45] presents the transfer of the grand en-
+semble formalism [44] to canonical systems. Considering
+global rotational invariance leads to (classical) spin-orbit
+
+J&pJ&p9
+coupling of torque identities [44].
+We return to steady states and demonstrate that the
+seemingly entirely formal functional relationships do in
+fact apply to real systems. We present in the following
+new computational methodology that we use to demon-
+strate the functional point of view. We will also demon-
+strate that the sum rules (26) and (27) are highly valuable
+in providing checks for numerical results.
+VII.
+MACHINE LEARNING THE KINEMATIC
+MAP
+Machine learning proves itself to be an increasingly
+useful tool in a variety of settings in soft matter, rang-
+ing from soft matter characterization [82], engineering of
+colloidal self-assembly [83], to the inverse design of soft
+materials [84]. Pivotal studies were addressed at colloidal
+structure detection [85], the identification of combinato-
+rial rules in mechanical metamaterials [86], the learning
+of many-body interaction potentials for spherical [87] and
+for anisotropic particles [88], and the prediction of the
+dynamics of supercooled liquids from their static proper-
+ties [89].
+More specifically, in the context of classical density
+functional theory, an early and pioneering study formu-
+lated a neural-network approach to liquid crystal order-
+ing in confinement [90].
+Free energy density function-
+als were obtained for one-dimensional fluids from a con-
+volutional neural network [91] and an analytical form
+of an excess free energy functional was generated from
+an equation learning network [92]. Cats et al. [93] re-
+cently used machine learning to improve the standard
+mean-field approximation of the excess Helmholtz free-
+energy functional for a three-dimensional Lennard-Jones
+(LJ) system at a supercritical temperature. These signif-
+icant reserach efforts were devoted to tailoring analytical
+forms of model free energy functionals, by training cer-
+tain key components such as spatial convolution kernels,
+and much insight into the inner workings of excess free
+energy functionals was gained [91–93].
+However, here we proceed very differently and more-
+over do so out-of-equilibrium. We use the LJ model and
+the identical planar geometry as in Ref. [93], such that
+the density profile ρ(x) depends only on a single posi-
+tion coordinate x. We consider steady states and retain
+planar symmetry by considering flow that is directed in
+the x-direction, such that the current J(x) = J(x)ex,
+where J(x) is the magnitude of the current and ex is the
+unit vector in the x-direction. Both the density profile
+ρ(x) and the velocity field v(x) = J(x)/ρ(x) are indepen-
+dent of time. The continuity equation (5) then implies
+0 = ∂ρ(x)/∂t = −∂[v(x)ρ(x)]/∂x, from which one ob-
+tains by spatial integration ρ(x)v(x) = J0 = const. Here
+the value of J0 determines the intensity of the flow; we
+recall the illustration shown in Fig. 1.
+We base the machine learning procedure on a convolu-
+tional neural network, as was done e.g. in Ref. [91], and
+following Refs. [91–93] we use many-body computer sim-
+ulations to provide training, validation, and test data.
+In contrast to these equilibrium studies though, in or-
+der to address the nonequilibrium problem we need to
+represent the physical time evolution on the many-body
+trajectory level. We use the recently developed highly
+performant adaptive BD algorithm [6] and apply it to
+the three-dimensional LJ fluid.
+As laid out above, in
+order to address situations of planar symmetry we drive
+the system only along the ex-direction. The specific form
+of the driving force field fext(x)ex is however irrelevant,
+as the training data only serves to extract the intrinsic
+kinematic functional relationship.
+In order to cover a sufficiently broad range of flow sit-
+uations, we represent the external force field as a trun-
+cated Fourier series fext(x) = �nmax
+n=0 An cos(2πnx/L),
+where L is the size of the cubic simulation box with pe-
+riodic boundary conditions and An are random ampli-
+tudes with zero mean and uniform distribution inside of
+a given finite interval. We truncate at order nmax = 5
+such that the length scale L/(2πnmax) is comparable to
+the LJ molecular size σ. Ten percent of our simulation
+runs are carried out in equilibrium, i.e. for A0 = 0. We
+use N = 500 LJ particles inside of a cubic simulation
+box of size L = 10σ. The temporal duration of each run
+is 1000τ, where τ = σ2/D0 is the Brownian time scale.
+After initialization the system is randomized for 1τ at a
+very high temperature. Then we wait for 100τ to allow
+the system to reach a steady state and then collect data
+during the remaining time. In total we use 1000 such sim-
+ulation runs; these are subdivided for purposes of train-
+ing (520), validation (280) and testing (200).
+A more
+detailed account will be given elsewhere.
+Our aim is to machine-learn and hence to explicitly
+demonstrate the kinematic map, ρ(r), v(r) → fint(r) in
+steady state. We present the learning algorithm with in-
+puts ρ(x), v(x) and targets fint(x). The data for these
+three fields are from building steady state averages via
+the adaptive BD over the corresponding one-body oper-
+ators. We recall the microscopic definition of the inter-
+particle one-body force density Fint(r) via Eq. (7) and
+we refer the reader to Appendix A of Ref. [51] for a de-
+scription of several methods to sample the current in
+BD and hence obtain the overdamped velocity profile
+v(r). Finally, we use the standard counting method for
+the density profile ρ(r), although more efficient “force
+sampling” methods [94–97] exist. At this stage we nei-
+ther impose adiabatic-superadiabatic splitting (11), nor
+structure-flow splitting (18), nor do we use any analyti-
+cal model form of the functional relationship. We rather
+work on the level of the bare one-body simulation data,
+generated in the above described randomized uniaxial
+flow situations of the desired planar symmetry.
+We refer to the result of this procedure as the machine-
+learned internal force field f ⋆
+int(x, [ρ, v]). This represents
+a “surrogate model” in the sense of the terminology of the
+machine learning community. By construction this data
+structure depends functionally on the density profile and
+
+10
+on the velocity profile. Importantly the external force
+field fext(x), as given by the above described randomized
+Fourier series, has not been used in the training, which
+was rather based solely on the intrinsic force field and
+its kinematic dependence on the density profile and the
+velocity field.
+In order to test the validity of the functional relation-
+ship and to address the question whether f ⋆
+int(x, [ρ, v])
+indeed represents the true fint(r, t, [ρ, v]) of power func-
+tional theory restricted to the present planar and steady
+situation, we consider a toy flow situation as an appli-
+cation.
+We choose the density profile to consist of a
+single (co)sinusoidal deviation from the bulk, ρ(x) =
+[0.5 + 0.2 cos(2πx/L)]σ−3.
+In order for the system to
+be in steady state, the velocity then necessarily needs to
+satisfy v(x) = J0/ρ(x), where the strength of the current
+J0 = const is a free parameter.
+We proceed in two ways.
+First, we check for self-
+consistency. Therefore we solve the force density balance
+relationship (6) for the external force field, which yields
+the explicit result:
+fext(x) = kBTρ′(x) + γv(x) − f ⋆
+int(x, [ρ, v]),
+(29)
+where ρ′(x) = ∂ρ(x)/∂x. As is explicit in Eq. (29), in-
+putting the toy state ρ(x), v(x) on the right hand side
+yields a concrete machine learning prediction for the ex-
+ternal force field on the left hand side. We then input
+this result for fext(x) as the driving force field in a single
+adaptive BD simulation run and expect this procedure to
+reproduce the density and velocity profile of the toy state.
+The reproductive success will however materialize only
+provided that i) the functional kinematic dependence ac-
+tually exists and that ii) it is accurately represented by
+the neural network.
+The results, shown in Fig. 2, demonstrate the accom-
+plishment of the reconstruction of the toy state.
+This
+establishes that the machine learned functional provides
+a numerically very highly accurate representation of the
+true internal force functional. We take this validation via
+the machine learning to be a practical, data-science-level
+verification of the existence of the power functional kine-
+matic map. We recall the original formal construction
+[42, 53] and its subsequent confirmation via custom flow
+[51, 52].
+Turning to the physics of the compressional flow, we
+use the adiabatic-superadiabatic decomposition (11) to-
+gether with the flow-structure splitting (18) to analyze
+both the machine-learned functional f ⋆
+int(x, [ρ, v]) as well
+as the direct simulation results. As anticipated, both flow
+and structural force fields have nontrivial spatial varia-
+tion, see Fig. 2. The flow force primarily contains viscous
+effects that stem from the dissipation that the compres-
+sional and extensional regions of the flow pattern gener-
+ate. The structural force field becomes more strongly in-
+homogeneous and also larger in magnitude upon increas-
+ing the amplitude of the flow. This trend is necessary
+to provide a balance for the increasingly asymmetric and
+growing external force field, which in turn is required to
+keep the density profile unchanged upon increasing the
+throughput through the prescribed density wave.
+The
+power functional predictions (21) and (22) capture these
+effects reasonably well given the simplicity of the ana-
+lytical expressions, see the insets in Fig. 2. We find our
+numerical results to satisfy the Noether sum rules (26)
+and (27) to very good accuracy.
+It remains to point out the stark contrast with the
+standard DDFT (1), which gives a trivial null result in
+the present setup by construction: the density profile re-
+mains unchanged upon increasing flow, and so does the
+adiabatic force field. So the DDFT provides no mecha-
+nism to account for the nonequilibrium physics.
+VIII.
+CONCLUSIONS
+For the purpose of assessing the status of the DDFT
+equation of motion (1) we have first described two exact
+limits that this approximation reproduces: the dynamics
+of the noninteracting diffusive ideal gas [see Eq. (4)] and
+the spatially inhomogeneous static equilibrium limit [see
+Eq. (2)]. On general grounds one expects the DDFT to
+perform well when the situation under consideration is
+close to one of these limits. In particular near the static
+case this is nontrivial, as the system might be dense and
+spatially highly structured, as evident by a strongly inho-
+mogeneous density profile. Provided that the dynamics
+are driven weakly enough via a time-dependent external
+potential then the DDFT can be a highly useful device,
+which enables one to describe the temporal evolution as
+a chain of equilibrium states, labelled by time.
+In general the contributions beyond the equilibrium
+physics will however be relevant.
+On the level of the
+formally exact one-body equation of motion (14), the su-
+peradiabatic force field fsup(r, t) will then contribute and
+potentially very significantly so. Together with the adi-
+abatic force field, which follows from the equilibrium ex-
+cess free energy functional via −∇δFexc[ρ]/δρ(r, t), their
+sum constitutes the full interparticle forces. These are
+coarse-grained, in a microscopically sharp way, to the
+one-body level of dynamical correlation functions. We
+have argued i) that power functional theory is a con-
+crete formal structure that allows to obtain fsup(r, t)
+from a generating functional and ii) that simple approx-
+imate forms already capture much relevant nonequilib-
+rium physics and they do so in a transparent and sys-
+tematic way.
+We have described and exemplified for uniaxial steady
+compressional flow of the three-dimensional Lennard-
+Jones fluid the kinematic functional map that governs
+the exact nonequilibrium dynamics on the one-body level
+of dynamic correlation functions. As this description is
+based on a single position coordinate and a single time
+variable, it is of both conceptual and practical simplicity.
+As described by power functional theory the superadia-
+batic interparticle force field functionally depends on the
+density and the velocity field, i.e. fsup(r, t, [ρ, v]), for over-
+
+11
+damped Brownian motion. The functional dependence is
+causal, i.e. on the values of the density profile and velocity
+field at previous times, in general up to an initial state.
+The superadiabatic force field carries this kinematic de-
+pendence, i.e. on the history of ρ(r, t) and v(r, t), but
+crucially it is independent of the external force field that
+drives the system.
+We have explicitly demonstrated the functional map
+ρ(r, t), v(r, t) → fint(r, t) by establishing this functional
+relationship via machine learning the intrinsic force field.
+Using the force balance then gives direct access to the
+form of the required external force field via Eq. (29). The
+machine-learned model of the functional map hence en-
+ables “instant custom flow” at negligible computational
+cost at the time of use. We recall that the custom flow
+method [51, 52] is based on the kinematic functional map,
+such that from knowing the kinematic one-body fields,
+the external force field that is necessary to generate the
+given time evolution follows straightforwardly from the
+exact force balance (6).
+An analytical approach to one-body functional maps
+leads to the simple structure of velocity gradient forms
+for the viscous and structural superadiabatic forces, as
+exemplified in Eqs. (16) and (17) for compressional flow,
+i.e. for velocity fields with nonvanishing divergence. As
+we have shown, the resulting predictions for the flow
+force (21) and for the structural force field (22) represent
+a reasonable description of the simulation data and its
+representation via the machine-learned functional. We
+attribute the remaining differences to higher-order terms
+[59] which we have not addressed here for simplicity. As
+we have shown, our results from direct simulation, from
+machine learning, and from the analytical approxima-
+tions, satisfy exact global Noether sum rules.
+We have restricted our discussion to a single and rela-
+tively easily accessible type of nonequilibrium dynamics,
+that of stationary uniaxial compressional flow that rep-
+resents a model steady (batch) sedimentation situation.
+The power functional approach allows to go much fur-
+ther, including the treatment of viscoelasticity [56], as
+arising from superadiabatic memory, deconfinement un-
+der shear [57], the dynamic decay of the van Hove pair
+correlation function as governed by drag, viscous and
+structural forces [68, 69], and the complex forms of both
+flow and structural forces that arise under spatially com-
+plex forms of driving [59]. Time-dependent uniaxial flow
+is relevant in a variety of situations, including colloidal
+stratification [98, 99] and sedimentation [100].
+Although power functional theory operates on the one-
+body level of dynamical correlation functions, two-body
+correlation functions are accessible both formally via the
+nonequilibrium Ornstein-Zernike route [42] and explic-
+itly by the dynamical test particle limit. The latter is
+the dynamic generalization of Percus’ static test parti-
+cle limit [61], which identifies two-point correlation func-
+tions, such as g(r) as also recently shown to be intimat-
+edly related to thermal Noether invariance at second or-
+der [50], with one-body density profiles in an external
+potential. This is set equal to the interparticle pair po-
+tential.
+The dynamical test-particle limit goes further
+in that it describes the test particle via its own dynami-
+cal degrees of freedom, which are coupled to those of all
+other particles in the system. The concept was originally
+formulated as an approximation within DDFT [62, 63]
+and formally exactly within power functional theory [66].
+Two-body superadiabatic effects were shown via simula-
+tion work to be significant [67–69] and they arise natu-
+rally in an exact formulation of the test particle dynamics
+[66]. The test particle limit allowed for a rationalization
+of the dynamical pair structure as e.g. experimentally
+observed in two-dimensional colloids [9]. Recently an ap-
+proach to DDFT based on the two-body level was formu-
+lated [101].
+In event-driven BD simulations superadiabatic forces
+were shown to consist of drag, viscous, and structural
+contributions [68, 69]; see Ref. [42] for an extended dis-
+cussion. The physics of active particles [70–74] is very
+significantly governed by a vigorous interplay between su-
+peradiabatic and adiabatic forces, both of which are very
+strong, as the tendency of these systems to self-compress
+leads naturally to very high local densities.
+Furthermore,
+relevant and interesting microscopic
+models that go beyond the simple fluid paradigm of a
+pair potential, such as the monatomic water model by
+Molinero and Moore [102, 103] and the three-body gel
+by Saw et al. [104, 105], are accessible. Despite the com-
+plexity of both its defining Hamiltonian and the intricate
+transient network structure, the inhomogeneous viscous
+response of the three-body gel was recently demonstrated
+[60] to be surprisingly well captured by a simple power
+functional flow approximation.
+We finally recall that
+superadiabatic effects transcend overdamped dynamics,
+and are relevant both in quantum dynamics [42, 80, 81]
+and in classical molecular dynamics [42, 78, 79].
+While we have restricted ourselves to discussing the
+point of view of functional relationships, it would be in-
+teresting to explore in future work possible cross connec-
+tions to other theoretical approaches, such as Onsager’s
+variational principle for soft matter [106–109], stochastic
+thermodynamics [110], large deviation theory [111, 112],
+mode-coupling theory [113, 114], generalized hydrody-
+namics [115], as well as to the physics of nonequilibrium
+phase transitions [116] and of Brownian solitons [117].
+ACKNOWLEDGMENTS
+This work is supported by the German Research Foun-
+dation (DFG) via Project No. 436306241.
+
+12
+[1] S. R. Nagel, Experimental soft-matter science, Rev.
+Mod. Phys. 89, 025002 (2017).
+[2] R. Evans, D. Frenkel, and M. Dijkstra, From simple
+liquids to colloids and soft matter, Phys. Today 72, 38
+(2019).
+[3] J. P. Hansen and I. R. McDonald, Theory of Simple
+Liquids, 4th ed. (Academic Press, London, 2013).
+[4] R. Evans, The nature of the liquid-vapour interface and
+other topics in the statistical mechanics of non-uniform,
+classical fluids, Adv. Phys. 28, 143 (1979).
+[5] P. C. Hohenberg and B. I. Halperin, Theory of dynamic
+critical phenomena, Rev. Mod. Phys. 49, 435 (1977).
+[6] F. Samm¨uller and M. Schmidt, Adaptive Brownian dy-
+namics, J. Chem. Phys. 155, 134107 (2021).
+[7] C. P. Royall, J. Dzubiella, M. Schmidt, and A. van
+Blaaderen, Non-equilibrium sedimentation of colloids on
+the particle scale, Phys. Rev. Lett. 98, 188304 (2007).
+[8] M. Bier, R. van Roij, M. Dijkstra, and P. van der Schoot,
+Self-diffusion of particles in complex fluids: temporary
+cages and permanent barriers, Phys. Rev. Lett. 101,
+215901 (2008).
+[9] D. Stopper, A. L. Thorneywork, R. P. A. Dullens, and
+R. Roth, Bulk dynamics of Brownian hard disks: Dy-
+namical density functional theory versus experiments on
+two-dimensional colloidal hard spheres, J. Chem. Phys.
+148, 104501 (2018).
+[10] R. Evans, Density functionals in the theory nonuniform
+fluids, in Fundamentals of Inhomogeneous Fluids, edited
+by D. Henderson (Dekker, New York, 1992).
+[11] For an overview of new developments in classical den-
+sity functional theory, see:
+R. Evans, M. Oettel, R.
+Roth, and G. Kahl, New developments in classical den-
+sity functional theory, J. Phys.: Condens. Matter 28,
+240401 (2016).
+[12] R. Evans, M. C. Stewart, and N. B. Wilding, A unified
+description of hydrophilic and superhydrophobic sur-
+faces in terms of the wetting and drying transitions of
+liquids, Proc. Nat. Acad. Sci. 116, 23901 (2019).
+[13] M. K. Coe, R. Evans, and N. B. Wilding, Density deple-
+tion and enhanced fluctuations in water near hydropho-
+bic solutes: identifying the underlying physics, Phys.
+Rev. Lett. 128, 045501 (2022).
+[14] M. K. Coe, R. Evans, and N. B. Wilding, Understanding
+the physics of hydrophobic solvation, arXiv:2212.04967
+[15] D. Martin-Jimenez, E. Chac´on, P. Tarazona, and R.
+Garcia, Atomically resolved three-dimensional struc-
+tures of electrolyte aqueous solutions near a solid sur-
+face, Nat. Commun. 7, 12164 (2016).
+[16] J. Hern´andez-Mu˜noz, E. Chac´on, and P. Tarazona, Den-
+sity functional analysis of atomic force microscopy in a
+dense fluid, J. Chem. Phys. 151, 034701 (2019).
+[17] P. Cats, R. Evans, A. H¨artel, and R. van Roij, Primi-
+tive model electrolytes in the near and far field: Decay
+lengths from DFT and simulations, J. Chem. Phys. 154,
+124504 (2021).
+[18] Y. Rosenfeld, Free-energy model for the inhomogeneous
+hard-sphere fluid mixture and density-functional theory
+of freezing, Phys. Rev. Lett. 63, 980 (1989)..
+[19] R. Roth, Fundamental measure theory for hard-sphere
+mixtures:
+a review, J. Phys.:
+Condens. Matter 22,
+063102 (2010).
+[20] R. Roth, R. Evans, A. Lang, and G. Kahl, Fundamen-
+tal measure theory for hard-sphere mixtures revisited:
+the White Bear version, J. Phys.: Condens. Matter 14,
+12063 (2002).
+[21] H. Hansen-Goos and R. Roth, Density functional the-
+ory for hard-sphere mixtures: the White Bear version
+mark II, J.Phys.: Condens. Matter 18, 8413 (2006).
+[22] U. M. B. Marconi and P. Tarazona, Dynamic density
+functional theory of fluids, J. Chem. Phys. 110, 8032
+(1999).
+[23] A. J. Archer and R. Evans, Dynamical density func-
+tional theory and its application to spinodal decompo-
+sition, J. Chem. Phys. 121, 4246 (2004).
+[24] G. K.-L. Chan and R. Finken, Time-dependent density
+functional theory of classical fluids, Phys. Rev. Lett. 94,
+183001 (2005).
+[25] P. Espa˜nol and H. L¨owen, Derivation of dynamical
+density functional theory using the projection operator
+technique, J. Chem. Phys. 131, 244101 (2009).
+[26] U. M. B. Marconi and S. Melchionna, Phase-space ap-
+proach to dynamical density functional theory, J. Chem.
+Phys. 126, 184109 (2007).
+[27] J. Dzubiella and C. N. Likos, Mean-field dynamical den-
+sity functional theory, J. Phys.: Condens. Matter 15,
+L147 (2003).
+[28] J. F. Lutsko and M. Oettel, Reconsidering power func-
+tional theory, J. Chem. Phys. 155, 094901 (2021).
+[29] G. Szamel, An alternative, dynamic density functional-
+like theory for time-dependent density fluctuations in
+glass-forming fluids, J. Chem. Phys. 156, 191102 (2022).
+[30] B. D. Goddard, R. D. Mills-Williams, M. Ottobre,
+and G. Pavliotis, Well-posedness and equilibrium be-
+haviour of overdamped dynamic density functional the-
+ory, arXiv:2002.11663
+[31] A. J. Archer, Dynamical density functional theory for
+dense atomic liquids, J. Phys.: Condens. Matter 18,
+5617 (2006).
+[32] A. J. Archer, Dynamical density functional theory for
+molecular and colloidal fluids: A microscopic approach
+to fluid mechanics, J. Chem. Phys. 130, 014509 (2009).
+[33] R. Stierle and J. Gross, Hydrodynamic density func-
+tional theory for mixtures from a variational principle
+and its application to droplet coalescence, J. Chem.
+Phys. 155, 134101 (2021).
+[34] B. D. Goddard, A. Nold, N. Savva, G. A. Pavliotis,
+and S. Kalliadasis, General dynamical density func-
+tional theory for classical fluids, Phys. Rev. Lett. 109,
+120603 (2012).
+[35] M. Rex and H. L¨owen, Dynamical density functional
+theory for colloidal dispersions including hydrodynamic
+interactions, Eur. Phys. J. E 28, 139 (2009).
+[36] J. Dzubiella, and A. Moncho-Jord´a, Controlling the mi-
+crostructure and phase behavior of confined soft colloids
+by active interaction switching, Phys. Rev. Lett. 125,
+078001 (2020).
+[37] M. Bley, J. Dzubiella, and A. Moncho-Jord´a, Active bi-
+nary switching of soft colloids: stability and structural
+properties, Soft Matter 17, 7682 (2021).
+[38] B. D. Goddard, B. Gooding, G. A. Pavliotis, and H.
+Short, Noisy bounded confidence models for opinion dy-
+namics: the effect of boundary conditions on phase tran-
+
+13
+sitions, IMA J. Appl. Math. 87, 80 (2022).
+[39] M. te Vrugt, J. Bickmann, and R. Wittkowski, Effects
+of social distancing and isolation on epidemic spreading
+modeled via dynamical density functional theory, Nat.
+Commun. 11, 5576 (2020).
+[40] M. te Vrugt, H. L¨owen, and R. Wittkowski, Classical
+dynamical density functional theory: from fundamen-
+tals to applications, Adv. Phys. 69, 121 (2020).
+[41] M. te Vrugt and R. Wittkowski, Perspective: New direc-
+tions in dynamical density functional theory, J. Phys.:
+Condens. Matter 35, 041501 (2023).
+[42] M. Schmidt, Power functional theory for many-body dy-
+namics, Rev. Mod. Phys. 94, 015007 (2022).
+[43] T. Schilling, Coarse-grained modelling out of equilib-
+rium, Phys. Rep. 972, 1 (2022).
+[44] S. Hermann and M. Schmidt, Noether’s theorem in sta-
+tistical mechanics, Commun. Phys. 4, 176 (2021).
+[45] S. Hermann and M. Schmidt, Why Noether’s theorem
+applies to statistical mechanics, J. Phys.:
+Condens.
+Matter 34, 213001 (2022) (Topical Review).
+[46] S. M. Tschopp, F. Samm¨uller, S. Hermann, M. Schmidt,
+and J. M. Brader, Force density functional theory
+in- and out-of-equilibrium, Phys. Rev. E 106, 014115
+(2022).
+[47] F. Samm¨uller, S. Hermann, and M. Schmidt, Should
+classical density functional theory be based on forces?
+A comparative study, arxiv:2212.01780 (2022).
+[48] S. Hermann and M. Schmidt, Force balance in thermal
+quantum many-body systems from Noether’s theorem,
+J. Phys. A: Math. Theor. 55, 464003 (2022). (Claritons
+and the Asymptotics of ideas: the Physics of Michael
+Berry).
+[49] S. Hermann and M. Schmidt, Variance of fluctuations
+from Noether invariance, Commun. Phys. 5, 276 (2022).
+[50] F. Samm¨uller, S. Hermann, D. de las Heras, and M.
+Schmidt, What is liquid, from Noether’s perspective?
+(to be published).
+[51] D. de las Heras, J. Renner, and M. Schmidt, Custom
+flow in overdamped Brownian dynamics, Phys. Rev. E
+99, 023306 (2019).
+[52] J. Renner, M. Schmidt, and D. de las Heras, Custom
+flow in molecular dynamics, Phys. Rev. Res. 3, 013281
+(2021).
+[53] M. Schmidt and J. M. Brader, Power functional theory
+for Brownian dynamics, J. Chem. Phys. 138, 214101
+(2013).
+[54] A. Fortini,
+D. de las Heras,
+J. M. Brader,
+and
+M. Schmidt, Superadiabatic forces in Brownian many-
+body dynamics, Phys. Rev. Lett. 113, 167801 (2014).
+[55] N. C. X. Stuhlm¨uller, T. Eckert, D. de las Heras, and
+M. Schmidt, Structural nonequilibrium forces in driven
+colloidal systems, Phys. Rev. Lett. 121, 098002 (2018).
+[56] L. L. Treffenst¨adt and M. Schmidt, Memory-induced
+motion reversal in Brownian liquids, Soft Matter 16,
+1518 (2020).
+[57] N. Jahreis and M. Schmidt, Shear-induced deconfine-
+ment of hard disks, Col. Pol. Sci. 298, 895 (2020).
+[58] D. de las Heras and M. Schmidt, Velocity gradient
+power functional for Brownian dynamics, Phys. Rev.
+Lett. 120, 028001 (2018).
+[59] D. de las Heras and M. Schmidt, Flow and structure
+in nonequilibrium Brownian many-body systems, Phys.
+Rev. Lett. 125, 018001 (2020).
+[60] F. Samm¨uller, D. de las Heras, and M. Schmidt, Inho-
+mogeneous steady shear dynamics of a three-body col-
+loidal gel former, J. Chem. Phys. (to appear in the Spe-
+cial Topic on Colloidal Gels); arXiv:2210.07679.
+[61] J. K. Percus, Approximation methods in classical sta-
+tistical mechanics, Phys. Rev. Lett. 8, 462 (1962).
+[62] A. J. Archer, P. Hopkins, and M. Schmidt, Dynamics in
+inhomogeneous liquids and glasses via the test particle
+limit, Phys. Rev. E 75, 040501(R) (2007).
+[63] P. Hopkins, A. Fortini, A. J. Archer, and M. Schmidt,
+The van Hove distribution function for Brownian hard
+spheres: Dynamical test particle theory and computer
+simulations for bulk dynamics, J. Chem. Phys. 133,
+224505 (2010).
+[64] D. Stopper, R. Roth and H. Hansen-Goos, Communi-
+cation: Dynamical density functional theory for dense
+suspensions of colloidal hard spheres, J. Chem. Phys.
+143, 181105 (2015).
+[65] D. Stopper, K. Marolt, R. Roth, and H. Hansen-Goos,
+Modeling diffusion in colloidal suspensions by dynam-
+ical density functional theory using fundamental mea-
+sure theory of hard spheres, Phys. Rev. E 92, 022151
+(2015).
+[66] J. M Brader and M. Schmidt, Power functional theory
+for the dynamic test particle limit, J. Phys.: Condens.
+Matter 27, 194106 (2015).
+[67] T. Schindler and M. Schmidt, Dynamic pair correlations
+and superadiabatic forces in a dense Brownian liquid, J.
+Chem. Phys. 145, 064506 (2016).
+[68] L. L. Treffenst¨adt and M. Schmidt, Universality in
+driven and equilibrium hard sphere liquid dynamics,
+Phys. Rev. Lett. 126, 058002 (2021).
+[69] L. L. Treffenst¨adt, T. Schindler, M. Schmidt, Dynamic
+decay and superadiabatic forces in the van Hove dy-
+namics of bulk hard sphere fluids, SciPost Phys. 12,
+133 (2022).
+[70] S. Hermann, D. de las Heras, and M. Schmidt, Non-
+negative interfacial tension in phase-separated active
+Brownian particles,
+Phys. Rev. Lett. 123,
+268002
+(2019).
+[71] S. Hermann, P. Krinninger, D. de las Heras, and M.
+Schmidt, Phase coexistence of active Brownian parti-
+cles, Phys. Rev. E 100, 052604 (2019).
+[72] P. Krinninger, M. Schmidt, and J. M. Brader, Nonequi-
+librium phase behaviour from minimization of free
+power dissipation, Phys. Rev. Lett. 117, 208003 (2016).
+[73] P. Krinninger and M. Schmidt, Power functional theory
+for active Brownian particles: general formulation and
+power sum rules, J. Chem. Phys. 150, 074112 (2019).
+[74] S. Hermann, D. de las Heras, and M. Schmidt, Phase
+separation of active Brownian particles in two dimen-
+sions: Anything for a quiet life, Mol. Phys. e1902585
+(2021).
+[75] D. de las Heras, M. Schmidt, Full canonical information
+from grand potential density functional theory, Phys.
+Rev. Lett. 113, 238304 (2014).
+[76] D. de las Heras, J. M. Brader, A. Fortini, M. Schmidt,
+Particle conservation in dynamical density functional
+theory, J. Phys.: Condens. Matter 28, 244024 (2016).
+[77] T. Schindler, R. Wittmann, and J. M. Brader, Particle-
+conserving dynamics on the single-particle level, Phys.
+Rev. E 99, 012605 (2019).
+[78] M. Schmidt, Power functional theory for Newtonian
+many-body dynamics, J. Chem. Phys. 148, 044502
+
+14
+(2018).
+[79] J. Renner, M. Schmidt, and D. de las Heras, Shear and
+bulk acceleration viscosities in simple fluids, Phys. Rev.
+Lett. 128, 094502 (2022).
+[80] M. Schmidt, Quantum power functional theory for
+many-body dynamics, J. Chem. Phys. 143, 174108
+(2015).
+[81] M. Br¨utting, T. Trepl, D. de las Heras, and M. Schmidt,
+Superadiabatic forces via the acceleration gradient in
+quantum many-body dynamics, Molecules 24, 3660
+(2019).
+[82] P. S. Clegg, Characterising soft matter using machine
+learning, Soft Matter, 17, 3991 (2021).
+[83] M. Dijkstra and E. Luijten, From predictive modelling
+to machine learning and reverse engineering of colloidal
+self-assembly, Nature Materials 20, 762 (2021)
+[84] G. M. Coli, E. Boattini, L. Filion, M. Dijkstra, Inverse
+design of soft materials via a deep learning-based evo-
+lutionary strategy, Sci. Adv. 8, eabj6731 (2022)
+[85] E. Boattini, M. Dijkstra, and L. Filion, Unsupervised
+learning for local structure detection in colloidal sys-
+tems, J. Chem. Phys. 151, 154901 (2019).
+[86] R. van Mastrigt, M. Dijkstra, M. van Hecke, and
+C. Coulais, Machine learning of implicit combinato-
+rial rules in mechanical metamaterials, Phys. Rev. Lett.
+129, 198003 (2022).
+[87] G. Campos-Villalobos, E. Boattini, L. Filion, and M.
+Dijkstra, Machine learning many-body potentials for
+colloidal systems, J. Chem. Phys. 155, 174902 (2021).
+[88] G. Campos-Villalobos, G. Giunta, S. Mar´ın-Aguilar,
+M. Dijkstra,
+Machine-learning effective many-body
+potentials for anisotropic particles using orientation-
+dependent symmetry functions, J. Chem. Phys. 157,
+024902 (2022).
+[89] S. Ciarella, M. Chiappini, E. Boattini, M. Dijkstra,
+and L. M. C. Janssen, Dynamics of supercooled liquids
+from static averaged quantities using machine learning,
+arXiv:2212.09338
+[90] T. Santos-Silva, P. I. C. Teixeira, C. Anquetil-Deck, and
+D. J. Cleaver, Neural-network approach to modeling liq-
+uid crystals in complex confinement, Phys. Rev. E 89,
+053316 (2014).
+[91] S.-C. Lin and M. Oettel, A classical density functional
+from machine learning and a convolutional neural net-
+work, SciPost Phys. 6, 025 (2019).
+[92] S.-C. Lin, G. Martius, and M. Oettel, Analytical clas-
+sical density functionals from an equation learning net-
+work, J. Chem. Phys. 152, 021102 (2020).
+[93] P. Cats, S. Kuipers, S. de Wind, R. van Damme, G. M.
+Coli, M. Dijkstra, and R. van Roij, Machine-learning
+free-energy functionals using density profiles from sim-
+ulations, APL Mater. 9, 031109 (2021).
+[94] B. Rotenberg, Use the force! Reduced variance estima-
+tors for densities, radial distribution functions, and lo-
+cal mobilities in molecular simulations, J. Chem. Phys.
+153, 150902 (2020).
+[95] D. Borgis, D., R. Assaraf, B. Rotenberg, and R. Vuilleu-
+mier, Computation of pair distribution functions and
+three-dimensional densities with a reduced variance
+principle, Mol. Phys. 111, 3486 (2013).
+[96] D. de las Heras and M. Schmidt, Better than counting:
+Density profiles from force sampling, Phys. Rev. Lett.
+120, 218001 (2018).
+[97] J. Renner, M. Schmidt, and D. de las Heras, Bet-
+ter than counting: Orientational distribution functions
+from torque sampling, arXiv:2212.11576
+[98] B. He, I. Martin-Fabiani, R. Roth, G. I. T´oth, and A.
+J. Archer, Dynamical density functional theory for the
+drying and stratification of binary colloidal dispersions,
+Langmuir 37, 1399 (2021).
+[99] M. Kundu and M. P. Howard, Dynamic density func-
+tional theory for drying colloidal suspensions: Compar-
+ison of hard-sphere free-energy functionals, J. Chem.
+Phys. 157, 184904 (2022).
+[100] J. Sui, M. Doiab and Y. Ding, Dynamics of the float-
+ing nematic phase formation in platelet suspension with
+thickness polydispersity by sedimentation, Soft Matter
+14, 8956 (2018).
+[101] S. M. Tschopp and J. M. Brader, First-principles su-
+peradiabatic theory for the dynamics of inhomogeneous
+fluids, J. Chem. Phys. 157, 234108 (2022).
+[102] V. Molinero and E. B. Moore, Water modeled as an in-
+termediate element between carbon and silicon, J. Phys.
+Chem. B 113, 4008–4016 (2009).
+[103] M. K. Coe, R. Evans, and N. B. Wilding, The coexis-
+tence curve and surface tension of a monatomic water
+model, J. Chem. Phys. 156, 154505 (2022).
+[104] S. Saw, N. L. Ellegaard, W. Kob, and S. Sastry, Struc-
+tural relaxation of a gel modeled by three body interac-
+tions, Phys. Rev. Lett. 103, 248305 (2009).
+[105] S. Saw, N. L. Ellegaard, W. Kob, and S. Sastry, Com-
+puter simulation study of the phase behavior and struc-
+tural relaxation in a gel-former modeled by three-body
+interactions, J. Chem. Phys. 134, 164506 (2011).
+[106] M. Doi, Onsager’s variational principle in soft matter,
+J. Phys.: Condens. Matter 23, 284118 (2011).
+[107] M. Doi, Onsager principle as a tool for approximation,
+Chinese Phys. B 24, 020505 (2015).
+[108] H. Wang, T. Qian, and X. Xu, Onsager’s variational
+principle in active soft matter, Soft Matter 17, 3634
+(2021).
+[109] X. Wang, J. Dobnikar, and D. Frenkel, Numerical test
+of the Onsager relations in a driven system, Phys. Rev.
+Lett. 129, 238002 (2022).
+[110] U. Seifert, Stochastic thermodynamics, fluctuation the-
+orems and molecular machines, Rep. Prog. Phys. 75,
+126001 (2012).
+[111] R. L. Jack and P. Sollich, Large Deviations and Ensem-
+bles of Trajectories in Stochastic Models, Prog. Theo.
+Phys. Suppl. 184, 304 (2010).
+[112] R. L. Jack and P. Sollich, Effective interactions and large
+deviations in stochastic processes, Europ. Phys. J. Spec.
+Top. 224, 2351 (2015).
+[113] L. M. C. Janssen, Mode-coupling theory of the glass
+transition: a primer, Front. Phys. 6, 97 (2018).
+[114] L. M. C. Janssen and D. R. Reichman, Microscopic dy-
+namics of supercooled liquids from first principles, Phys.
+Re. Lett. 115, 205701 (2015).
+[115] G. Mazzuca, T. Grava, T. Kriecherbauer, K. T.-R.
+McLaughlin, C. B. Mendl, H. Spohn, Equilibrium space-
+time correlations of the toda lattice on the hydrody-
+namic scale, arXiv:2301.02431.
+[116] D. Lips, A. Ryabov, and P. Maass, Brownian asym-
+metric simple exclusion process, Phys. Rev. Lett. 121,
+160601 (2018).
+[117] A. P. Antonov, A. Ryabov, and P. Maass, Solitons in
+overdamped Brownian dynamics, Phys. Rev. Lett. 129,
+
+15
+080601 (2022).
+

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@@ -0,0 +1,2672 @@
+Optimal multiple FSO transceiver configuration for using on High-altitude
+platforms
+Dieu Linh Truong ∗1 and The Ngoc Dang †2
+1School of Information and Communication Technology, Hanoi University of Science and Technology,
+Vietnam
+2Department of Wireless Communications, Posts and Telecommunication Institute of Technology,
+Vietnam
+January 23, 2023
+Abstract
+Free-space optical (FSO) communication requires light of
+sight (LoS) between the transmitter and the receiver.
+For
+long-distance communication, many research projects have
+been conducted towards using a network composed of high-
+altitude platforms (HAPs) flying at an elevation of 20 km to
+carry intermediate FSO transceivers that forward data be-
+tween ground stations.
+The clear environment at high el-
+evations prevents terrestrial obstacles from cutting the LoS
+between the transceivers.
+An FSO transceiver on a HAP
+can communicate with ground stations within a small area
+owing to its limited beam size. We suggest using multiple
+FSO transceivers on a HAP to extend its ground coverage.
+However, the use of too many FSO transceivers may quickly
+exhaust the onboard energy of the HAP. As a result, HAP
+must be lowered to recharge frequently.
+In this study, we first propose a configuration of multiple
+FSO transceivers to widen the ground coverage of a HAP.
+We then propose a set of closed-form expressions to calculate
+the extended coverage. Finally, to implement a HAP network
+using multiple FSO transceivers, we seek the optimal config-
+uration of multiple FSO transceivers that minimizes the to-
+tal cost of the HAP network, including amortization, energy,
+and maintenance costs. The simulation results show that the
+proposed multiple FSO transceiver configuration clearly in-
+creases the ground coverage of a HAP and significantly re-
+duces the cost of the HAP network.
+Keywords— Free Space Optics, High-altitude platform, Beam
+size optimization, HAP based FSO network
+1
+Introduction
+Free-space optical (FSO) communication uses light propagation
+in free space to transmit data. In recent years, this technology
+has emerged as a promising choice for short-distance high-speed
+communication between endpoints with a clear light of sight (LoS).
+∗linhtd@soict.hust.edu.vn
+†ngocdt@ptit.edu.vn
+Commercial FSO transmitters available in the market at prices of
+thousands of dollars can operate at 1.25 − 10 Gbps over 1 − 2
+kilometers, for example, the SONABeam series of fSona [1].
+To reach a long distance, a multi-hop FSO system can be used,
+where data are transmitted through intermediate FSO transceivers
+[2], [3]. To avoid obstacles that cut the LoS between terrestrial
+FSO transceivers, researchers from academia and industry have
+proposed placing intermediate FSO transceivers of the multi-hop
+FSO system on high-altitude platforms (HAPs).
+High-altitude
+platforms are flying objects that operate at altitudes of 17–24 km
+in the stratosphere. Several HAP models have been proposed and
+piloted previously. Some projects continue until recently, such as
+the Loon Project of Google [4], the UAV project of Facebook [5],
+and the Stratobus project of Thales Alenia Space [6].
+A multi-hop FSO system using a HAP network is described
+in [7] and illustrated in Figure 1. According to this model, FSO
+transceivers on the ground (so-called ground FSO nodes) are re-
+grouped into clusters to become the serving zones of HAPs.
+A
+HAP has an FSO transceiver looking down to exchange data
+with the ground FSO nodes of the cluster under it.
+This FSO
+transceiver is called serving FSO transceiver. A HAP also carries
+several FSO transceivers pointing towards other HAPs for inter-
+HAP communication. These FSO transceivers are known as inter-
+HAP FSO transceivers.
+Although the ITU recommends a HAP footprint width of ap-
+proximately 500 km in radius, experimental projects show much
+smaller coverage areas [8].
+Nevertheless, a network of multiple
+HAPs can cover a country entirely. For example, a constellation
+of 16 HAPs with multiple radio frequency antennas was considered
+to cover Japan [9].
+An end-to-end data-switching scheme for a multi-hop FSO sys-
+tem using HAP was proposed in [7].
+Since the communication
+between a HAP and the ground is point-to-multipoint, the serv-
+ing FSO transceiver on the HAP controls multiple accesses from
+ground FSO nodes under it using the WDM technique.
+Each
+ground node is assigned a separate wavelength for up and down
+communication. An IP router on the HAP aggregates IP packets
+heading toward a common cluster within a single flow. The flow
+will be carried by one or more continuous lightpaths between the
+source and destination HAPs. The number of lightpaths is deter-
+mined according to the size of the flow and the transport capacity
+of a wavelength. A WDM switch is installed on each HAP to route
+1
+arXiv:2301.08642v1  [cs.NI]  20 Jan 2023
+
+these lightpaths over the HAP network on a wavelength-switched
+basis. In Figure 1, the blue path HAP1-HAP2-HAP4-HAP5 and
+the red path HAP1-HAP2-HAP3 are two flows.
+1
+2
+2
+1
+1
+1
+1
+2
+3
+3
+3
+3
+3
+WDM switch
+IP router 
+IP router
+WDM switch
+1
+2
+2
+p-HAP-2
+1
+3
+3
+p-HAP-1
+toward HAP-2 of cluster-2
+toward HAP-1 of cluster-1 
+ HAP2
+Inter-HAP FSO transceiver
+Ground FSO node
+Serving FSO
+3
+1
+2
+A cluster
+A cluster
+transceiver
+ HAP1
+ HAP3
+ HAP4
+ HAP5
+Serving zone of HAP 1
+Serving zone of HAP 2
+inter-HAP link
+inter-HAP link
+Figure 1: Multi-hop FSO communication system using HAP.
+In terrestrial FSO communications, the light beams are usually
+set to be very narrow for low transmission energies. However, for
+HAP and ground communication, the serving FSO transceiver of
+the HAP must project a sufficiently wide laser beam for covering
+distributed ground FSO nodes.
+A single serving FSO transceiver has a relatively small foot-
+print owing to the low capacity of the current laser source, and
+the limited sensibility and aperture sizes of ground receivers. The
+calculation in Section 2.1 shows that with a laser source of 1 Watt,
+required received power at receivers of -41.1dBm, and receiver
+aperture radius of 2 m, a single serving FSO transceiver at an
+elevation of 20 km can cover a ground area of 6.691 km radius
+only (see Table 3).
+To extend the coverage of a HAP, we propose using multiple
+serving FSO transceivers arranged in a bundle, as shown in Fig-
+ure 2. Each serving FSO transceiver points in a slightly different
+direction to cover a particular ground area that overlaps other ar-
+eas to create a continuous coverage region. Given a ground region
+to be served, using HAPs with multiple serving FSO transceivers
+reduces the number of required HAPs compared to using HAPs
+with a single serving FSO transceiver. However, the expenditure
+for serving FSO transceivers increases. Therefore, the number of
+serving FSO transceivers to be used on a HAP should be carefully
+considered.
+Regarding the communication between ground nodes and a
+HAP, the multiple serving FSO transceiver model still uses the
+WDM technique, where each ground node is assigned a unique
+wavelength within its cluster to communicate with its HAP. The
+number of ground nodes to be served by a HAP is restricted by
+the number of wavelengths offered by the WDM technique.
+In this study, we focus on identifying the optimal configuration
+of multiple serving FSO transceivers to achieve a minimal-cost
+HAP network for serving a set of ground FSO nodes. The optimal
+configuration should define the number of serving FSO transceivers
+to be set up on a HAP and the beam width for each transceiver.
+The cost of the HAP network includes the investment, energy, and
+maintenance costs.
+Compared with the previous study in reference [7], the current
+research differs in two aspects.
+First, the current research pro-
+Figure 2: A HAP with multiple serving FSO transceivers and
+its footprint.
+poses the use of multiple serving FSO transceivers on each HAP
+instead of a single serving FSO transceiver, as in [7]. Second, the
+current research identifies the optimal beam widths for serving
+FSO transceivers, whereas in [7], the beam widths are predefined.
+The current study also differs from that in [10], where beam size
+was optimized for an inter-HAP link, which is a point-to-point link.
+The remainder of this paper is organized as follows. First, we
+analyze the single and multiple serving FSO transceivers configu-
+rations in Section 2 to determine their ground coverage sizes and
+constraints on transmitter beams. In Section 3, we state the prob-
+lem of designing a minimal-cost HAP-based FSO network, which is
+the target of the optimization of multiple serving FSO transceiver
+configuration. Then, in Section 4, we define a HAP energy con-
+sumption formula and show that solar energy is necessary for keep-
+ing the HAP working in space for a long period. We also present
+a constraint that a HAP must respect to relying uniquely on so-
+lar energy. In Section 5, we present the algorithms for identifying
+the optimal multiple serving FSO transceiver configuration and
+its footprint radius. Section 6 presents the process designing the
+minimal cost HAP-based FSO network using the optimal multi-
+ple serving FSO transceiver configuration. Section 7 presents the
+simulation results. Finally, Section 8 concludes the paper.
+2
+Serving FSO transceiver configu-
+rations
+2.1
+Single serving FSO transceiver configu-
+ration
+In this section, the allowable beam width and ground coverage of
+a single serving FSO transceiver are determined. The beam size
+is restricted to ensure that the received power at a ground point
+within the beam footprint is detectable by receivers.
+2
+
+Figure 3: Surface of the part of sphere blocked by solid angle
+α is calculated as the sum of the surface of all ribbons around
+the sphere when the solid angle varies from α to 0.
+Assume that the transmitter source radiates within a solid angle
+α and that the radiation density is uniform in all directions within
+the solid angle at a distance r from the source. The radiation den-
+sity at distance r is inversely proportional to the surface of the part
+of the sphere radius r blocked by the solid angle α. To calculate
+this surface, we divide the sphere into thin ribbons corresponding
+to open angles of d(α/2). The width of a ribbon is rd(α/2), as
+shown in Figure 3. The radius of the ribbon at zenith angle α/2 is
+r sin(α/2). Thus, the ribbon surface is 2πr sin(α/2)rd(α/2). The
+surface of the part of the sphere blocked by the solid angle α is the
+sum of the surfaces of all ribbons when zenith angle varies from α
+to 0, as follows:
+� 0
+α
+2πr sin (α
+2 )rd(α
+2 ) = 2πr2(1 − cos (α
+2 ))
+Let Ur be the radiation density at distance r and Ptx be the
+transmitted power at the source. We deduce:
+Ur =
+Ptx
+2πr2(1 − cos (α/2))
+(1)
+Let P rx
+j
+be the received power at ground FSO node j.
+The
+received power is proportional to the radiation density and the
+received aperture of the ground node. It is:
+P rx
+j
+=
+e−σLjULjAR
+(2)
+where
+• Lj is the distance between ground FSO node j and its serving
+HAP Hi (see Figure 4),
+• σ is the attenuation coefficient of the links between the HAP
+and ground,
+• ULj is radiation density at distance Lj from the source,
+• AR is the aperture area of the receiver. Let Rrx be the receiver
+aperture radius, then, AR = πR2
+rx.
+In (2), the first term represents the attenuation of laser power
+through the atmosphere, which is described by the exponential
+Beer–Lambert Law [11].
+Figure 4: Received power on border nodes of a coverage area
+is the smallest amongst all nodes in the area.
+By substituting ULj from (1) into (2), we obtain the received
+power at node j as follows:
+P rx
+j
+= e−σLj × Ptx × R2
+rx
+2L2
+j
+×
+1
+1 − cos (α/2)
+(3)
+The power received at node j must not be less than the required
+level of the receiver, denoted by ρrx. It is obvious that point j at
+the border of the ground coverage area receives the least power
+because it is the furthest from the source (see Figure 4). Hence,
+all points in the coverage areas of HAP Hi receive sufficient power
+if and only if the border points receive at least the required power;
+that is,
+P rx
+j
+= e−σH/ cos ( α
+2 ) PtxR2
+rx cos2 ( α
+2 )
+2H2(1 − cos ( α
+2 )) ≥ ρrx
+(4)
+where Lj is substituted by H/ cos( α
+2 ) for border node j.
+Solving inequation (4) yields the beam width of the single serv-
+ing FSO transceiver configuration. Corresponding to beam width
+α, the ground coverage radius of the configuration is:
+Ri = H tan(α
+2 )
+(5)
+Lemma 1. Function P rx
+j
+decreases with α ∈ [0..π].
+Proof of Lemma 1 is given in Appendix A.
+Figure 5 shows the received power at the border of the cover-
+age area with different receiver aperture radius Rrx. This figure
+confirms that P rx
+j
+decrease with an increase in α.
+Let αmax be the value for α that makes P rx
+j (αmax) = ρrx; then
+according to Lemma 1,
+P rx
+j (α) ≥ P rx
+j (αmax) = ρrx, ∀α ∈ [0..αmax]
+thus all α ∈ [0..αmax] satisfy constraint (4).
+Calculations using the parameters given in Table 1 show that
+when Rrx = 2 m, αmax = 37° and the coverage radius is 6.691 km.
+When Rrx = 4 m, αmax
+= 67° and the coverage radius is
+13.237 km.
+3
+
+Ribbon surface= 2πr sin(α/2) r d(α/2)
+Kd(a/2)
+r.sin(a/2)
+a/2
+SourceHAP H;
+α
+H
+Received power Prx
+R;
+Nodej
+coverage area of HAP H 0
+ 2
+ 4
+ 6
+ 8
+ 10
+ 0
+ 20
+ 40
+ 60
+ 80
+ 100
+ 120
+ 140
+ 160
+ 180
+Received power at coverage border (10-8 W)
+Beam size α(degree)
+Rrx=0.125m
+Rrx=1m
+Rrx=2m
+Rrx=4m
+Required at receiver (Prx)
+Figure 5: Received power at the coverage border of the single
+serving FSO transceiver configuration with different receiver
+apertures.
+2.2
+Multiple serving FSO transceiver config-
+uration
+The ground coverage of a HAP can be widened by combining sev-
+eral serving FSO transceivers. Different combinations are possible.
+In this research, we study a straightforward configuration in which
+a principal serving FSO transceiver is in the center projecting
+light perpendicular to the ground, and several identical supplemen-
+tary serving FSO transceivers are set evenly around the principal
+one (Figure 6). Each supplementary transceiver projects slanted
+beams to extend the coverage in one direction. This arrangement
+is referred to as mFSO configuration. Usually, the transmitters in
+a bundle are considered to project signals in parallel. However,
+because of the large principal beam, the supplementary serving
+FSO transceiver projection directions are far from being perpen-
+dicular to the ground, and their footprints are ellipses instead of
+circles.
+To create a continuous coverage region, the footprint of the
+principal serving FSO transceiver and those of the supplemen-
+tary serving FSO transceivers should overlap.
+Therefore, there
+should be a sufficiently large number of supplementary serving
+FSO transceivers to cover entirely the contour of the principal foot-
+print. The extended coverage area is defined as the largest circle
+covered by these footprints (Figure 6). The principal transceiver is
+responsible for the region defined by its footprint. A supplemen-
+tary serving FSO transceiver is responsible for the part limited
+by its footprint, principal coverage circle, and extended coverage
+circle.
+Let α be always the beam width of the principal serving
+FSO transceiver.
+To ensure that ground nodes under principal
+coverage receive sufficient power, α should still respect constraint
+(4), as in the single serving FSO transceiver configuration.
+Let the beam width of a supplementary serving FSO transceiver
+be β. In the responsible area of the supplementary transceiver, the
+points on the extended coverage circle are the farthest from the
+supplementary transceiver; thus, they receive the least power. If
+these points receive at least ρrx, all other points receive sufficient
+power.
+It is easy to note that the footprints of the neighboring supple-
+Figure 6: footprint of multiple FSO transceiver (mFSO) con-
+figuration.
+mentary serving FSO transceivers join each other on the extended
+coverage circle. Let J be such a joint point, the power J receives
+from the supplementary FSO transceiver is defined similar to (3)
+but with beam width β, which is
+P rx
+J
+= e−σ×LJ × Ptx × R2
+rx
+4L2
+J
+×
+2
+1 − cos (β/2)
+(6)
+Thus, β is constrained by the condition P rx
+J
+≥ ρrx, which gives:
+e−σ×LJ
+PtxR2
+rx
+2L2
+J(1 − cos (β/2)) ≥ ρrx
+(7)
+Let us denote the extended coverage radius by Rext then
+LJ =
+�
+H2 + R2
+ext
+(8)
+Appendix B presents detailed calculations of LJ and Rext. The
+calculations yielded the following results
+Rext = H2 tan( ξ+α
+2 ) − tan( α
+2 )(1 − tan2( ξ+α
+2 ))
+1 − tan2( ξ+α
+2 ) + 2 tan( ξ+α
+2 ). tan( α
+2 )
+(9)
+where
+tan(ξ + α
+2
+) = tan(γ) + tan(θ)
+1 − tan(γ). tan(θ). cos( π
+m)
+tan(γ) = tan(α
+2 ). cos( π
+m)
+tan(θ) =
+�
+sin2( β
+2 ) − sin2( α
+2 ). sin2( π
+m)
+cos( β
+2 )
+(10)
+(11)
+(12)
+and m is the number of supplementary FSO transceivers set
+around the principal one.
+We can remark that Rext and thus LJ depend on α, β and m.
+Hereafter, Rext is sometimes denoted by Rext(α, m, β) and LJ by
+LJ(α, m, β) to express these dependencies.
+4
+
+Principle coverage circle
+0
+K'
+K
+2T
+m
+Extended coverage circle3
+Problem of designing minimal cost
+HAP network
+There are several costs in a HAP network, such as investment, en-
+ergy, and maintenance costs. Based on the expected life duration
+and maintenance cycle of a HAP, these costs can be distributed
+by day as 1) daily amortization cost representing investment cost,
+2) average daily maintenance cost, and 3) daily energy cost. Con-
+sequently, the problem of minimizing network cost becomes min-
+imizing the daily network cost, which comprises these three com-
+ponents.
+Following variables are introduced for formulating mathemati-
+cally the daily network cost:
+• K: Number of HAPs in the network. The HAPs are indexed
+by i ∈ 1..K.
+• niF
+i : Number of FSO transceivers used on HAPi for inter-
+HAP communications.
+• nsF
+i : Number of serving FSO transceiver of HAPi.
+Let ζday
+H
+and ζday
+F
+be constants that express the daily amortiza-
+tion costs of a HAP and an FSO transceiver, respectively. These
+costs are defined as the ratio of the prices of the HAP or FSO
+transceiver to their expected lifetime duration. Then, the overall
+daily amortization cost of the HAP network is:
+Kζday
+H
++ (
+K
+�
+i=1
+nsF
+i
++
+K
+�
+i=1
+niF
+i )ζday
+F
+(13)
+To evaluate the daily maintenance and energy costs, we need to
+consider the HAP design. HAPs are classified into two categories
+based on the underlying physical principle that provides the lifting
+force for the HAPs: aerodynamic (the HAP is heavier than air)
+and aerostatic (the HAP is lighter than air).
+While aerostatic
+platforms use buoyancy to float in the air, aerodynamic platforms
+use dynamic forces created by movement through the air [8]. In
+general, both aerostatic and aerodynamic systems require a “flying
+energy” to keep the HAP relatively stable for maintaining FSO
+communication between HAPs and that between HAPs and FSO
+ground nodes. An aerodynamic system requires a large propulsion
+power to move. Aerostatic systems typically consume less energy
+than aerodynamic systems do. To be able to operate for a long
+duration in space, HAPs are mainly unmanned.
+HAPs are equipped with different energy resources such as on-
+site production (e.g., solar energy harvested by solar panels) or
+rechargeable energy (e.g., batteries or fuel cells brought from the
+ground).
+Solar energy-based HAPs can operate continuously in
+space until they are lowered for maintenance purpose. Recharge-
+able energy-based HAPs are lowered once the reserved energy is
+depleted. In brief, the continuous in-space working duration of a
+HAP is limited by its available energy, which is relatively fixed by
+the HAP design, its energy consumption level, which varies de-
+pending on the payload weight and communication of the HAP,
+and its maintenance cycle.
+We define the maintenance cost of a HAP as the expense of low-
+ering the HAP to perform technical maintenance, energy recharge
+on the ground, and then reinstall it in space.
+Let di be the number of days on which HAPi can operate con-
+tinuously in space. Let ζmtn be constant expressing the cost of
+one time lowering a HAP, maintaining it, recharging it, and then
+reinstalling it in space. The daily maintenance cost of the HAP
+network is
+K
+�
+i=1
+ζmtn
+di
+(14)
+Regarding the daily energy cost, we consider solar energy to be
+free, whereas the solar panel cost is counted in the cost of the HAP.
+The cost of rechargeable energy is part the maintenance cost. As a
+result, the energy cost does not explicitly represent the total cost.
+Nonetheless, the energy consumption level of a HAP affects its
+in-space working duration di; therefore, we analyze this in Section
+4.
+Combining (13) and (14), we obtain the following overall daily
+cost of the HAP network:
+Cost = Kζday
+H
++ (
+K
+�
+i=1
+nsF
+i
++
+K
+�
+i=1
+niF
+i )ζday
+F
++
+K
+�
+i=1
+ζmtn
+di
+(15)
+The problem of minimizing daily cost of the HAP network is
+stated as follows.
+• Given input parameters including
+– NFSO: Set of ground FSO nodes and their coordinates.
+The number of nodes in the set is denoted as |NFSO|,
+– M:
+Data traffic to be carried between ground FSO
+nodes. This is the list of bandwidth demands between
+the ground nodes.
+• Outputs to seek are
+– A HAP network with HAP locations and inter-HAP
+links,
+– Beam width to set to each serving FSO transceiver.
+• Optimization objective is
+– Minimizing the daily cost expressed in (15) of the HAP
+network.
+The following two remarks drive us to conduct further analyses
+in subsequent sections.
+First, if a HAP has self-sufficient solar
+energy, its in-space working duration di is not limited by its energy
+consumption but depends uniquely on the maintenance cycle of the
+HAP, which is usually constant. In Section 4, we show the daily
+energy consumption of a HAP and the constraint that a HAP
+needs to respect to rely solely on solar energy.
+Second, the cost of the HAP network increases with an increase
+in the number of FSO transceivers and HAPs.
+The number of
+HAPs can be reduced by increasing ground coverage. To increase
+ground coverage, more serving FSO transceivers can be used on
+each HAP, but this introduces greater energy consumption and ex-
+tra amortization cost. Section 5 focuses on identifying the optimal
+configuration for serving FSO transceivers on a HAP to achieve a
+minimal HAP network cost.
+4
+Daily energy consumption of a
+HAP with payload
+Several parameters affect the power consumption of a HAP. The
+descriptions and notations of these parameters are listed in section
+Energy parameters of Table 1. Most parameters were set based on
+industrial experimental projects such as the Loon project [4], Stra-
+tobus project [6], and other studies listed in the reference column.
+Section 7.1 presents the choice of parameter values in detail.
+5
+
+Param.
+nota-
+tions
+Descriptions
+Values
+References
+Cost related parameters
+ζday
+H
+Daily amortization cost of a HAP.
+100
+ζday
+F
+Daily amortization cost of an FSO transceiver on HAP.
+10
+ζmtn
+Cost of one-time maintenance of a HAP including lowing it down,
+1000
+maintenance, charging and reinstall it in the stratosphere.
+Dm
+Maintenance cycle.
+365 days
+[6]
+Energy parameters
+Esolar
+Minimum daily harvested solar energy by a HAP.
+42 - 290 kWh
+[12]
+ρavion
+Power consumed by the avionic part of a HAP to carry an unit of mass.
+2 W/kg
+ρHCM
+F
+Power for heating, cooling, and management for each FSO on HAP.
+20 W
+[4]
+ρPAT
+Power consumed by a PAT system.
+15W
+[13]
+ρinter
+F
+Power consumed by inter-HAP FSO transceivers for laser source (0.1 W),
+heating/cooling/management (20 W) and PAT (15 W).
+35.1 W
+[4]
+Inter-HAP FSO link parameters
+C2
+n
+Atmosphere structure parameter.
+5.0 × 10−18m−2/3
+-
+Attenuation coefficient.
+3.5 × 10−6m−1
+[4]
+-
+Coupling loss.
+45 dBm
+-
+Transmitted power of an inter-HAP FSO transceiver.
+0.1 W
+[4]
+-
+Receiver aperture diameter of an inter-HAP FSO transceiver.
+0.037 m
+[4]
+-
+Beam width of an inter-HAP FSO transmitter.
+280 µrad
+[4]
+HAP-ground link parameters and variables
+σ
+Attenuation coefficient.
+3.5 × 10−6m−1
+ρFSO
+tx
+Transmitted power of the laser source of a serving FSO transceiver.
+1 Watt
+Rrx
+Receiver aperture radius of a ground FSO transceiver.
+0.05 m
+SONABeam [1]
+ρrx
+Required received power at a ground FSO transceiver.
+7.76.10−8 W
+-41.1 dBm in [4]
+Other parameters
+H
+Elevation of HAPs.
+20 km
+LHH
+Maximum length of an inter-HAP link so that its BER is under δ.
+88 km
+δ
+BER threshold for inter-HAP links and lightpaths between HAPs.
+W
+The number of wavelengths in WDM technique.
+40; 80
+µH
+Platform mass excluding FSO transceivers.
+28.5 kg; 500 kg
+[4]
+µF
+FSO transceiver mass.
+6.3 kg
+[4]
+Table 1: Parameters. Greek characters are used for denoting constant parameters.
+Let us consider the power consumption of a single HAP Hi
+that has m serving FSO transceiver and niF
+i
+inter-HAP FSO
+transceivers. The power consumption includes:
+• P avion
+Hi
+: Power draw of avionic part for maintaining Hi with
+payload in space.
+• P down
+Hi
+: Power draw of all serving FSO transceivers on HAP
+Hi.
+This power includes the heating/cooling/management
+power, laser transmitted power of all serving FSO transceivers
+on the HAP, and the power consumed by the Pointing Acqui-
+sition and Tracking (PAT) system of the HAP.
+• P inter
+Hi
+: Power draw of all inter-HAP FSO transceivers on
+HAP Hi for inter-HAP communication. The power includes
+the heating/cooling/management, and PAT power for each
+inter-HAP FSO transceiver.
+Inter-HAP FSO transceivers
+are oriented towards different remote HAPs; therefore, each
+transceiver must have a PAT system.
+The total daily energy consumption (by 24 hours) of Hi is
+Econsum = (P avion
+Hi
++ P down
+Hi
++ P inter
+Hi
+) × 24
+(16)
+To breakdown further P avion
+Hi
+, P down
+Hi
+, and P inter
+Hi
+, we introduce
+following parameters:
+• ρavion: Power consumed by the avionic part of the HAP to
+carry a unit of mass.
+• ρFSO
+tx
+: Transmitted power of each serving FSO transceiver.
+Because the current power of laser source is limited to 1 W,
+which is very small in comparison with the power consumed
+by other factors on the HAP, we consider that ρFSO
+tx
+= 1 W,
+regardless of the beam width of the serving FSO transceiver.
+• ρHCM
+F
+: Power draw for heating, cooling, and management. It
+is also considered constant for each serving FSO transceiver
+and is set to ρHCM
+F
+= 20 W, according to reference [4].
+• ρPAT : Power draw for Pointing, Acquisition and Tracking
+activity; it is another constant and is set to ρPAT = 15 W
+[13]. A HAP system uses a single PAT for its set of serving
+FSO transceivers.
+• ρinter
+F
+: Power draw of a single inter-HAP FSO transceiver
+including communication, heating, cooling, management, and
+6
+
+PAT. According to [4], 0.1 W laser power is sufficient for an
+inter-HAP communication of 100 km distance. In this study,
+we limited the inter-HAP link length to less than 100 km and
+considered the laser power constantly 0.1 W regardless of the
+distance. Therefore, ρinter
+F
+= ρHCM
+F
++ ρPAT + 0.1.
+• µH: Mass of the HAP.
+• µF: Mass of an FSO on the HAP.
+Assuming that P avion
+Hi
+is linearly proportional to the weight of
+the HAP by ρavion,
+P avion
+Hi
+= [µH + (nsF
+i
++ niF
+i )µF]ρavion
+(17)
+P down
+Hi
+is
+the
+sum
+of
+the
+power
+consumed
+by
+serving
+FSO transceivers and PAT activity of the HAP; thus,
+P down
+Hi
+= nsF
+i (ρFSO
+tx
++ ρHCM
+F
+) + ρPAT
+(18)
+P inter
+Hi
+is the sum of the power consumed by inter-HAP FSO
+transceivers; thus,
+P inter
+Hi
+= ρinter
+F
+.niF
+i
+(19)
+Substituting (17), (18), and (19) into (16), we obtain the daily
+power consumption of a HAP as
+Econsum = {[µH + (nsF
+i
++ niF
+i )µF]ρavion
++ nsF
+i (ρFSO
+tx
++ ρHCM
+F
+) + ρPAT
++ ρinter
+F
+niF
+i } × 24
+(20)
+4.1
+Necessity of solar energy and utilization
+constraint
+Current HAPs mainly use energy from solar panels mounted on
+HAP wings and/or energy from batteries or hydrogen fuel cells
+(HFC) onboard. Solar energy can be harvested and charged into
+batteries during the day for nighttime use. Harvested solar energy
+varies with year time and location. According to the experiments
+in [12], in York, UK, the harvested solar power is 42–80 kWh/day,
+and in Enugu, Nigeria, it is 290–545 kWh/day, depending on the
+size of the solar panel.
+Figure
+7
+depicts
+the
+total
+daily
+energy
+consumption
+of
+a HAP, calculated from (20), versus the number of serving
+FSO transceivers.
+Parameters were ρavion = 2 W/kg, ρPAT =
+15 W, HAP weights µH = 28.5 kg or 500 kg. The HAP carried
+10 inter-HAP FSO transceivers.
+The referenced daily solar en-
+ergy levels were the minimum daily solar energy levels in York
+and Enugu. From a certain number of serving FSO transceivers,
+a HAP consumes more energy than the harvested solar energy
+in York, and an HFC would be necessary. Owing to the limited
+payload capacity of a HAP, its HFC capacity is also very limited.
+According to [8], the current state-of-the-art fuel-cell density is
+approximately 1600 Wh/kg. A lightweight HAP, such as a Google
+balloon weights 28.5 kg, cannot carry heavy long-lasting fuel cells
+on board. The larger HAP Stratobus can carry up to 450 kg, but
+it weights already 7 tons leading to high energy consumption for
+flying. Even if the Stratobus payload capacity is reserved for the
+HFC, its energy would quickly run out within a few days.
+Based on this observation, we believe that long-duration flights
+should consider solar energy as the principal energy source. In this
+case, the power consumption of a HAP with payload must not
+ 0
+ 50000
+ 100000
+ 150000
+ 200000
+ 250000
+ 300000
+ 350000
+ 0
+ 20
+ 40
+ 60
+ 80
+ 100
+Total daily energy consumption (W-hr)
+Number of serving FSO transceivers
+µΗ=28.5 kg
+µΗ=500 kg
+Min solar energy at York
+Min solar energy at Enugu
+Figure 7: Energy consumption by a HAP with different num-
+ber of serving FSO transceivers in comparison with the min-
+imum harvested solar energy at York and Enugu. ρavion =
+2/kg W and ρPAT = 15 W.
+exceed the daily harvested solar energy. Let the daily harvested
+solar energy be Esolar; then,
+�
+[µH + (nsF
+i
++ niF
+i )µF]ρavion + ρPAT
++ nsF
+i (ρFSO
+tx
++ ρHCM
+F
+) + ρinter
+F
+niF
+i
+�
+≤ Esolar
+24
+(21)
+According to Figure 7, solar energy provision does not need to
+be very large. A solar energy level between the minimum harvested
+in York and Enugu allows a 500 kg HAP to carry at least 6 serving
+FSO transceivers. A HAP can carry hundreds FSO transceivers
+with more than 125 kWh solar energy. Therefore, it is realistic to
+rely on the solar energy. Hereafter, we consider that HAPs solely
+use solar energy.
+Despite self-sufficient solar energy, HAPs still need to be lowered
+periodically for maintenance, for example, after one year in the
+case of Stratobus [6]. Let us denote the maintenance cycle as a
+constant Dm. Then
+di = Dm,
+∀i ∈ 1..K
+(22)
+5
+Optimal mFSO configuration
+Using multiple serving FSO transceivers increases the expense of
+FSO transceivers, although it can reduce the expense of HAPs.
+This section aims to determine the mFSO configuration that min-
+imizes the HAP network cost defined in (15). We assume that all
+HAPs use identical mFSO configurations, that is, identical prin-
+cipal beam width α, supplementary beam width β and number of
+supplementary serving FSO transceivers m.
+Let us now consider the dependence of the HAP network cost on
+mFSO configuration. As each HAP has m supplementary serving
+FSO transceivers and uses only solar energy, the cost (15) becomes
+Cost = Kζday
+H
++ (Km +
+K
+�
+i=1
+niF
+i )ζday
+F
++ Kζmtn
+Dm
+Cost is a function of K, m and niF
+i . K depends on the coverage
+radius Rext(α, m, β) of the mFSO configuration. niF
+i , as the num-
+ber of inter-HAP links of HAP i, depends on the traffic demand
+7
+
+set M. Hence, Cost depends on mFSO configuration and M. It
+is difficult to determine the optimal mFSO configuration without
+considering M. To relax the dependance on M, we estimate Cost
+by a function that depends solely on mFSO configuration, that is,
+tuple (α, m, β); then try to find an instance (α, m, β) minimizing
+the estimated cost in expecting that the instance also drives the
+real cost to a minimum.
+5.1
+Cost estimation
+Figure 8: A ground area is divided into grid of square cells;
+each cell is circumscribed by a circle representing a serving
+zone of a HAP.
+First, we estimate the number of HAPs K.
+Samples of the
+estimation are datasets with uniformly distributed ground nodes.
+Let S be the surface of the ground area containing those nodes,
+and W the number of wavelengths in the WDM technique. We
+divide the ground zone S into a grid of square cells of size ℓ × ℓ,
+each one will be covered by a HAP (see Figure 8). To be served
+by a HAP, a cell must satisfy the following two conditions:
+1. A cell can contain at most W ground nodes because a HAP
+can use at most W wavelengths to serve ground nodes. Owing
+to the uniform distribution of ground nodes, we have
+ℓ2
+S |NFSO| ≤ W
+2. A cell must be contained inside by a circle radius equivalent
+to the extended radius Rext of a HAP
+ℓ ≤
+√
+2Rext
+The maximum number of HAPs required to cover region S is the
+number of cells. Let this number be ˆK, then,
+ˆK = S
+ℓ2 = ⌈max {|NFSO|
+W
+,
+S
+2R2
+ext
+}⌉
+(23)
+Hence, ˆK is an overestimation of the number of HAPs.
+Next, we estimate the value of niF
+i .
+Let V be the maximum
+number of inter-HAP links that a HAP may have. Then
+niF
+i
+≤ V, ∀i.
+Finally, Cost can be overestimated as:
+�
+Cost = ˆK
+�
+ζday
+H
++ (m + V + 1)ζday
+F
++ ζmtn
+Dm
+�
+(24)
+�
+Cost is a function of Rext(α, m, β) and m while V is a parameter
+of the estimator. The estimation is more precise when V is set close
+to the actual number of inter-HAP links of a HAP, and coarser
+otherwise.
+5.2
+Algorithms finding optimal configura-
+tion
+Given α and m, a larger β results in a larger Rext, and thus a
+smaller ˆK and �
+Cost. Therefore, β should be set to the largest value
+according to (7) for a given α and m. It is worth noting that the
+value of β does not affect the solar energy consumption because the
+laser power ρFSO
+tx
+is small and is considered constant. Determining
+the optimal configuration becomes finding the optimal values of α
+and m.
+Algorithm 1 Find the optimal mFSO configuration
+1: function Find-optimal-mFSO
+2:
+niF
+i ← V
+3:
+cMin ← ∞
+▷ cost min
+4:
+αMax ← maximum α by (4)
+5:
+for α = αMax . . . 0 do
+6:
+mMax ← calculated by (25)
+▷ max m
+7:
+mOpt ← 0
+▷ optimal m
+8:
+for m = 0 . . . mMax do
+9:
+β ← Beta-max(α, m)
+▷ max β
+10:
+Calculate Rext(α, m, β) using (9),(10),(11) (12)
+11:
+Calculate �
+Cost(α, m, β) using (24)
+12:
+if �
+Cost < cmin then
+13:
+cmin ← �
+Cost
+14:
+αOpt ← α
+▷ optimal α
+15:
+mOpt ← m
+▷ optimal m
+16:
+βOpt ← β
+▷ optimal β
+17:
+end if
+18:
+end for
+19:
+end for
+20:
+return αOpt, mOpt, βOpt
+21: end function
+Algorithm 2 Find the maximum β given α, m
+1: function Beta-max(α, m)
+2:
+for β = 0 . . . 180 do
+3:
+Calculate Rext(α, m, β) using (9),(10),(11) (12)
+4:
+Calculate LJ using (8)
+5:
+Calculate P rx
+J
+using (6)
+6:
+if P rx
+J
+¡ρrx then ▷ Looking for the first β violate
+constraint (7)
+7:
+return β-1
+▷ the previous trial β was the
+maximum
+8:
+end if
+9:
+end for
+10: end function
+Following an exhaustive search approach, we examine all possi-
+ble values of α and m to seek for the pair that minimizes �
+Cost
+in (24).
+The search range of α is from 0° to the maximum
+8
+
+Serving zone
+of a HAPvalue set by constraint (4). The number of supplementary serving
+FSO transceivers m is also limited. Indeed, since the number of
+inter-HAP links of a HAP can go up to V as set in Section 5.1, and
+nsF
+i
+= m + 1, ∀i, then from the energy constraint (21), we deduce
+the upper bound for m:
+m ≤
+Esolar
+24
+− (VµFρavion + Vρinter
+F
++ µHρavion + ρPAT)
+µFρavion + ρHCM
+F
++ ρFSO
+tx
+− 1 (25)
+Algorithm 1 implements the exhaustive search idea. First, two
+nested loops scan all possible values of α satisfying constraint (4)
+and all possible values of m satisfying (25) to find the pair that
+minimizes �
+Cost in (24). For each pair (α, m), the largest value
+of β according to constraint (7) is selected using Algorithm 2.
+The optimal mFSO configuration is reported by the algorithms as
+(αOpt, mOpt, βOpt).
+Algorithm 2 finds the maximum β that satisfies constraint (7)
+for a given pair of (α, m) by testing the possible values of β in-
+creasingly from 0 until the received power P rx
+J
+at the border of
+the extended coverage area reaches the required received power
+ρrx. The received power P rx
+J
+is calculated using the set of equa-
+tions (6), (8), (9),(10),(11), and (12).
+In the implementation of both algorithms, α and β step by 1°
+after each iteration. Finer stepping allows obtaining more accu-
+rate results. However, even with 1° stepping, the variation in the
+optimal Rext is only a few hundred meters, which is negligible in
+comparison to the absolute value of Rext which is in the range of
+6-30 kilometers.
+The complexity of Algorithm 1 is O(m) because α ≤ π. The
+complexity of Algorithm 2 is constant because β ≤ π.
+6
+Design HAP network topology
+This section presents the HAP network design using the optimal
+configuration identified above. Let denote Linter as the number
+of inter-HAP links. Since �K
+i=1 niF
+i
+is the total number of inter-
+HAP FSO transceivers, it is equal to 2Linter. The network cost
+becomes:
+Cost = Kζday
+H
++ (K(m + 1) + 2Linter)ζday
+F
++ K ζmtn
+Dm
+and is equivalent to
+Cost = K
+�
+ζday
+H
++ (m + 1)ζday
+F
++ ζmtn
+Dm
+�
++ 2Linterζday
+F
+(26)
+The cost is proportional to the number of HAPs K and the
+number of inter-HAP links Linter.
+We consider that the daily
+amortization cost of a HAP is much greater than that of an FSO
+transceiver; thus, the coefficient of K is much greater than the
+coefficient of Linter in Cost. Consequently, K should be prioritized
+to minimize over Linter. Therefore, the topology design is broken
+into following two steps:
+i) ground nodes are clustered into equal radius circles that will
+become serving zones of HAPs in such a way that the number
+of clusters is the smallest for minimizing K;
+ii) corresponding HAPs are located at the centers of clusters but
+at an elevation of 20 km and are interconnected by the fewest
+number of inter-HAP links, Linter.
+A HAP network topology design algorithm was proposed in [7]
+following these two steps. In this algorithm, the clustering radius
+was not determined but was left as an input of the algorithm. In
+the current study, we set the clustering radius as the extended
+coverage radius Rext of the optimal mFSO configuration to drive
+towards a HAP network with minimal Cost. The main steps of the
+Figure 9: HAP network design flowchart.
+HAP network design process are presented in Figure 9, where the
+steps taken from [7] are shown in color. The process is explained
+as follows:
+• Initialize V, the maximum number of inter-HAP links of a
+HAP, by a constant.
+• Calculate the optimal mFSO configuration using Algorithm
+1, and set the clustering radius as its Rext.
+• Apply the clustering algorithm proposed in [7] to distribute
+ground nodes into clusters of radius Rext while keeping the
+number of ground nodes in each cluster under W. Each cluster
+becomes a serving zone of a HAP. The HAP is located at the
+center of the cluster but at an elevation of 20 km.
+• Bandwidth demands between ground nodes belonging to dif-
+ferent serving zones are bundled into lightpaths between
+corresponding HAPs, creating the inter-HAP traffic matrix
+MHAP .
+• Apply HAP topology design algorithm proposed in [7] to build
+the HAP topology.
+The algorithm begins with an empty
+topology. It finds a route for each lightpath demand of MHAP
+9
+
+Init V
+Find optimal conf.
+(α, m, β) and Rext
+Clustering ground nodes
+with Rext radius [7]
+V=V+1
+Calculate inter-HAP
+Design HAP topo [7]
+M
+HAF
+No
+is
+routed entirely
+Report HAP topofrom a full-mesh graph linking all HAPs within communica-
+tion distance limit LHH. Each time a lightpath uses an inter-
+HAP link that has not yet been included in the current HAP
+topology, the link is incorporated into the topology. The link
+in the topology is prioritized for use in building routes for the
+next lightpath demands.
+• Once all lightpath demands in MHAP are routed, the final
+topology is achieved. Otherwise, routing may fail due to the
+low connectivity between HAPs. In this case, V is increased
+by one, and the process is repeated until all lightpath de-
+mands in MHAP are routed.
+7
+Simulation results
+The algorithms for finding the optimal mFSO configuration were
+implemented and integrated with the topology designed algorithm
+described in Section 6. We performed simulations with practical
+parameters and evaluated the efficiency of mFSO configuration
+compared to the single serving FSO transceiver configuration.
+7.1
+Parameter values
+The simulation parameters are listed in Table 1. The values of
+these parameters were chosen according to experiments reported
+in the literature. This subsection explains the choices of the pa-
+rameter values.
+Cost-related parameters:
+The cost-related parameters are set
+such that the daily amortization cost of a HAP is significantly
+greater than that of an FSO transceiver, and the one-time mainte-
+nance cost is significantly higher than the daily amortization cost
+of a HAP. The maintenance cycle of a HAP is set as Dm = 1 year
+according to published information on Stratobus [6].
+Energy-related parameters:
+• Esolar - daily harvested solar energy.
+We considered daily
+solar energy levels between the minimum daily solar energy
+values in York and Enugu reported in [12], which were 42
+kWh and 290 kWh, respectively.
+• ρavion - power consumed by the avionic part of a HAP to
+carry a unit of mass. Although the power-to-mass ratio can
+be estimated as 6 W/kg according to [12], the published power
+rates of real systems are smaller. For aerodynamic systems
+such as Zephir-S, Zephir-T [14], and Phasa-35 [15], ρavion
+varies from 2.68 -3.04 W/kg. Indeed, Zephir-S weighs 80 kg
+(75 kg platform and 5 kg payload) and consumes 243 W,
+Zephir-T weighs 160 kg (140 kg platform and 20 kg payload)
+and consumes 429 W, and Phasa-35 weighs 165 kg (150 kg
+platform and 15 kg payload) and consumes 459 W. Aerostatic
+systems consume even less power. The Stratobus weighs 7000
+kg and consumes 5 kW when it carries a 250 kg payload
+and 8 kW when it carries 450 kg [6]. Thus, the power-to-
+mass ratio of Stratobus is between 0.69 and 1.07 W/kg only.
+Therefore, in this simulation ρavion was set to 2 W/kg.
+• ρPAT - power consumed by a PAT system; it was set to 15 W
+according to [13].
+• ρHCM
+F
+- power consumed for heating, cooling, and manage-
+ment; it was set to 20 W according to [4].
+• ρinter
+F
+- power consumed by an inter-HAP FSO transceiver;
+it was set to 35.1 W including laser power, ρHCM
+F
+and ρPAT.
+Inter-HAP link parameters: These parameters were set to values
+similar to those provided in the Loon project [4].
+HAP-ground FSO link parameters: The attenuation coefficient
+of an FSO link between a HAP and a ground node is set identical
+to that of inter-HAP links. The required received power ρrx at a
+ground node was set according to [4]. The aperture radius Rrx of
+a ground FSO receiver was set according to the commercial FSO
+transceiver SONABeam [1].
+Other parameters:
+• δ - BER threshold for inter-HAP links and lightpaths. We
+set δ = 10−3 because errors with that BER can be corrected
+using current Forward Error Correction (FEC) techniques.
+• LHH - the maximum allowable distance between two HAPs
+such that the BER of an inter-HAP link is less than δ = 10−3.
+Using the inter-HAP FSO link parameters listed in Table 1,
+the calculation yielded LHH = 88 km.
+• µH - platform mass; it varies significantly from one design
+to another. The Loon balloon weighs just 28.5 kg while the
+Stratobus weighs 7000 kg. With ρavion = 2 W/kg, a HAP
+weighing more than 7000 kg already consumes 326 kWh/day
+to carry itself, which is more than the maximum harvested
+solar energy, leading to no remaining energy to carry FSO
+transceivers. Therefore, µH = 500 kg was used in the simula-
+tions.
+• µF - mass of an FSO transceiver on HAPs. It was set accord-
+ing to the FSO transceiver used in the Loon project, which
+weighs 6.3 kg [4]. This value is consistent with the weights
+between 8 and 10 kg of commercial terrestrial SONABeam
+FSO transceivers [1].
+• W - the number of wavelengths per FSO link. It was set to
+40 or 80 according to the current WDM technique.
+The test dataset contained 19 test cases, each with 400 – 2800
+ground nodes.
+The ground FSO node locations were randomly
+generated on a square surface of 100 × 100 km, which is the size
+of a large metropolis.
+The test cases had different numbers of
+ground nodes, reflecting different ground node densities. The traf-
+fic requirement M contained demands randomly generated between
+ground FSO nodes such that the total incoming or outgoing traffic
+of a ground FSO node did not exceed 1 Gbps, which is the capacity
+of a single wavelength.
+Initially, V was set to 10.
+The optimal multiple serving
+FSO transceiver configuration (α, m, β) was calculated using Algo-
+rithms 1 and 2. The extended radius Rext of the optimal configura-
+tions was calculated using (9) and was then used as the clustering
+radius in the HAP topology design step.
+With Esolar = 42 kWh and W = 40, V must be increased to 12
+to get all demands in MHAP routed successfully for all test cases.
+With all other Esolar and W values, the topology design algorithm
+successfully routed all demands in MHAP for all test cases right
+with initial V = 10.
+Figure 10 illustrates the HAP locations and their footprints
+calculated using the proposed algorithms for a test case of 1005
+ground FSO nodes, Esolar = 75 kWh, and W = 80.
+7.2
+mFSO configuration versus single serv-
+ing FSO transceiver configuration
+Table 3 lists the maximum beam width αmax according to (4)
+and the maximum ground coverage radius of the single serving
+FSO transceiver configuration when the receiver aperture radius
+10
+
+Esolar = 42 kWh
+Esolar = 50 ∼ 290 kWh
+W = 40, V = 12
+W = 80, V = 10
+W = 40, V = 10
+W = 80, V = 10
+|NFSO|
+α
+m
+β
+Rext
+Cost
+α
+m
+β
+Rext
+Cost
+α
+m
+β
+Rext
+Cost
+α
+m
+β
+Rext
+Cost
+(1)
+(2)
+(3)
+(4)
+(5)
+(6)
+(7)
+(8)
+(9)
+(10)
+(11)
+(12)
+(13)
+(14)
+(15)
+(16)
+(17)
+(18)
+(19)
+(20)
+(21)
+480
+37
+0
+-
+6691
+15308
+37
+0
+-
+6691
+13488
+37
+13
+16
+11929
+10010
+37
+13
+16
+11929
+9470
+588
+37
+0
+-
+6691
+16639
+37
+0
+-
+6691
+14519
+37
+13
+16
+11929
+9304
+37
+13
+16
+11929
+8964
+763
+37
+0
+-
+6691
+18495
+37
+0
+-
+6691
+15815
+37
+13
+16
+11929
+9990
+37
+13
+16
+11929
+9510
+854
+37
+0
+-
+6691
+19141
+37
+0
+-
+6691
+16341
+37
+13
+16
+11929
+10493
+37
+13
+16
+11929
+9933
+998
+37
+0
+-
+6691
+19101
+37
+0
+-
+6691
+16701
+37
+13
+16
+11929
+10855
+37
+13
+16
+11929
+10215
+1005
+37
+0
+-
+6691
+19068
+37
+0
+-
+6691
+16308
+37
+13
+16
+11929
+11138
+37
+13
+16
+11929
+10478
+1150
+37
+0
+-
+6691
+19666
+37
+0
+-
+6691
+16926
+37
+13
+16
+11929
+10915
+37
+13
+16
+11929
+10275
+1345
+37
+0
+-
+6691
+20644
+37
+0
+-
+6691
+17564
+37
+13
+16
+11929
+11741
+37
+13
+16
+11929
+10395
+1477
+37
+0
+-
+6691
+20752
+37
+0
+-
+6691
+17612
+37
+12
+16
+11539
+13115
+37
+13
+16
+11929
+10335
+1523
+37
+0
+-
+6691
+21895
+37
+0
+-
+6691
+18595
+37
+12
+16
+11539
+14053
+37
+13
+16
+11929
+10375
+1675
+37
+0
+-
+6691
+21735
+37
+0
+-
+6691
+18535
+37
+11
+16
+11042
+14128
+37
+13
+16
+11929
+10495
+1736
+37
+0
+-
+6691
+22461
+37
+0
+-
+6691
+19301
+37
+11
+16
+11042
+14874
+37
+13
+16
+11929
+10455
+1911
+37
+0
+-
+6691
+22481
+37
+0
+-
+6691
+19021
+37
+10
+16
+10395
+14869
+37
+13
+16
+11929
+10495
+2009
+37
+0
+-
+6691
+22641
+37
+0
+-
+6691
+19321
+37
+10
+16
+10395
+15575
+37
+13
+16
+11929
+10595
+2135
+37
+0
+-
+6691
+22761
+37
+0
+-
+6691
+19221
+37
+10
+16
+10395
+16493
+37
+13
+16
+11929
+10575
+2304
+37
+0
+-
+6691
+22881
+37
+0
+-
+6691
+19301
+37
+9
+16
+9524
+18192
+37
+13
+16
+11929
+10655
+2325
+37
+0
+-
+6691
+22368
+37
+0
+-
+6691
+18948
+37
+9
+16
+9524
+18555
+37
+13
+16
+11929
+10675
+2491
+37
+0
+-
+6691
+22761
+37
+0
+-
+6691
+19401
+37
+8
+16
+8946
+18660
+37
+13
+16
+11929
+10655
+2753
+37
+0
+-
+6691
+23346
+37
+0
+-
+6691
+19926
+37
+8
+16
+8946
+20284
+37
+13
+16
+11929
+11178
+Table 2: Optimal configurations and costs of all test cases with Rrx = 2 m.
+Receiver aperture
+Maximum beam
+Maximum
+radius Rrx (m)
+width αmax
+coverage radius (m)
+2
+37 °
+6691
+4
+67 °
+13237
+Table 3: Maximum beam width and coverage radius of single
+serving FSO transceiver configuration.
+was varied. Table 4 lists the extended coverage radius of the max-
+imum mFSO configuration for different solar energy levels and
+receiver aperture radii. The maximum mFSO configuration was
+obtained using the largest principal beam αmax, largest m accord-
+ing to (25), and largest β according to (7), given αmax and m.
+The coverage radius of the maximum mFSO configuration was ex-
+tended approximately twice in comparison with that of single FSO
+transceiver configuration, except for Esolar = 42kWh. When solar
+energy level increased, the maximum m increased; thus, the ex-
+tended coverage radius increased. However, when m was already
+large, the extention increased slowly with m.
+Additionally, the
+maximum extended coverage was much larger when Rrx = 4 than
+Rrx = 2m because a receiver can accept weaker signals with larger
+apertures.
+To compare the network costs incurred by the two configura-
+tions, we examined the detailed results in Table 2.
+The table
+lists the optimal mFSO con��gurations and network costs. When
+Esolar = 42kWh, the optimal number of supplementary serving
+FSO transceivers is m = 0; thus, the configuration uses a single
+serving FSO transceiver. Therefore, these cases were used as ref-
+erences for single serving FSO transceiver configuration.
+When
+Esolar > 50kWh, all optimal configurations were truly mFSO,
+and the results were identical for all solar energy levels.
+The
+numbers indicate that mFSO configuration offered significantly
+Esolar
+Max Rext (m)
+(kWh)
+Max m
+Rrx = 2 (m)
+Rrx = 4 (m)
+42
+6
+6691
+13237
+50
+16
+12174
+25582
+75
+47
+13559
+28403
+100
+78
+13678
+28845
+125
+109
+13711
+28969
+150
+140
+13724
+29020
+175
+171
+13731
+29047
+200
+202
+13735
+29062
+225
+233
+13738
+29071
+250
+264
+13739
+29077
+275
+295
+13740
+29082
+290
+314
+13741
+29084
+Table 4: Maximum extended coverage radius of mFSO con-
+figuration when V = 10.
+lower costs (listed in columns 16th and 21th) than those of single
+serving FSO transceiver configuration (listed in columns 6th and
+11th) for the same test cases and number of wavelengths W. The
+costs resulting from mFSO configuration were as low as 54–87% of
+those resulting from single serving FSO transceiver configuration.
+These numbers confirm that when there is sufficient solar energy,
+mFSO configuration is definitively a better choice than single serv-
+ing FSO configuration.
+11
+
+ 0
+ 20
+ 40
+ 60
+ 80
+ 100
+ 120
+ 0
+ 20
+ 40
+ 60
+ 80
+ 100
+y-axis
+x-axis
+Figure 10: Footprints of HAPs with mFSO configuration ob-
+tained from the topology design for a test case of 1005 ground
+FSO nodes when Esolar = 75 kwh, W = 80. A circle repre-
+sents an extended coverage area of a HAP. Small points in-
+side the circle are ground nodes and the dot at the center of
+the circle is the projected location of its serving HAP on the
+ground.
+7.3
+Factors impact optimal mFSO configu-
+ration
+Comparing the values of the optimal extended coverage radius in
+Table 2 and the maximum extended coverage radius in Table 4,
+we can see that the optimal extended coverage radius was gener-
+ally not the maximum. This is reasonable because the maximum
+configuration uses an excessive number of supplementary serving
+FSO transceivers.
+Low solar energy may render mFSO configuration impossible.
+Indeed, Esolar = 42 kWh could afford maximally 6 supplemen-
+tary serving FSO transceivers (see Table 4), which was too few to
+entirely cover the contour of the principal coverage area. Thus,
+single FSO transceiver configuration was the unique choice.
+When the solar energy level exceeds 50 kWh, its exact value does
+not affect the optimal configuration. The simulation showed that
+the optimal configurations were identical for all solar energy levels
+from 50 kWh/day and above. This is explained by the fact that
+a greater solar energy level allows to accept configurations with
+large coverage but may be more expensive because of using more
+supplementary serving FSO transceivers. As a result, large con-
+figurations were not selected as optimal configurations. In other
+words, increasing solar energy does not necessarily improve the
+HAP network cost.
+Since the optimal multiple serving FSO transceiver configura-
+tions were identical for all Esolar ≥ 50 kWh, all other numerical
+results related to topology design and routing with these solar en-
+ergy levels were identical and are presented as single results in
+subsequent figures.
+The coverage of the optimal configurations decreased when the
+ground nodes became denser. Indeed, test cases with large num-
+bers of ground nodes had greater ground node densities, and
+columns 13th and 16th of Table 2 shows that the optimal m and
+Rext decreased when the density increased. The reason is that,
+with a greater ground node density, is a small ground region al-
+ready contains W ground nodes, which is the maximum serving
+capacity of a HAP. Therefore, a HAP could serve only a small
+zone and required only a few supplementary FSO transceivers to
+cover the zone.
+7.4
+Numbers of HAPs and inter-HAP links
+0
+20
+40
+60
+80
+100
+120
+140
+160
+0
+500
+1000
+1500
+2000
+2500
+3000
+Number of HAPs
+Number of ground FSO nodes
+ˆK for Esolar=42 kWh
+K for Esolar=42 kWh
+ˆK for Esolar ≥ 50 kWh kWh
+K for Esolar ≥ 50 kWh
+Lower bound
+(a) W=40
+0
+20
+40
+60
+80
+100
+120
+140
+160
+0
+500
+1000
+1500
+2000
+2500
+3000
+Number of HAPs
+Number of ground FSO nodes
+ˆK for Esolar=42 kWh
+K for Esolar=42 kWh
+ˆK for Esolar ≥ 50 kWh kWh
+K for Esolar ≥ 50 kWh
+Lower bound
+(b) W=80
+Figure 11: Number of HAPs and lower bound with (a) W =
+40 and (b) W = 80 in different solar energy levels.
+Since each HAP can serve at most W ground FSO nodes, a lower
+bound for the number of HAPs is:
+nLB
+HAP = |NFSO|
+W
+(27)
+Figure 11 shows the number of HAPs, the estimated number of
+HAPs ˆK and lower bound nLB
+HAP when (a) W = 40 and (b) W = 80.
+With Esolar ≥ 50 kWh, the actual number of HAPs was almost
+identical to ˆK in both subfigures.
+Furthermore, when W = 40
+12
+
+ 0
+ 100
+ 200
+ 300
+ 400
+ 500
+ 600
+ 0
+ 500
+ 1000
+ 1500
+ 2000
+ 2500
+ 3000
+Number of inter-HAP links
+Number of ground FSO nodes
+Esolar=42 kWh
+Esolar ≥ 50 kWh
+(a) W=40
+ 0
+ 100
+ 200
+ 300
+ 400
+ 500
+ 600
+ 0
+ 500
+ 1000
+ 1500
+ 2000
+ 2500
+ 3000
+Number of inter-HAP links
+Number of ground FSO nodes
+Esolar=42 kWh
+Esolar ≥ 50 kWh
+(b) W=80
+Figure 12: Number of inter-HAP links when (a) W = 40 and
+(b) W = 80 for different solar energy levels.
+and Esolar ≥ 50 kWh, the number of HAPs approached the lower
+bound starting from test cases with 1000 ground nodes or above.
+This implies that the number of HAPs was almost optimal.
+Figure 12 presents the absolute numbers of inter-HAP links.
+The number of inter-HAP links increased with the number of
+ground nodes, because the network size and traffic demand in-
+creased. The number of inter-HAP links clearly decreased when
+the wavelength density increased from W = 40 to W = 80. In
+other words, denser WDM technique helps reduce the number of
+inter-HAP FSO transceivers and consequently the network cost.
+mFSO configuration allows reducing significantly both the num-
+bers of HAPs and inter-HAP links. Indeed, according to Figure
+11, the number of HAPs was much smaller with Esolar ≥ 50
+kWh where mFSO configuration was used, in comparison with
+Esolar = 42 kWh, where single serving FSO configuration was
+used. A similar phenomenon is observed in Figure 12 for the num-
+ber of inter-HAP links.
+0
+5000
+10000
+15000
+20000
+25000
+30000
+35000
+40000
+0
+500
+1000
+1500
+2000
+2500
+3000
+Cost
+Number of ground FSO nodes
+�
+Cost of Esolar = 42 kWh
+Cost of Esolar = 42 kWh
+�
+Cost of Esolar ≥ 50 kWh
+Cost of Esolar ≥ 50 kWh
+(a) W=40
+0
+5000
+10000
+15000
+20000
+25000
+30000
+35000
+40000
+0
+500
+1000
+1500
+2000
+2500
+3000
+Cost
+Number of ground FSO nodes
+�
+Cost of Esolar = 42 kWh
+Cost of Esolar = 42 kWh
+�
+Cost of Esolar ≥ 50 kWh
+Cost of Esolar ≥ 50 kWh
+(b) W=80
+Figure 13: Real costs and overestimated costs with W = 40
+and W = 80.
+7.5
+Quality of cost estimation
+Figure 13 presents the estimated and actual costs for different
+solar energy levels and wavelength densities. The estimated cost
+was very close to the actual cost, mostly for Esolar ≥ 50kWh and
+W = 40.
+Parameter V, the threshold of the number of inter-HAP links
+of a HAP, affects the quality of the cost estimation. To evaluate
+the choice of V, we compared it with the number of inter-HAP
+links that a HAP finally has. Figure 14 shows the average number
+of inter-HAP links per HAP. When there were 40 wavelengths
+per link, the average number of inter-HAP links per HAP varied
+between 5.7 and 9.3 for Esolar ≥ 50 kWh and V = 10, and between
+8.8 and 11.8 for Esolar = 42 kWh while V raised up to 12. Hence,
+the value of V was close to the actual number of inter-HAP links
+required by a HAP. However, when there were 80 wavelengths per
+link, the average number of Inter-HAP links per HAP was reduced
+to between 4.4 and 8.4, which is slightly far from the threshold
+V = 10. A smaller V may help better estimate of the optimal cost
+in these cases.
+13
+
+ 0
+ 5
+ 10
+ 15
+ 20
+ 0
+ 500
+ 1000
+ 1500
+ 2000
+ 2500
+ 3000
+Average number of inter-HAP links per HAP
+Number of ground FSO nodes
+Esolar=42 kWh
+Esolar ≥ 50 kWh
+(a) W=40
+ 0
+ 5
+ 10
+ 15
+ 20
+ 0
+ 500
+ 1000
+ 1500
+ 2000
+ 2500
+ 3000
+Average number of inter-HAP links per HAP
+Number of ground FSO nodes
+Esolar=42 kWh
+Esolar ≥ 50 kWh
+(b) W=80
+Figure 14: Number of inter-HAP links per HAP when (a)
+W = 40 and (b) W = 80 for different solar energy levels.
+8
+Conclusions
+Using mFSO configuration widens a HAP footprint, however, its
+application is constrained by the available solar energy of the HAP.
+Moreover, mFSO configuration may imply an extra investment
+cost due to additional serving FSO transceivers in comparison with
+single FSO transceiver configuration. This study focused on de-
+termining the optimal mFSO configuration. First, we proposed
+a set of closed-form expressions for computing the coverage of
+an mFSO configuration in terms of beam widths of the princi-
+pal and supplementary transceivers and number of supplementary
+FSO transceivers. Second, we proposed an algorithm to determine
+the optimal mFSO configuration that minimizes the total HAP
+network cost. Third, we designed a HAP network topology using
+the optimal configuration to achieve a minimal final cost.
+The simulation results showed that mFSO significantly ex-
+tended the HAP footprint. With the testing dataset, the extended
+footprint radii were generally two times larger than the single FSO
+transceiver footprint radii, leading to a four-fold larger coverage
+surface. The network cost with the optimal mFSO configuration
+was as low as 54% of the network cost when using a single serving
+FSO transceiver on a HAP.
+Acknowledgements
+This research was funded by the Vietnam National Foundation for
+Science and Technology Development (NAFOSTED) under grant
+number 102.02-2018.305.
+References
+[1] fSONA,
+��SONABeam
+2500-E+
+model
+specifications.”
+http://fsona.com. Accessed Jan. 2022.
+[2] A. Acampora and S. Krishnamurthy, “A broadband wireless
+access network based on mesh-connected free-space optical
+links,” IEEE Personal Communications, vol. 6, no. 5, pp. 62–
+65, 1999.
+[3] J. Zhang, “Proposal of free space optical mesh network ar-
+chitecture for broadband access,” in 2002 IEEE Interna-
+tional Conference on Communications. Conference Proceed-
+ings. ICC 2002 (Cat. No.02CH37333), vol. 4, pp. 2142–2145
+vol.4, 2002.
+[4] B. Moision, B. Erkmen, E. Keyes, T. Belt, O. Bowen,
+D. Brinkley, P. Csonka, M. Eglington, A. Kazmierski, N. hy-
+ong Kim, J. Moody, T. Tu, and W. Vermeer, “Demonstration
+of free-space optical communication for long-range data links
+between balloons on Project Loon,” in Free-Space Laser Com-
+munication and Atmospheric Propagation XXIX (H. Hem-
+mati and D. M. Boroson, eds.), vol. 10096, pp. 259 – 272,
+International Society for Optics and Photonics, SPIE, 2017.
+[5] C. Chen, A. Grier, M. Malfa, E. Booen, H. Harding, C. Xia,
+M. Hunwardsen, J. Demers, K. Kudinov, G. Mak, B. Smith,
+A. Sahasrabudhe, F. Patawaran, T. Wang, A. Wang, C. Zhao,
+D. Leang, J. Gin, M. Lewis, D. Nguyen, and K. Quirk,
+“High-speed optical links for UAV applications,” in Free-
+Space Laser Communication and Atmospheric Propagation
+XXIX (H. Hemmati and D. M. Boroson, eds.), vol. 10096,
+pp. 316 – 324, International Society for Optics and Photon-
+ics, SPIE, 2017.
+[6] Thales
+group,
+“What’s
+up
+with
+stratobus.”
+https://www.thalesgroup.com/en/worldwide/space/news/whats-
+stratobus, 2017. Accessed Jan. 2022.
+[7] D. L. Truong, X. V. Dang, and T. N. Dang, “Survivable free
+space optical mesh network using high-altitude platforms,”
+Optical Switching and Networking, vol. 47, p. 100716, 2023.
+[8] G. Karabulut Kurt,
+M. G. Khoshkholgh,
+S. Alfattani,
+A. Ibrahim, T. S. J. Darwish, M. S. Alam, H. Yanikomeroglu,
+and A. Yongacoglu, “A Vision and Framework for the High
+Altitude Platform Station (HAPS) Networks of the Future,”
+IEEE Communications Surveys & Tutorials, vol. 23, no. 2,
+pp. 729–779, 2021.
+[9] R. Miura and M. Oodo, “Wireless Communications System
+Using Stratospheric Platforms: R and D Program on Telecom
+and Broadcasting System Using High Altitude Platform Sta-
+tions,” Journal of the Communication Research Laboratory,
+vol. 48, pp. 33–48, Dec. 2001.
+14
+
+[10] V. V. Mai and H. Kim, “Beam size optimization and adap-
+tation for high-altitude airborne free-space optical commu-
+nication systems,” IEEE Photonics Journal, vol. 11, no. 2,
+pp. 1–13, 2019.
+[11] A. A. Farid and S. Hranilovic, “Outage capacity optimization
+for free-space optical links with pointing errors,” Journal of
+Lightwave Technology, vol. 25, no. 7, pp. 1702–1710, 2007.
+[12] S. C. Arum, D. Grace, P. D. Mitchell, M. D. Zakaria, and
+N. Morozs, “Energy management of solar-powered aircraft-
+based high altitude platform for wireless communications,”
+Electronics, vol. 9, no. 1, 2020.
+[13] F. Fidler, M. Knapek, J. Horwath, and W. R. Leeb, “Optical
+Communications for High-Altitude Platforms,” IEEE Journal
+of Selected Topics in Quantum Electronics, vol. 16, pp. 1058–
+1070, Sep. 2010.
+[14] Airbus,
+“Zephir:
+Persistance
+and
+flexibility.”
+https://lf5422.com/wp-
+content/uploads/2018/08/0296 18 2 zephyr datasheet e horizontal a4.pdf,
+2018. Accessed Jan. 2022.
+[15] BAE Systems, “Phasa-35.” http://prismaticltd.co.uk/products/phasa-
+35/, 2018. Accessed Jan. 2022.
+A
+Proof of Lemma 1
+Proof. Let x = cos(α/2), a = σH, and b =
+PtxR2
+rx
+2H2
+then
+P rx
+j (x) = e−a/x
+bx2
+(1 − x)
+(28)
+Calculate the derivative of P rx
+j (x) we get
+P
+′rx
+j
+(x) = e−a/x
+�
+a
+1 − x + 2x − x2
+(1 − x)2
+�
+b
+(29)
+Thus, the derivative of P rx
+j (α) is
+P
+′rx
+j
+(α) = P
+′rx
+j
+(x).(− sin(α))
+(30)
+Beam α is limited between [0..π] because it orients to the ground.
+Thus, x ∈ [0..1].
+Consequently, 1 − x > 0 and 2x − x2 > 0.
+In addition, a, b > 0, then P
+′rx
+j
+(x) > 0 for all x ∈ [0..1].
+Be-
+cause − sin(α) < 0, ∀α ∈ [0..π], thus, P
+′rx
+j
+(α) < 0. Consequently,
+P rx
+j (α) decreases with α.
+B
+Calculation of extended coverage
+radius of mFSO configuration
+This section identifies formulas that calculate the extended cover-
+age radius of an mFSO configuration characterized by the princi-
+pal beam width α, supplementary beam width β and number of
+supplementary beams m.
+Conventionally, the coverage provided by a bundle of transmit-
+ters is calculated as if the transmitters project perpendicular to the
+ground. In mFSO configuration, the principal beam in the center
+is large, and it pushes the supplementary serving FSO transceiver
+projection directions far from perpendicular to the ground. These
+supplementary beams form oblique cones that intersect with the
+ground plane in ellipses. Considering of the elliptical form adds
+more complexity to the calculation.
+In Figure 15, H denotes the position of a HAP, and its projec-
+tion on the ground plane is O, thus HO = H. The principal beam
+forms a right circular cone whose axis is HO. The cone intersects
+the ground plane by a circle of radius Rα, which defines the prin-
+cipal footprint. The beam of a supplementary FSO transceiver is
+an oblique cone intersecting the ground plane by an ellipse that
+defines the corresponding supplementary footprint. The cone of
+the supplementary beam intersects with the cone of the principal
+beam by two lines: HK and HK′ where K and K′ are the two
+intersection points of the principal and supplementary footprints.
+Thus, OK = OK′ = Rα.
+m supplementary FSO transceivers are arranged evenly around
+the principal transceiver, each of which is responsible for extending
+the coverage within an angle of 2π/m from the center O.
+The
+responsible angle of the supplementary FSO transceiver in Figure
+15 is defined by rays −−→
+OK and −−→
+OK′. Thus, �
+KOK′ = 2π/m.
+Ray −−→
+OK intersects with the supplementary beam cone at J,
+then OJ is the radius of the extended coverage region. Readers
+refer to Figure 6 for a complete view of the extended coverage
+circle and the positions of K, K′ and J on the ground.
+Figure 15: Computation of the distance from supplementary
+FSO transceivers and the border of extended coverage area
+LJ in function of Beta.
+Since the principal beam width is α, then �
+OHK = α/2.
+Let the base plane containing K and K′ of the supplementary
+beam cone cuts the cone axis at T, the primary cone axis HO at P,
+and HJ at J1. Then �
+THK = β/2. In addition, the supplementary
+cone intersects with this base plane by a circle containing K, K′
+with center T. Let Rβ be the radius of the circle, then TK =
+TK′ = Rβ.
+Let M be the midpoint of KK′ then H, O, T, M belong to the
+same plane.
+Let ξ = �
+KHJ. The extended coverage radius is Rext = OJ =
+15
+
+H
+β/2
+LY
+a/2
+Supplementarycone base plane
+K'
+R
+Pilm
+a
+Supplementary foot print
+Principal
+foot print
+GroundHO tan(�
+OHJ) = H tan( α
+2 + ξ). Thus,
+Rext = H. tan
+�
+2(ξ + α
+2
+) − α
+2
+�
+Rext = H2 tan( ξ+α
+2 ) − tan( α
+2 )(1 − tan2( ξ+α
+2 ))
+1 − tan2( ξ+α
+2 ) + 2 tan( ξ+α
+2 ). tan( α
+2 )
+(31)
+B.1
+Calculation of tan( ξ+α
+2 )
+Let N be the midpoint of KJ1. As K and J1 are at the intersection
+of the supplementary cone and its base plane, HK = HJ1, HN ⊥
+KJ1, and HN is the angle bisector of �
+KHJ1. Therefore, �
+NHK =
+ξ/2, thus �
+NHP = ξ+α
+2 . In addition, since KO is on the base plane
+of the principal cone, HO ⊥ KO.
+Thus, △PNH and △POK
+are similar right triangles. Consequently, �
+OKP = �
+NHP = ξ+α
+2 .
+Furthermore,
+tan(ξ + α
+2
+) = OP
+OK = OP
+Rα
+(32)
+Let �
+OHM = γ and �
+THM = θ Then �
+OHT = θ + γ.
+Because MO is on the base plan of the principal cone, MO ⊥
+HO. In addition, as PT is on the base plane of the supplementary
+cone whose axis is HT then HT ⊥ PT. Consequently, △PTH and
+△POM are similar right triangles. We can deduce that �
+PMO =
+�
+PHT = θ + γ. Therefore,
+tan(θ + γ) = OP
+OM =
+OP
+Rα. cos( π
+m)
+Combining with (32) we deduce :
+tan(ξ + α
+2
+) = tan(θ + γ). cos( π
+m)
+(33)
+Thus
+tan(ξ + α
+2
+) =
+tan(γ) + tan(θ)
+1 − tan(γ). tan(θ). cos( π
+m)
+(34)
+Since γ = �
+OHM then, tan(γ) = MO
+HO .
+From right triangle △OMK we have MO = OK. cos( π
+m).
+From right triangle △HOK we have HO = OK/ tan( α
+2 ).
+Thus
+tan(γ) = tan(α
+2 ). cos( π
+m)
+(35)
+It remains to calculate tan (θ).
+B.2
+Calculation of tan (θ)
+Look at the right triangle △HTM, we can see that:
+tan(θ) = TM
+TH
+(36)
+Since K and K′ are on a circle centered at T, and M is the
+midpoint of KK′ then △TMK is a right triangle, then
+TM =
+�
+TK2 − KM 2 =
+�
+R2
+β − R2α. sin2( π
+m)
+(37)
+Easy to find that △THK is another right triangle then
+TH = TK/ tan(β
+2 ) = Rβ/ tan(β
+2 )
+(38)
+Replacing (37) and (38) in to (36) we get
+tan(θ)
+=
+�
+R2
+β − R2α. sin2( π
+m)
+Rβ/ tan( β
+2 )
+=
+tan(β
+2 )
+�
+1 − (Rα
+Rβ )2. sin2( π
+m)
+(39)
+From right triangle △HTK we obtain Rβ = HK sin( β
+2 ).
+From right triangle △HOK we obtain Rα = HK sin( α
+2 ).
+Replacing these values to (39), we obtain:
+tan(θ) =
+�
+sin2( β
+2 ) − sin2( α
+2 ). sin2( π
+m)
+cos( β
+2 )
+(40)
+Substituting the values of tan(γ) in (35) and tan(θ) in (40) into
+(34), we obtain tan( ξ+α
+2 ). Subsequently, replacing the obtained
+tan( ξ+α
+2 ) to (31) we get Rext.
+16
+

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+Noname manuscript No.
+(will be inserted by the editor)
+HyRSM++: Hybrid Relation Guided Temporal Set Matching for
+Few-shot Action Recognition
+Xiang Wang · Shiwei Zhang · Zhiwu Qing · Zhengrong Zuo · Changxin Gao ·
+Rong Jin · Nong Sang
+Received: date / Accepted: date
+Abstract Few-shot action recognition is a challenging but
+practical problem aiming to learn a model that can be eas-
+ily adapted to identify new action categories with only a few
+labeled samples. Recent attempts mainly focus on learning
+deep representations for each video individually under the
+episodic meta-learning regime and then performing tempo-
+ral alignment to match query and support videos. However,
+they still suffer from two drawbacks: (i) learning individ-
+ual features without considering the entire task may result
+in limited representation capability, and (ii) existing align-
+ment strategies are sensitive to noises and misaligned in-
+stances. To handle the two limitations, we propose a novel
+Hybrid Relation guided temporal Set Matching (HyRSM++)
+approach for few-shot action recognition. The core idea of
+HyRSM++ is to integrate all videos within the task to learn
+discriminative representations and involve a robust match-
+ing technique. To be specific, HyRSM++ consists of two key
+components, a hybrid relation module and a temporal set
+matching metric. Given the basic representations from the
+feature extractor, the hybrid relation module is introduced
+to fully exploit associated relations within and cross videos
+in an episodic task and thus can learn task-specific embed-
+dings. Subsequently, in the temporal set matching metric, we
+carry out the distance measure between query and support
+Xiang Wang · Zhiwu Qing · Zhengrong Zuo · Changxin Gao (Corre-
+sponding author) · Nong Sang
+Key Laboratory of Ministry of Education for Image Processing and
+Intelligent Control, School of Artificial Intelligence and Automation,
+Huazhong University of Science and Technology
+E-mail: {wxiang, qzw, zhrzuo, cgao, nsang}@hust.edu.cn
+Shiwei Zhang
+Alibaba Group
+E-mail: zhangjin.zsw@alibaba-inc.com
+Rong Jin
+Twitter
+E-mail: rongjinemail@gmail.com
+videos from a set matching perspective and design a bidi-
+rectional Mean Hausdorff Metric to improve the resilience
+to misaligned instances. In addition, we explicitly exploit
+the temporal coherence in videos to regularize the matching
+process. In this way, HyRSM++ facilitates informative cor-
+relation exchanged among videos and enables flexible pre-
+dictions under the data-limited scenario. Furthermore, we
+extend the proposed HyRSM++ to deal with the more chal-
+lenging semi-supervised few-shot action recognition and un-
+supervised few-shot action recognition tasks. Experimental
+results on multiple benchmarks demonstrate that our method
+consistently outperforms existing methods and achieves
+state-of-the-art performance under various few-shot set-
+tings. The source code is available at https://github.
+com/alibaba-mmai-research/HyRSMPlusPlus.
+Keywords Few-shot Action Recognition · Set Match-
+ing · Semi-supervised Few-shot Action Recognition ·
+Unsupervised Few-shot Action Recognition
+1 Introduction
+Recently, the development of large-scale video bench-
+marks [8, 23, 6, 13, 24] and deep networks [88, 51, 18,
+89, 65, 52] have significantly boosted the progress of ac-
+tion recognition. To achieve this success, we typically re-
+quire large amounts of manually labeled data. However, ac-
+quiring these labeled examples consumes a lot of manpower
+and time, which actually limits further applications of this
+task. In this case, researchers look to alternatives to achieve
+action classification without extensive costly labeling. Few-
+shot action recognition is a promising direction to reduce
+manual annotations and thus has attracted much attention
+recently [112, 105]. It aims at learning to classify unseen
+action classes with extremely few annotated examples.
+arXiv:2301.03330v1  [cs.CV]  9 Jan 2023
+
+2
+Xiang Wang et al.
+...
+CNN
+CNN
+CNN
+...
+0.8
+0.1
+Query video
+...
+Support set
+Hybrid relation module
+Support: make coffee
+Query: make coffee
+Support: make coffee
+Query: make coffee
+
+
+Metric space 
+(support)
+Metric space 
+(quey)
+“pour water”
+“pour coffee powder”
+Temporal alignment
+Temporal set matching
+(b) 
+Time line
+Matching line
+
+
+(a)
+Pull
+Push
+Pull
+Push
+Fig. 1 (a) Concept of the proposed hybrid relation module. We adaptively produce task-specific video embeddings by extracting relevant discrim-
+inative patterns cross videos in an episodic task. (b) Example of make coffee, the current temporal alignment metrics tend to be strict, resulting in
+an incorrect match on misaligned videos. In contrast, the proposed temporal set matching metric involving set matching technique and temporal
+coherence regularization is more flexible in finding the best correspondences.
+To solve the few-shot data-scarcity problem, popu-
+lar attempts [112, 7, 68, 106] are mainly based on the
+metric-based meta-learning technique [86], in which a com-
+mon embedding space is first learnt via episodic training
+and then an explicit or implicit alignment metric is em-
+ployed to calculate the distances between the query (test)
+videos and support (reference) videos for classification in
+an episodic task. Typically, Ordered Temporal Alignment
+Module (OTAM) [7] adopts a deep feature extractor to con-
+vert an input video into a frame feature sequence indepen-
+dently and explicitly explores the ordered temporal align-
+ment path between support and query videos in this feature
+space. Temporal-Relational CrossTransformer (TRX) [68]
+learns a deep embedding space and tries to exhaustively con-
+struct temporally-corresponding sub-sequences of actions to
+compare. Some recent works [33, 94, 108, 62] propose to
+design multi-level metrics for few-shot action recognition.
+Although these methods have achieved remarkable per-
+formance, there are still two limitations: individual feature
+learning and inflexible matching strategy. First, discrimina-
+tive interactive clues cross videos in an episode are ignored
+when each video is considered independently during repre-
+sentation learning. As a result, these methods actually as-
+sume the learned representations are equally effective on
+different episodic tasks and maintain a fixed set of video fea-
+tures for all test-time tasks, i.e., task-agnostic, which hence
+might overlook the most discriminative dimensions for the
+current task. Existing work also shows that the task-agnostic
+methods tend to suffer inferior generalization in other fields,
+such as image recognition [47, 101], NLP [66, 57], and in-
+formation retrieval [53]. Second, actions are usually com-
+plicated and involve many subactions with different orders
+and offsets, which may cause the failure of existing tempo-
+ral alignment metrics. For example, as shown in Figure 1(b),
+to make coffee, you can pour water before pour coffee pow-
+der, or in a reverse order, hence it is hard for recent temporal
+alignment strategies to find the right correspondences. Thus
+a more flexible metric is required to cope with the misalign-
+ment.
+Inspired by the above observations, we thus solve the
+few-shot action recognition problem by developing a novel
+Hybrid Relation guided temporal Set Matching algorithm,
+dubbed HyRSM++, which is architecturally composed of a
+hybrid relation module and a temporal set matching metric.
+In the hybrid relation module, we argue that the considerable
+relevant relations within and cross videos are beneficial to
+generate a set of customized features that are discriminative
+for a given task. To this end, we first apply an intra-relation
+function to strengthen structural patterns within a video via
+modeling long-range temporal dependencies. Then an inter-
+relation function operates on different videos to extract rich
+semantic information to reinforce the features which are
+more relevant to query predictions, as shown in Figure 1(a).
+By this means, we can learn task-specific embeddings for
+the few-shot task. On top of the hybrid relation module,
+we design a novel temporal set matching metric consist-
+ing of a bidirectional Mean Hausdorff Metric and a tem-
+poral coherence regularization to calculate the distances be-
+tween query and support videos, as shown in Figure 1(b).
+The objective of the bidirectional Mean Hausdorff Metric
+is to measure video distance from the set matching per-
+spective. Concretely, we treat each video as a set of frames
+and alleviate the strictly ordered constraints to acquire bet-
+ter query-support correspondences. Furthermore, to exploit
+long-range temporal order dependencies, we explicitly im-
+pose temporal coherence regularization on the input videos
+for more stable measurement without introducing extra net-
+work parameters. In this way, by combining the hybrid re-
+lation module and temporal set matching metric, the pro-
+posed HyRSM++ can sufficiently integrate semantically re-
+lational representations within the entire task and provide
+flexible video matching in an end-to-end manner. We evalu-
+ate the proposed HyRSM++ on six challenging benchmarks
+and achieve remarkable improvements again current state-
+of-the-art methods.
+Although the intuition of HyRSM++ is straightforward,
+it is elaborately designed for few-shot action recognition.
+Can our HyRSM++ be applied to the more challenging
+
+HyRSM++: Hybrid Relation Guided Temporal Set Matching for Few-shot Action Recognition
+3
+semi-supervised or unsupervised action recognition tasks
+even if the settings are entirely different? To answer this
+question, we extend HyRSM++ to the semi-supervised and
+unsupervised objectives with minor task adaptation modi-
+fications, and experimental results indicate that HyRSM++
+can be well adapted to different scenarios well and achieves
+impressive performance.
+In summary, we make the following four contributions:
+(1) We propose a novel hybrid relation module to cap-
+ture the intra- and inter-relations inside the episodic task,
+yielding task-specific representations for different tasks.
+(2) We reformulate the query-support video pair distance
+metric as a set matching problem and develop a bidirectional
+Mean Hausdorff Metric, which can be robust to complex ac-
+tions. To utilize long-term temporal order cues, we further
+design a new temporal coherence regularization on videos
+without adding network parameters.
+(3) We conduct extensive experiments on six challeng-
+ing datasets to verify that the proposed HyRSM++ achieves
+superior performance over the state-of-the-art methods.
+(4) We show that the proposed HyRSM++ can be di-
+rectly extended to the more challenging semi-supervised
+few-shot action recognition and unsupervised few-shot ac-
+tion recognition task with minor modifications.
+In this paper, we have extended our preliminary CVPR-
+2022 conference version [91] in the following aspects. i)
+We integrate the temporal coherence regularization and set
+matching strategy into a temporal set matching metric so
+that the proposed metric can explicitly leverage temporal
+order information in videos and match flexibly. Note that
+temporal coherence regularization does not introduce ad-
+ditional parameters and will not increase the burden of in-
+ference. ii) We conduct more comprehensive ablation stud-
+ies to verify the effectiveness and efficiency of the pro-
+posed HyRSM++. iii) We clearly improve the few-shot
+action recognition performance over the previous version.
+Experimental results also manifest that HyRSM++ signifi-
+cantly surpasses existing competitive methods and achieves
+state-of-the-art performance. iv) We show that the proposed
+HyRSM++ can be easily extended to the more challeng-
+ing semi-supervised few-shot recognition and unsupervised
+few-shot action recognition tasks.
+2 Related Work
+In the literature, there are some techniques related to this
+paper, mainly including few-shot image classification, set
+matching, temporal coherence, semi-supervised few-shot
+learning, unsupervised few-shot learning, and few-shot ac-
+tion recognition. In this section, we will briefly review them
+separately.
+Few-shot Image Classification.
+Recently, the research
+of few-shot learning [17, 55, 56] has proceeded roughly
+along with the following directions: data augmentation,
+optimization-based, and metric-based. Data augmentation is
+an intuitive method to increase the number of training sam-
+ples and improve the diversity of data. Mainstream strategies
+include spatial deformation [70, 67] and semantic feature
+augmentation [9, 100]. Optimization-based methods learn
+a meta-learner model that can quickly adopt to a new task
+given a few training examples. These algorithms include the
+LSTM-based meta-learner [74], learning efficient model ini-
+tialization [19], and learning stochastic gradient descent op-
+timizer [50]. Metric-based methods attempt to address the
+few-shot classification problem by ”learning to compare”.
+This family of approaches aims to learn a feature space and
+compare query and support images through Euclidean dis-
+tance [76, 101, 99], cosine similarity [86, 98], or learnable
+non-linear metric [80, 29, 47]. Our work is more closely re-
+lated to the metric-based methods [47, 101] that share the
+same spirit of learning task-specific features, whereas we fo-
+cus on solving the more challenging few-shot action recog-
+nition task with diverse spatio-temporal dependencies. In
+addition, we will further point out the differences and con-
+duct performance comparisons in the experimental section.
+Set Matching. The objective of set matching is to accu-
+rately measure the similarity of two sets, which have re-
+ceived much attention over the years. Set matching tech-
+niques can be used to efficiently process complex data struc-
+tures [2, 72, 3] and has been applied in many computer vi-
+sion fields, including face recognition [63, 93, 92], object
+matching [73, 107], etc. Among them, Hausdorff distance
+is an important alternative to handle set matching problems.
+Hausdorff distance and its variants have been widely used
+in the field of image matching and achieved remarkable re-
+sults [34, 16, 35, 107, 82, 79]. Inspired by these great suc-
+cesses, we introduce set matching into the few-shot action
+recognition field for the first time.
+Temporal Coherence. Videos naturally involve temporal
+continuity, and there is much effort to effectively explore
+how to leverage this property [11, 22, 27, 58]. Inverse Dif-
+ference Moment (IDM) [11] is a commonly used measure of
+local homogeneity, which assumes that in a sequence, two
+elements are more similar if they are located next to each
+other. The idea of IDM has been widely applied to texture
+feature extraction [60], face recognition [59], and unsuper-
+vised representation learning [22, 27] and achieved remark-
+able performance. In this paper, we focus on constraining
+the few-shot matching process by exploiting temporal co-
+herence.
+Semi-supervised Few-shot Learning. In practical appli-
+cation scenarios, there are usually many unlabeled samples.
+Semi-supervised few-shot learning considers learning new
+concepts in the presence of extra unlabeled data. Ren et
+al. [71] first introduce the challenging semi-supervised few-
+shot learning paradigm and refine the prototypes by adopt-
+
+4
+Xiang Wang et al.
+ing a soft k-means on unlabeled data. LST [49] proposes a
+novel recursive-learning-based self-training strategy for ro-
+bust convergence of the inner loop. TransMatch [103] de-
+velops a new transfer learning framework by incorporat-
+ing MixMatch [4] and existing few-shot learning methods.
+PTN [31] employs the Poisson learning model to obtain in-
+formative presentations between the labeled and unlabeled
+data. PLCM [32] and iLPC [44] focus on cleaning predicted
+pseudo-labels and generating accurate confidence estima-
+tion. In the field of semi-supervised few-shot action recog-
+nition, LIM [113] utilizes a label-independent memory to
+preserve a feature bank and produces class prototypes for
+query classification.
+Unsupervised Few-shot Learning. The objective of un-
+supervised few-shot learning is to utilize unlabeled samples
+to construct meta-tasks for few-shot training. CACTUs [30]
+and UFLST [36] construct many tasks by clustering em-
+beddings and optimize the meta-learning process over the
+constructed tasks. UMTRA [38] generates artificial tasks
+by randomly sampling support examples from the training
+set and produces corresponding queries by augmentation.
+ULDA [69] and AAL [1] follow this paradigm to randomly
+group augmented images for meta-learning and point out the
+importance of data augmentation. More recently, MetaU-
+VFS [64] presents the first unsupervised meta-learning al-
+gorithm for few-shot action recognition and adopts a two-
+stream 2D and 3D CNN model to explore spatial and tem-
+poral features via contrastive learning.
+Few-shot Action Recognition.
+The difference between
+few-shot action recognition and the previous few-shot learn-
+ing approaches is that it deals with more complex higher
+dimensional video data instead of two-dimensional images.
+The existing methods mainly focus on metric-based learn-
+ing. OSS-Metric Learning [40] adopts OSS-Metric of video
+pairs to match videos. TARN [5] learns an attention-based
+deep-distance measure from an attribute to a class center
+for zero-shot and few-shot action recognition. CMN [112]
+utilizes a multi-saliency embedding algorithm to encode
+video representations. AMeFu-Net [20] uses depth infor-
+mation to assist learning. Xian et al. [95] propose to learn
+a generative adversarial network and produce video fea-
+tures of novel classes for generalization. Coskun et al. [12]
+leverage object-object interaction, hand grasp, optical flow,
+and hand trajectory to learn an egocentric few-shot classi-
+fier. OTAM [7] preserves the frame ordering in video data
+and estimates distances with ordered temporal alignment.
+ARN [105] introduces a self-supervised permutation invari-
+ant strategy for spatio-temporal modeling. ITANet [106]
+proposes a frame-wise implicit temporal alignment strategy
+to achieve accurate and robust video matching. TRX [68]
+matches actions by matching plentiful tuples of different
+sub-sequences. More recently, STRM [84] makes use of lo-
+cal and global enrichment mechanism for spatio-temporal
+modeling based on TRX [68] and enforces class-separability
+at different phase. Some works [33, 94, 108, 62] propose
+to design multi-level metrics for few-shot action recogni-
+tion. Note that most above methods focus on learning video
+embedding independently. Unlike these previous methods,
+our HyRSM++ improves the transferability of embedding
+by learning intra- and inter-relational patterns that can bet-
+ter generalize to unseen classes.
+3 Method
+In this section, we first formulate the definition of the
+few-shot action recognition task. Then we present our Hy-
+brid Relation guided temporal Set Matching (HyRSM++)
+method.
+3.1 Problem formulation
+Few-shot action recognition aims to obtain a model that can
+generalize well to new classes when limited labeled video
+data is available. To make training more faithful to the test
+environment, we adopt the episodic training manner [86] for
+few-shot adaptation as in previous work [86, 7, 68, 106]. In
+each episodic task, there are two sets, i.e., a support set S
+and a query set Q. The support set S contains N × K sam-
+ples from N different action classes, and each class contains
+K support videos, termed the N-way K-shot problem. The
+goal is to classify the query videos in Q into N classes with
+these support videos.
+3.2 HyRSM++
+Pipeline. The overall architecture of HyRSM++ is illus-
+trated in Figure 2. For each input video sequence, we first
+divide it into T segments and extract a snippet from each
+segment, as in previous methods [88, 7]. This way, in an
+episodic task, the support set can be denoted as S
+=
+{s1, s2, ..., sN×K}, where si = {s1
+i , s2
+i , ..., sT
+i }. For sim-
+plicity and convenience, we discuss the process of the N-
+way 1-shot problem, i.e., K = 1, and consider that the
+query set Q contains a single video q. Then we apply
+an embedding model to extract the feature representations
+for each video sequence and obtain the support features
+Fs = {fs1, fs2, ..., fsN } and the query feature fq, where
+fsi = {f 1
+i , f 2
+i , ..., f T
+i } and fq = {f 1
+q , f 2
+q , ..., f T
+q }. After
+that, we input Fs and fq to the hybrid relation module to
+learn task-specific features, resulting in ˜Fs and ˜fq. Finally,
+the enhanced representations ˜Fs and ˜fq are fed into the set
+matching metric to generate matching scores. Based on the
+output scores, we can train or test the total framework.
+
+HyRSM++: Hybrid Relation Guided Temporal Set Matching for Few-shot Action Recognition
+5
+Support set
+Query video
+Backbone
+Intra-relation
+Intra-relation
+Intra-relation
+Intra-relation
+A
+Inter-relation modeling
+Hybrid relation module
+0.1
+0.2
+0.7
+A
+Avg-pooling
+E
+Expend
+Concatenate
+Convolution
+Temporal set matching metric
+Backbone
+Backbone
+Backbone
+A
+A
+A
+E
+E
+E
+E
+Pull
+Push
+Fig. 2 Schematic illustration of the proposed Hybrid Relation guided temporal Set Matching (HyRSM++) approach on a 3-way 1-shot problem.
+Given an episode of video data, a feature embedding network is first employed to extract their feature vectors. Then, A hybrid relation module is
+followed to integrate rich information within each video and cross videos with intra-relation and inter-relation functions. Finally, the task-specific
+features are fed forward into a temporal set matching metric for matching score prediction. Best viewed in color.
+Hybrid relation module. Given the features Fs and fq
+output by the embedding network, current approaches, e.g.,
+OTAM [7], directly apply a classifier C in this feature space.
+They can be formulated as:
+yi = C(fsi, fq)
+(1)
+where yi is the matching score between fsi and fq. During
+training, yi = 1 if they belong to the same class, otherwise
+yi = 0. In the testing phase, yi can be adopted to predict
+the query label. From the perspective of probability theory,
+it makes decisions based on the priors fsi and fq:
+yi = P((fsi, fq)|fsi, fq)
+(2)
+which is a typical task-agnostic method. However, the task-
+agnostic embedding is often vulnerable to overfit irrelevant
+representations [29, 47] and may fail to transfer to unseen
+classes not yet observed in the training stage.
+Unlike the previous methods, we propose to learn task-
+specific features for each target task. To achieve this goal, we
+introduce a hybrid relation module to generate task-specific
+features by capturing rich information from different videos
+in an episode. Specifically, we elaborately design the hybrid
+relation module H in the following form:
+˜fi = H(fi, G); fi ∈ [Fs, fq], G = [Fs, fq]
+(3)
+That is, we improve the feature fi by aggregating seman-
+tic information cross video representations, i.e., G, in an
+episodic task, allowing the obtained task-specific feature ˜fi
+to be more discriminative than the isolated feature. For ef-
+ficiency, we further decompose hybrid relation module into
+two parts: intra-relation function Ha and inter-relation func-
+tion He.
+The intra-relation function aims to strengthen structural
+patterns within a video by capturing long-range temporal de-
+pendencies. We express this process as:
+f a
+i = Ha(fi)
+(4)
+here f a
+i
+∈ RT ×C is the output of fi through the intra-
+relation function and has the same shape as fi. Note that the
+intra-relation function has many alternative implements, in-
+cluding multi-head self-attention (MSA), Transformer [85],
+Bi-LSTM [25], Bi-GRU [10], etc., which is incredibly flex-
+ible and can be any one of them.
+Based on the features generated by the intra-relation
+function, an inter-relation function is deployed to semanti-
+cally enhance the features cross different videos:
+f e
+i = He
+i (f a
+i , Ga) =
+|Ga|
+�
+j
+(κ(ψ(f a
+i ), ψ(f a
+j )) ∗ ψ(f a
+j ))
+(5)
+where Ga = [F a
+s , f a
+q ], ψ(·) is a global average pooling layer,
+and κ(f a
+i , f a
+j ) is a learnable function that calculates the se-
+mantic correlation between f a
+i and f a
+j . The potential logic
+is that if the correlation score between f a
+i and f a
+j is high,
+i.e., κ(f a
+i , f a
+j ), it means they tend to have the same seman-
+tic content, hence we can borrow more information from f a
+j
+to elevate the representation f a
+i , and vice versa. In the same
+way, if the score κ(f a
+i , f a
+i ) is less than 1, it indicates that
+some irrelevant information in f a
+i should be suppressed.
+In this way, we can improve the feature discrimination
+by taking full advantage of the limited samples in each
+episodic task. The inter-relation function also has similar
+implements with the intra-relation function but with a dif-
+ferent target. After the inter-relation function, we employ
+an Expend-Concatenate-Convolution operation to aggregate
+
+6
+Xiang Wang et al.
+information, as shown in Figure 2, where the output feature
+˜fi has the same shape as f e
+i . In the form of prior, our method
+can be formulated as:
+yi = P(( ˜fsi, ˜fq)|H(fsi, G), H(fq, G)); G = [Fs, fq]
+(6)
+Intuitively, compared with Equation 2, it can be conducive
+to making better decisions because more priors are provided.
+In particular, the hybrid relation module is a plug-and-play
+unit. In the experiment, we will fully explore different con-
+figurations of the hybrid relation module and further inves-
+tigate its insertablility.
+Temporal set matching metric. Many prior few-shot
+action recognition algorithms usually impose a strict tempo-
+ral alignment strategy on generated video representations for
+few-shot classification. However, they suffer from causing
+some failed matches when encountering misaligned video
+instances. Instead, we develop a flexible metric based on set
+matching that explicitly discovers optimal frame matching
+pairs for the ability to be insensitive to misalignment. Con-
+cretely, the proposed temporal set matching metric contains
+two parts, bidirectional Mean Hausdorff Metric (Bi-MHM)
+and temporal coherence regularization, respectively. We will
+describe them in detail below.
+Given the relation-enhanced features ˜Fs and ˜fq, we
+present a novel metric to enable efficient and flexible match-
+ing. In this metric, we treat each video as a set of T frames
+and reformulate distance measurement between videos as
+a set matching problem, which is robust to complicated
+instances, whether they are aligned or not. Specifically,
+we achieve this goal by modifying the Hausdorff distance,
+which is a typical set matching approach. The standard
+Hausdorff distance D can be formulated as:
+d( ˜fi, ˜fq) = max
+˜
+f a
+i ∈ ˜fi
+( min
+˜
+f bq ∈ ˜
+fq
+��� ˜f a
+i − ˜f bq
+���)
+d( ˜fq, ˜fi) = max
+˜
+f b
+q ∈ ˜
+fq
+( min
+˜
+f a
+i ∈ ˜fi
+��� ˜f bq − ˜f a
+i
+���)
+D = max(d( ˜fi, ˜fq), d( ˜fq, ˜fi))
+(7)
+where ˜fi ∈ RT ×C contains T frame features, and
+��·
+�� is a
+distance measurement function, which is the cosine distance
+in our method.
+However, the previous methods [102, 21, 111, 16]
+pointed out that Hausdorff distance can be easily affected
+by noisy examples, resulting in inaccurate measurements.
+Hence they employ a directed modified Hausdorff distance
+that robust to noise as follows:
+dm( ˜fi, ˜fq) = 1
+Ni
+�
+˜
+f a
+i ∈ ˜fi
+( min
+˜
+f b
+q ∈ ˜
+fq
+��� ˜f a
+i − ˜f bq
+���)
+(8)
+where Ni is the length of ˜fi, and equal to T in this paper.
+Hausdorff distance and its variants achieve great success in
+image matching [82, 16, 34] and face recognition [21, 79].
+We thus propose to introduce the set matching strategy into
+the few-shot action recognition field and further design a
+novel bidirectional Mean Hausdorff Metric (Bi-MHM):
+Db = 1
+Ni
+�
+˜
+f a
+i ∈ ˜fi
+( min
+˜
+f bq ∈ ˜
+fq
+��� ˜f a
+i − ˜f bq
+���)+
+1
+Nq
+�
+˜
+f bq ∈ ˜
+fq
+( min
+˜
+f a
+i ∈ ˜fi
+��� ˜f bq − ˜f a
+i
+���)
+(9)
+where Ni and Nq are the lengths of the support feature ˜fi
+and the query feature ˜fq respectively.
+The proposed Bi-MHM is a symmetric function, and the
+two items are complementary to each other. From Equa-
+tion 9, we can find that Db can automatically find the best
+correspondencies between two videos, e.g., ˜fi and ˜fq. Note
+that our Bi-MHM is a non-parametric classifier and does not
+involve numerous non-parallel calculations, which helps to
+improve computing efficiency and transfer ability compared
+to the previous complex alignment classifiers [7, 68]. More-
+over, the hybrid relation module and Bi-MHM can mutually
+reinforce each other, consolidating the correlation between
+two videos collectively.
+The Bi-MHM approach described above assumes video
+sequence representations belonging to the same action have
+the same set structure in the feature space and does not
+explicitly utilize temporal order information. However, it
+would be much more general to take the inherent temporal
+information in videos into account. For this reason, we take
+advantage of the temporal coherence that naturally exists in
+sequential video data and construct a temporal coherence
+regularization to further constrain the matching process by
+incorporating temporal order information.
+IDM [11] is a commonly used means that can exploit
+temporal coherence within videos, which can be formulated
+as:
+I( ˜fi) =
+T
+�
+a=1
+T
+�
+b=1
+1
+(a − b)2 + 1 ·
+��� ˜f a
+i − ˜f b
+i
+���
+(10)
+where ˜fi is the input video feature, T is the temporal length
+of the video, and the above loss encourages frames that are
+close in time to be close in the feature space as well. In addi-
+tion, there is another way to use temporal order information
+in the literature [22, 59]:
+I( ˜fi; ˜f a
+i , ˜f b
+i ) =
+�
+�
+�
+��� ˜f a
+i − ˜f b
+i
+��� ,
+if |a − b| = 1
+max(0, m −
+��� ˜f a
+i − ˜f b
+i
+���)
+if |a − b| > 1
+(11)
+where m is the size of the margin. Equation 11 utilizes
+the video coherence property by pulling two frame features
+closer if they are adjacent, pushing farther apart by one mar-
+gin m if they are not adjacent. Through observation, we can
+
+HyRSM++: Hybrid Relation Guided Temporal Set Matching for Few-shot Action Recognition
+7
+see that in Equation 10, all frames are pulled close regardless
+of time distance. In Equation 11, all frame features are sep-
+arated by a margin m if they are not adjacent to the current
+frame, i.e., all pairs are treated equally. The above two man-
+ners do not fully exploit the smooth and continuous changes
+of the video. To this end, we propose a novel form to mine
+temporal coherence property:
+I( ˜fi; ˜f a
+i , ˜f b
+i ) =
+�
+�
+�
+1
+(a−b)2+1 ·
+��� ˜f a
+i − ˜f b
+i
+��� ,
+if |a − b| ≤ δ
+max(0, mab −
+��� ˜f a
+i − ˜f b
+i
+���)
+if |a − b| > δ
+(12)
+where δ is a window size and mab = 1 − e− (|a−b|−δ)2
+2σ2
+for smooth temporal coherence. Compared with the origi-
+nal forms, our proposed temporal coherence regularization
+can better reflect the continuous change of video and thus
+lead to better performance.
+In the training phase, we take the negative distance for
+each class as logit. Then we utilize the same cross-entropy
+loss as in [7, 68], the auxiliary semantic loss [46, 54] and the
+temporal coherence regularization to jointly train the model.
+The auxiliary semantic loss refers to the cross-entropy loss
+on the real action classes, which is widely used to improve
+training stability and generalization. During inference, we
+select the support class closest to the query for classification.
+3.3 Extended applications of HyRSM++
+3.3.1 Semi-supervised few-shot action recognition
+The objective of semi-supervised few-shot action recogni-
+tion [113] is to fully explore the auxiliary information from
+unlabeled video data to boost the few-shot classification.
+Compared with the standard supervised few-shot setting, in
+addition to the support set S and query set Q, an extra un-
+labeled set U is also included in a semi-supervised few-shot
+task to alleviate data scarcity. We demonstrate that the pro-
+posed HyRSM++ can build a bridge between labeled and
+unlabeled examples, leading to higher classification perfor-
+mance.
+Given an unlabeled set U, a common practice in semi-
+supervised learning literature [110, 104, 77] is to adopt the
+Pseudo Labeling technique [45], which assumes that the de-
+cision boundary usually lies in low-density areas and data
+samples in a high-density area have the same label. Sim-
+ilarly, traditional semi-supervised few-shot learning meth-
+ods [71, 49] usually produce pseudo labels for unlabeled
+data based on the known support set, and then the gener-
+ated high-confidence pseudo-label data is augmented into
+the support set. In this paper, we follow this paradigm and
+utilize HyRSM++ to leverage unlabeled examples. Since
+Algorithm 1 HyRSM++ for semi-supervised few-shot ac-
+tion recognition
+Require: A labeled support set S, an auxiliary unlabeled set U, and a
+query set Q
+Ensure: Optimized few-shot classifier HyRSM++
+1: Enter support set S and unlabeled set U into HyRSM++ and obtain
+the category prediction of U based on Equation 9;
+2: According to the prediction distribution, select the high-confidence
+samples to generate pseudo-labels and update S with the selected
+samples to get the augmented S
+′;
+3: Apply the augmented S
+′ and query set Q for supervised few-shot
+training as described in Section 3.2;
+noisy videos usually have higher losses in training, it is pos-
+sible to leverage the strong HyRSM++ to distinguish be-
+tween clean and noisy videos from the prediction scores.
+Based on this, we choose reliable pseudo-labeled samples
+in the unlabeled set by predictions and augment the support
+set with high-confidence pseudo-label data. Subsequently,
+we take advantage of the augmented support set to classify
+the query videos as in the supervised few-shot task. During
+the training stage, many semi-supervised few-shot tasks are
+sampled to optimize the whole model, as shown in Algo-
+rithm 1. For inference, the evaluation process is also con-
+ducted by sampling 10,000 episodic tasks.
+3.3.2 Unsupervised few-shot action recognition
+Unlike the previously described settings involving labelled
+data, unsupervised few-shot action recognition aims to use
+unlabeled data to construct few-shot tasks and learn adap-
+tations to different tasks. We further extend HyRSM++ to
+this unsupervised task and verify its capability of transfer-
+ring prior knowledge to learn to deal with unseen tasks effi-
+ciently.
+To perform unsupervised few-shot learning, construct-
+ing few-shot tasks is the first step. However, there are no
+label annotations that can be directly applied for few-shot
+learning in the challenging unsupervised setting. Following
+prior unsupervised few-shot algorithms [38, 36], we gener-
+ate few-shot tasks by first adopting existing unsupervised
+learning approaches to learn initialized feature embeddings
+of the input videos, and then leveraging deep clustering tech-
+niques to construct pseudo-classes of the videos. According
+to clustering results, we are able to produce few-shot tasks
+by sampling N-way K-shot episodes. We then use the con-
+structed few-shot tasks to train HyRSM++. During the test-
+ing phase, we sample 10,000 episodes from the test set to
+obtain the performance, and the label information is only
+used for evaluation.
+
+8
+Xiang Wang et al.
+Table 1 Comparison to recent few-shot action recognition methods on the meta-testing set of SSv2-Full, Kinetics, Epic-kitchens and HMDB51.
+The experiments are conducted under the 5-way setting, and results are reported as the shot increases from 1 to 5. ”-” means the result is not
+available in published works, and the underline indicates the second best result.
+Method
+Reference
+Dataset
+1-shot
+2-shot
+3-shot
+4-shot
+5-shot
+CMN++ [112]
+ECCV’18
+SSv2-Full
+34.4
+-
+-
+-
+43.8
+TRN++ [109]
+ECCV’18
+38.6
+-
+-
+-
+48.9
+OTAM [7]
+CVPR’20
+42.8
+49.1
+51.5
+52.0
+52.3
+TTAN [48]
+ArXiv’21
+46.3
+52.5
+57.3
+59.3
+60.4
+ITANet [7]
+IJCAI’21
+49.2
+55.5
+59.1
+61.0
+62.3
+TRX (Ω={1}) [68]
+CVPR’21
+38.8
+49.7
+54.4
+58.0
+60.6
+TRX (Ω={2, 3})[68]
+CVPR’21
+42.0
+53.1
+57.6
+61.1
+64.6
+STRM [84]
+CVPR’22
+43.1
+53.3
+59.1
+61.7
+68.1
+MTFAN [94]
+CVPR’22
+45.7
+-
+-
+-
+60.4
+Nguyen et al. [62]
+ECCV’22
+43.8
+-
+-
+-
+61.1
+Huang et al. [33]
+ECCV’22
+49.3
+-
+-
+-
+66.7
+HCL [108]
+ECCV’22
+47.3
+54.5
+59.0
+62.4
+64.9
+HyRSM
+CVPR’22
+54.3 (+5.0)
+62.2 (+6.7)
+65.1 (+6.0)
+67.9 (+5.5)
+69.0 (+0.9)
+HyRSM++
+-
+55.0 (+5.7)
+63.5 (+8.0)
+66.0 (+6.9)
+68.8 (+6.4)
+69.8 (+1.7)
+MatchingNet [86]
+NeurIPS’16
+Kinetics
+53.3
+64.3
+69.2
+71.8
+74.6
+MAML [19]
+ICML’17
+54.2
+65.5
+70.0
+72.1
+75.3
+Plain CMN [112]
+ECCV’18
+57.3
+67.5
+72.5
+74.7
+76.0
+CMN-J [113]
+TPAMI’20
+60.5
+70.0
+75.6
+77.3
+78.9
+TARN [5]
+BMVC’19
+64.8
+-
+-
+-
+78.5
+ARN [105]
+ECCV’20
+63.7
+-
+-
+-
+82.4
+OTAM [7]
+CVPR’20
+73.0
+75.9
+78.7
+81.9
+85.8
+ITANet [106]
+IJCAI’21
+73.6
+-
+-
+-
+84.3
+TRX (Ω={1}) [68]
+CVPR’21
+63.6
+75.4
+80.1
+82.4
+85.2
+TRX (Ω={2, 3}) [68]
+CVPR’21
+63.6
+76.2
+81.8
+83.4
+85.9
+STRM [84]
+CVPR’22
+62.9
+76.4
+81.1
+83.8
+86.7
+MTFAN [94]
+CVPR’22
+74.6
+-
+-
+-
+87.4
+Nguyen et al. [62]
+ECCV’22
+74.3
+-
+-
+-
+87.4
+Huang et al. [33]
+ECCV’22
+73.3
+-
+-
+-
+86.4
+HCL [108]
+ECCV’22
+73.7
+79.1
+82.4
+84.0
+85.8
+HyRSM
+CVPR’22
+73.7 (-0.9)
+80.0 (+0.9)
+83.5 (+1.1)
+84.6 (+0.6)
+86.1 (-1.3)
+HyRSM++
+-
+74.0 (-0.6)
+80.8 (+1.7)
+83.9 (+1.5)
+85.3 (+1.3)
+86.4 (-1.0)
+OTAM [7]
+CVPR’20
+Epic-kitchens
+46.0
+50.3
+53.9
+54.9
+56.3
+TRX [68]
+CVPR’21
+43.4
+50.6
+53.5
+56.8
+58.9
+STRM [84]
+CVPR’22
+42.8
+50.4
+54.9
+58.0
+59.2
+HyRSM
+CVPR’22
+47.4 (+1.4)
+52.9 (+2.3)
+56.4 (+1.5)
+58.8 (+0.8)
+59.8 (+0.6)
+HyRSM++
+-
+48.0 (+2.0)
+54.9 (+4.3)
+57.5 (+2.6)
+59.6 (+1.6)
+60.8 (+1.6)
+ARN [105]
+ECCV’20
+HMDB51
+45.5
+-
+-
+-
+60.6
+OTAM [7]
+CVPR’20
+54.5
+63.5
+65.7
+67.2
+68.0
+TTAN [48]
+ArXiv’21
+57.1
+-
+-
+-
+74.0
+TRX [68]
+CVPR’21
+53.1
+62.5
+66.8
+70.2
+75.6
+STRM [84]
+CVPR’22
+52.3
+62.5
+67.4
+70.9
+77.3
+MTFAN [94]
+CVPR’22
+59.0
+-
+-
+-
+74.6
+Nguyen et al. [62]
+ECCV’22
+59.6
+-
+-
+-
+76.9
+Huang et al. [33]
+ECCV’22
+60.1
+-
+-
+-
+77.0
+HCL [108]
+ECCV’22
+59.1
+66.5
+71.2
+73.8
+76.3
+HyRSM
+CVPR’22
+60.3 (+0.2)
+68.2 (+1.7)
+71.7 (+0.5)
+75.3 (+1.5)
+76.0 (-1.3)
+HyRSM++
+-
+61.5 (+1.4)
+69.0 (+2.5)
+72.7 (+1.5)
+75.4 (+1.6)
+76.4 (-0.9)
+4 Experiments
+In this section, the following key questions will be answered
+in detail: (1) Is HyRSM++ competitive to other state-of-
+the-art methods on challenging few-shot benchmarks? (2)
+What components play an integral role in HyRSM++ so that
+HyRSM++ can work well? (3) Can the proposed hybrid re-
+lation module be viewed as a simple plug-and-play unit and
+have the same effect for other methods? (4) Does the pro-
+posed temporal set matching metric have an advantage over
+other measure competitors? (5) Can HyRSM++ have stable
+performance in a variety of different video scenarios?
+4.1 Datasets and experimental setups
+Datasets. We evaluate our HyRSM++ on six standard public
+few-shot benchmarks. For the Kinetics [8], SSv2-Full [23],
+and SSv2-Small [23] datasets, we adopt the existing splits
+proposed by [7, 112, 106, 68], and each dataset consists
+
+HyRSM++: Hybrid Relation Guided Temporal Set Matching for Few-shot Action Recognition
+9
+MSA
+Transformer
+Bi-LSTM
+Bi-GRU
+Inter-relation
+MSA
+Transformer
+Bi-LSTM
+Bi-GRU
+Intra-relation
+54.3
+53.0
+53.6
+53.3
+54.3
+53.8
+53.6
+54.1
+50.6
+50.4
+51.4
+50.9
+51.8
+50.6
+50.8
+51.8
+50.5
+51.0
+51.5
+52.0
+52.5
+53.0
+53.5
+54.0
+Fig. 3 Comparison between different components in hybrid relation
+module on 5-way 1-shot few-shot action classification without tempo-
+ral coherence regularization. Experiments are conducted on the SSv2-
+Full dataset.
+MSA
+Transformer
+Bi-LSTM
+Bi-GRU
+Inter-relation
+MSA
+Transformer
+Bi-LSTM
+Bi-GRU
+Intra-relation
+55.0
+55.0
+54.6
+54.4
+54.5
+54.6
+54.4
+54.5
+50.7
+50.6
+51.7
+51.9
+52.1
+51.7
+52.3
+52.0
+51.0
+51.5
+52.0
+52.5
+53.0
+53.5
+54.0
+54.5
+55.0
+Fig. 4 Comparison between different components in hybrid relation
+module on 5-way 1-shot few-shot action classification with temporal
+coherence regularization. Experiments are conducted on the SSv2-Full
+dataset.
+of 64 and 24 classes as the meta-training and meta-testing
+set, respectively. For UCF101 [78] and HMDB51 [42], we
+verify our proposed methods by leveraging existing splits
+from [105, 68]. In addition to the above, we also utilize
+the egocentric Epic-kitchens [14, 13] dataset to evaluate
+HyRSM++.
+Implementation details. Following previous works [112, 7,
+68, 106], ResNet-50 [28] initialized with ImageNet [15] pre-
+trained weights is utilized as the feature extractor in our ex-
+periments. We sparsely and uniformly sample 8 (i.e., T = 8)
+frames per video to construct input frame sequence, which is
+in line with previous methods [7, 106]. In the training phase,
+we also adopt basic data augmentation such as random crop-
+ping and color jitter, and use Adam [39] optimizer to train
+our model. During the inference stage, we conduct few-shot
+action recognition evaluation on 10,000 randomly sampled
+episodes from the meta-testing set and report the mean ac-
+curacy. For many shot classification, e.g., 5-shot, we follow
+ProtoNet [76] and calculate the mean features of support
+videos in each class as the prototypes, and classify the query
+videos according to their distances against the prototypes.
+4.2 Comparison with state-of-the-art
+In this section, we validate the effectiveness of the proposed
+HyRSM++ by comparing it with state-of-the-art methods
+under various settings. As indicated in Table 1 and Ta-
+ble 2, the proposed HyRSM++ surpasses other advanced
+approaches significantly and is able to achieve new state-
+of-the-art performance. For instance, HyRSM++ improves
+the state-of-the-art performance from 49.2% to 55.0% un-
+der the 1-shot setting on SSv2-Full and consistently outper-
+forms our original conference version [91]. Specially, ex-
+tensively compared with current strict temporal alignment
+techniques [7, 106] and complex fusion methods [48, 68],
+HyRSM++ produces results that are superior to them un-
+der most different shots, which implies that our approach
+is considerably flexible and efficient. Note that the SSv2-
+Full and SSv2-Small datasets tend to be motion-based and
+generally focus on temporal reasoning. While Kinetics and
+UCF101 are partly appearance-related datasets, and scene
+understanding is usually essential. Besides, Epic-kitchens
+and HMDB51 are relatively complicated and might involve
+diverse object interactions. Extensively evaluated on these
+benchmarks, HyRSM++ provides excellent performance. It
+reveals that our HyRSM++ has strong robustness and gen-
+eralization for different scenes. From Table 2, we observe
+that HyRSM++ outperforms current state-of-the-art meth-
+ods on UCF101 and SSv2-Small under the 1-shot and 3-
+shot settings, which suggests that our HyRSM++ can learn
+rich and effective representations with extremely limited
+samples. It’s worth noting that under the 5-shot evaluation,
+our HyRSM++ yields 95.9% and 58.0% 5-shot performance
+on UCF101 and SSv2-Small, respectively, which is slightly
+behind STRM and HCL. We attribute this to STRM and
+HCL are ensemble methods that weight each sample with
+attention or use multiple metrics for few-shot classification,
+which makes them more suitable for multi-shots, while our
+HyRSM++ is a simple and general method without involves
+complex ensemble operations. Moreover, we also observe
+that with the introduction of temporal coherence regulariza-
+tion, HyRSM++ has a significant improvement compared to
+HyRSM, which verifies the effectiveness of exploiting tem-
+poral order information during the set matching process.
+4.3 Ablation study
+For ease of comparison, we use a baseline method Pro-
+toNet [76] that applies global-average pooling to backbone
+representations to obtain a prototype for each class. We will
+explore the role and validity of our proposed modules in de-
+tail below.
+Design choices of relation modeling.
+To systematically
+investigate the effect of different relation modeling opera-
+
+10
+Xiang Wang et al.
+Table 2 Results on 1-shot, 3-shot, and 5-shot few-shot classification on the UCF101 and SSv2-Small datasets. ”-” means the result is not available
+in published works, and the underline indicates the second best result.
+UCF101
+SSv2-Small
+Method
+Reference
+1-shot
+3-shot
+5-shot
+1-shot
+3-shot
+5-shot
+MatchingNet [86]
+NeurIPS’16
+-
+-
+-
+31.3
+39.8
+45.5
+MAML [19]
+ICML’17
+-
+-
+-
+30.9
+38.6
+41.9
+Plain CMN [112]
+ECCV’18
+-
+-
+-
+33.4
+42.5
+46.5
+CMN-J [113]
+TPAMI’20
+-
+-
+-
+36.2
+44.6
+48.8
+ARN [105]
+ECCV’20
+66.3
+-
+83.1
+-
+-
+-
+OTAM [7]
+CVPR’20
+79.9
+87.0
+88.9
+36.4
+45.9
+48.0
+TTAN [48]
+ArXiv’21
+80.9
+-
+93.2
+-
+-
+-
+ITANet [106]
+IJCAI’21
+-
+-
+-
+39.8
+49.4
+53.7
+TRX [68]
+CVPR’21
+78.2
+92.4
+96.1
+36.0
+51.9
+59.1
+STRM [84]
+CVPR’22
+80.5
+92.7
+96.9
+37.1
+49.2
+55.3
+MTFAN [94]
+CVPR’22
+84.8
+-
+95.1
+-
+-
+-
+Nguyen et al. [62]
+ECCV’22
+84.9
+-
+95.9
+-
+-
+-
+Huang et al. [33]
+ECCV’22
+71.4
+-
+91.0
+38.9
+-
+61.6
+HCL [108]
+ECCV’22
+82.5
+91.0
+93.9
+38.7
+49.1
+55.4
+HyRSM
+CVPR’22
+83.9 (-1.0)
+93.0 (+0.3)
+94.7 (-2.2)
+40.6 (+0.8)
+52.3 (+0.4)
+56.1 (-5.5)
+HyRSM++
+-
+85.8 (+0.9)
+93.5 (+0.8)
+95.9 (-1.0)
+42.8 (+3.0)
+52.4 (+0.5)
+58.0 (-2.6)
+Table 3 Ablation study under 5-way 1-shot and 5-way 5-shot settings
+on the SSv2-Full dataset. “TCR” refers to temporal coherence regular-
+ization.
+Intra-relation
+Inter-relation
+Bi-MHM
+TCR
+1-shot
+5-shot
+35.2
+45.3
+�
+41.2
+55.0
+�
+43.7
+55.2
+�
+44.6
+56.0
+�
+�
+45.3
+57.1
+�
+�
+48.1
+60.5
+�
+�
+48.3
+61.2
+�
+�
+�
+49.2
+62.8
+�
+�
+51.4
+64.6
+�
+�
+�
+52.4
+65.8
+�
+�
+�
+54.3
+69.0
+�
+�
+�
+�
+55.0
+69.8
+Table 4 Generalization of hybrid relation module. We conduct exper-
+iments on SSv2-Full.
+Method
+1-shot
+5-shot
+OTAM [7]
+42.8
+52.3
+OTAM [7]+ Intra-relation
+48.9
+60.4
+OTAM [7]+ Inter-relation
+46.9
+57.8
+OTAM [7]+ Intra-relation + Inter-relation
+51.7
+63.9
+tions in hybrid relation module, we vary the components to
+construct some variants and report the results in Figure 3
+and Figure 4. The comparison experiments are conducted
+on the SSv2-Full dataset under the 5-way 1-shot setting. We
+can observe that different combinations have quite distinct
+properties, e.g., multi-head self-attention (MSA) and Trans-
+former are more effective to model intra-class relations than
+Bi-LSTM and Bi-GRU. For example, utilizing multi-head
+5-way
+6-way
+7-way
+8-way
+9-way
+10-way
+Accuracy (%)
+Kinetics
+OTAM
+TRX
+STRM
+HyRSM
+HyRSM++
+50
+66
+54
+58
+70
+62
+5-way
+6-way
+7-way
+8-way
+9-way
+10-way
+Accuracy (%)
+SSv2-Full
+OTAM
+TRX
+STRM
+HyRSM
+HyRSM++
+25
+30
+40
+35
+50
+45
+55
+74
+Fig. 5 N-way 1-shot performance trends of our HyRSM++ and other
+state-of-the-art methods with different N on SSv2-Full. The compari-
+son results prove the superiority of our HyRSM++.
+Accuracy (%)
+(a) Frames
+42
+46
+50
+54
+2
+3
+4
+5
+6
+7
+8
+9
+10
+1
+2
+4
+8
+16
+32
+Accuracy (%)
+1-shot
+5-shot
+45
+50
+60
+55
+70
+65
+(b) Head number
+Fig. 6 (a) Performance on SSv2-Full using a different number of
+frames under the 5-way 1-shot setting. (b) The effect of the number
+of heads on SSv2-Full.
+self-attention to learn intra-relation produces at least 2.5%
+improvements than with Bi-LSTM. Nevertheless, compared
+with other recent algorithms [68, 106], the performance of
+each combination can still be improved, which strongly sug-
+gests the necessity of structure design for learning task-
+specific features. For simplicity, we choose the same struc-
+ture to explore intra-relation and inter-relation, and the con-
+
+HyRSM++: Hybrid Relation Guided Temporal Set Matching for Few-shot Action Recognition
+11
+20
+30
+40
+50
+60
+70
+SSv2-Full (Resnet-18)
+OTAM
+TRX
+HyRSM
+HyRSM++
+1-shot
+2-shot
+3-shot
+4-shot
+5-shot
+20
+30
+40
+50
+60
+70
+SSv2-Full (Resnet-34)
+OTAM
+TRX
+HyRSM
+HyRSM++
+1-shot
+2-shot
+3-shot
+4-shot
+5-shot
+Accuracy (%)
+50
+55
+60
+65
+70
+75
+80
+85
+Kinetics (Resnet-18)
+OTAM
+TRX
+HyRSM
+HyRSM++
+1-shot
+2-shot
+3-shot
+4-shot
+5-shot
+55
+60
+65
+70
+75
+80
+85
+Kinetics (Resnet-34)
+OTAM
+TRX
+HyRSM
+HyRSM++
+1-shot
+2-shot
+3-shot
+4-shot
+5-shot
+Accuracy (%)
+Accuracy (%)
+Accuracy (%)
+Fig. 7 Comparison of the backbone with different depths on the SSv2-
+Full and Kinetics datasets.
+Table 5 Comparative experiments on SSv2-Full using the Inception-
+v3 [81] feature extractor.
+Method
+1-shot
+2-shot
+3-shot
+4-shot
+5-shot
+OTAM [7]
+42.4
+46.6
+48.7
+49.2
+52.1
+TRX [68]
+37.7
+50.2
+55.5
+57.2
+60.1
+STRM [84]
+42.9
+53.9
+58.9
+62.3
+63.4
+HyRSM++
+53.3
+62.7
+65.3
+67.8
+69.3
+Table 6 Performance comparison on SSv2-Full with self-supervised
+initialization weights [97].
+Method
+1-shot
+2-shot
+3-shot
+4-shot
+5-shot
+OTAM [7]
+41.2
+45.9
+48.8
+50.1
+51.0
+TRX [68]
+37.5
+43.8
+49.9
+51.6
+52.1
+STRM [84]
+38.0
+46.2
+49.9
+53.4
+54.4
+HyRSM++
+50.9
+59.1
+62.6
+65.5
+66.4
+Table 7 Performance comparison with different relation modeling
+paradigms on SSv2-Full and Kinetics.
+Setting
+Method
+Dataset
+1-shot
+5-shot
+Support-only
+HyRSM
+SSv2-Full
+52.1
+67.2
+Support-only
+HyRSM++
+53.7
+68.8
+Support&Query
+HyRSM
+54.3
+69.0
+Support&Query
+HyRSM++
+55.0
+69.8
+Support-only
+HyRSM
+Kinetics
+73.4
+85.5
+Support-only
+HyRSM++
+73.5
+85.7
+Support&Query
+HyRSM
+73.7
+86.1
+Support&Query
+HyRSM++
+74.0
+86.4
+figuration of multi-head self-attention is adopted in the ex-
+periments.
+Analysis of the proposed components. Table 3 summa-
+rizes the ablation study of each module in HyRSM++. To
+evaluate the function of the proposed components, Pro-
+toNet [76] is taken as our baseline. From the ablation results,
+we can conclude that each component is highly effective.
+In particular, compared to the baseline, intra-relation mod-
+eling can respectively bring 6.0% and 9.7% performance
+52.3 
+51.2 
+49.7 
+49.1 
+48.0 
+47.2 
+64.6 
+61.3 
+59.2 
+56.3 
+53.1 
+50.4 
+68.1 
+62.3 
+60.8 
+58.5 
+55.9 
+52.0 
+69.8 
+66.2 
+65.1 
+64.5 
+62.3 
+60.0 
+0%
+10%
+20%
+30%
+40%
+50%
+Accuracy (%)
+5-way 5-shot
+OTAM
+TRX
+STRM
+HyRSM++
+45
+61
+49
+53
+65
+57
+42.8 
+41.4 
+40.3 
+39.0 
+37.1 
+35.7 
+42.0 
+38.5 
+35.8 
+33.2 
+31.3 
+28.9 
+43.1
+40.4
+37.8
+34.5
+32.1
+30.0 
+55.0 
+49.8 
+48.1 
+46.4 
+43.2 
+41.3 
+0%
+10%
+20%
+30%
+40%
+50%
+Accuracy (%)
+5-way 1-shot
+OTAM
+TRX
+STRM
+HyRSM++
+25
+30
+40
+35
+50
+45
+55
+69
+Noisy ratio
+Noisy ratio
+Fig. 8 Robustness comparison experiments in the presence of noisy
+samples. X% represents the proportion of noisy labels included in the
+dataset.
+Table 8 Comparison with recent temporal alignment methods on the
+SSv2-Full dataset under the 5-way 1-shot and 5-way 5-shot settings.
+Diagonal means matching frame by frame.
+Metric
+Bi-direction
+1-shot
+5-shot
+Diagonal
+-
+38.3
+48.7
+Plain DTW [61]
+-
+39.6
+49.0
+OTAM [7]
+�
+39.3
+47.7
+OTAM [7]
+�
+42.8
+52.3
+Bi-MHM
+�
+44.6
+56.0
+Temporal set matching metric
+�
+45.3
+57.1
+Table 9 Comparison of different set matching strategies on the SSv2-
+Full dataset.
+Metric
+Bi-direction
+1-shot
+5-shot
+Hausdorff distance
+�
+32.4
+38.2
+Hausdorff distance
+�
+34.5
+39.1
+Modified Hausdorff distance
+�
+44.2
+50.0
+Bi-MHM
+�
+44.6
+56.0
+Temporal set matching metric
+�
+45.3
+57.1
+Table 10 Generalization of temporal coherence regularization. We
+conduct experiments on SSv2-Full. ”Hard margin” represents the
+method described in Equation 11.
+Method
+1-shot
+5-shot
+OTAM [7]
+42.8
+52.3
+OTAM [7] + IDM
+43.7
+55.0
+OTAM [7] + Hard margin
+43.2
+55.3
+OTAM [7] + Temporal coherence regularization
+44.1
+55.8
+Bi-MHM
+44.6
+56.0
+Bi-MHM + IDM
+44.7
+56.3
+Bi-MHM + Hard margin
+44.7
+56.5
+Bi-MHM + Temporal coherence regularization
+45.3
+57.1
+gains on 1-shot and 5-shot, and inter-relation function boosts
+the performance by 8.5% and 9.9% on 1-shot and 5-shot.
+In addition, the proposed set matching metric improves 1-
+shot and 5-shot classification by 9.4% and 10.7%, respec-
+tively, which indicates the ability to find better correspond-
+ing frames in the video pair. Adding temporal coherence
+regularization to the set matching metric also achieves sta-
+
+12
+Xiang Wang et al.
+ble performance improvements. Moreover, stacking the pro-
+posed modules can further improve performance, indicating
+the complementarity between components. When consider-
+ing all the proposed modules together to form HyRSM++,
+the performance of 1-shot and 5-shot is improved to 55.0%
+and 69.8%, respectively, which strongly supports the impor-
+tance of learning task-related features and flexible metrics.
+Pluggability of hybrid relation module. In Table 4, we
+experimentally show that the hybrid relation module gen-
+eralizes well to other methods by inserting it into the re-
+cent OTAM [7]. In this study, OTAM with our hybrid re-
+lation module benefits from relational information and fi-
+nally achieves 8.9% and 11.6% gains on 1-shot and 5-shot.
+This fully evidences that mining the rich information among
+videos to learn task-specific features is especially valuable.
+N-way few-shot classification.
+In the previous experi-
+ments, all of our comparative evaluation experiments were
+carried out under the 5-way setting. In order to further ex-
+plore the influence of different N, in Figure 5, we com-
+pare N-way (N ≥ 5) 1-shot results on SSv2-Full and Ki-
+netics. Results show that as N increases, the difficulty be-
+comes higher, and the performance decreases. Neverthe-
+less, the performance of our HyRSM++ is still consistently
+ahead of the recent state-of-the-art STRM [84], TRX [68]
+and OTAM [7], which shows the feasibility of our method
+to boost performance by introducing rich relations among
+videos and the power of the set matching metric.
+Varying the number of frames. To demonstrate the scal-
+ability of HyRSM++, we also explore the impact of differ-
+ent video frame numbers on performance. Of note, previous
+comparisons are performed under 8 frames of input. Results
+in Figure 6(a) show that as the number of frames increases,
+the performance improves. HyRSM++ gradually tends to be
+saturated when more than 7 frames.
+Influence of head number. Previous analyses have shown
+that multi-head self-attention can focus on different patterns
+and is critical to capturing diverse features [41]. We investi-
+gate the virtue of varying the number of heads in multi-head
+self-attention on performance in Figure 6(b). Experimental
+results indicate that the effect of multi-head is remarkable,
+and the performance starts to saturate beyond a particular
+point.
+Varying depth of the backbone. The proposed HyRSM++
+is general and compatible with feature extractors of various
+capacities. The previous methods all utilize ResNet-50 as
+backbone by default for a fair comparison, and the impact
+of backbone’s depth on performance is still under-explored.
+As presented in Figure 7, we attempt to answer this question
+by adopting ResNet-18 and ResNet-34 pre-trained on Ima-
+geNet as alternative backbones. Results demonstrate that the
+deeper network clearly benefits from greater learning capac-
+ity and results in better performance. In addition, we notice
+Acc = 100%
+Acc = 100% 
+(+ hybrid relation module)
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+Acc = 40%
+Acc = 60%
+Acc = 60%
+Acc = 80%
+Acc = 60%
+Acc = 100% 
+(+ hybrid relation module)
+Acc = 100% 
+(+ hybrid relation module)
+Acc = 100% 
+(+ hybrid relation module)
+Acc = 100% 
+(+ hybrid relation module)
+Acc = 100% 
+(+ hybrid relation module)
+(a) Examples from SSv2-Full
+(b) Examples from Kinetics
+Fig. 9 Similarity visualization of how query videos (rows) match to
+support videos (columns). The boxes of different colors correspond to:
+correct match and incorrect match.
+Support
+Query
+(a) SSv2-Full: ”pretending to open something without actually opening it”
+(b) SSv2-Full: ”showing that something is empty”
+Support
+Query
+Support
+Query
+(c) Kinetics: ”cutting watermelon”
+Fig. 10 Visualization of matching results with the proposed set match-
+ing metric on SSv2-Full and Kinetics.
+that our proposed HyRSM++ consistently outperforms the
+competitors (i.e., OTAM and TRX), which indicates that our
+HyRSM++ is a generally effective framework.
+Influence of different backbones. To verify that our ap-
+proach is not limited to ResNet-like structures, we further
+perform experiments on Inception-v3 and report the results
+in Table 5. From the comparison, we note that HyRSM++ is
+significantly superior to other competitive algorithms. Com-
+pared with STRM [84], our proposed HyRSM++ leads to at
+least 5.5% performance gain under various settings.
+Impact of pretraining types. Supervised ImageNet initial-
+ization [15] is widely employed in many vision tasks [7,
+113, 90] and achieves impressive success. Recently, self-
+supervised techniques have also received widespread at-
+tention and revealed excellent application potential. In Ta-
+
+0.73
+0.063
+0.06
+0.082
+0.065
+0.16
+0.35
+0.21
+0.1 
+0.18
+0.067
+0.069
+0.67
+0.08
+0.11
+0.27
+0.038
+0.076
+0.42
+0.2 
+0.11
+0.27
+0.13
+0.09
+0.4 0.23
+0.25
+0.11
+0.19
+0.22
+0.18
+0.25
+0.24
+0.14
+0.19
+0.18
+0.18
+0.36
+0.091
+0.18
+0.23
+0.15
+0.11
+0.4
+0.1
+0.14
+0.32
+0.08
+0.11
+0.340.65
+0.029
+0.15
+0.088
+0.078
+0.092
+0.55
+0.04
+0.053
+0.27
+0.17
+0.076
+0.59
+0.094
+0.07
+0.11
+0.053
+0.042
+0.65
+0.14
+0.1
+0.039
+0.024
+0.087
+0.750.34
+0.16
+0.14
+0.23
+0.14
+0.17
+0.26
+0.19
+0.21
+0.18
+0.16
+0.2 
+0.27
+0.13
+0.24
+0.2 
+0.21
+0.16
+0.27
+0.16
+0.15
+0.23
+0.17
+0.18
+0.270.46
+0.064
+0.037
+0.38
+0.055
+0.035
+0.69
+0.021
+0.17
+0.082
+0.051
+0.042
+0.78
+0.06
+0.067
+0.09
+0.2
+0.18
+0.45
+0.087
+0.12
+0.18
+0.098
+0.13
+0.470.26
+0.13
+0.28
+0.21
+0.12
+0.13
+0.38
+0.23
+0.19
+0.069
+0.14
+0.16
+0.19
+0.2 
+0.32
+0.25
+0.13
+0.2 
+0.29
+0.12
+0.15
+0.09
+0.21
+0.19
+0.360.21
+0.25
+0.16
+0.25
+0.13
+0.24
+0.23
+0.16
+0.19
+0.18
+0.14
+0.16
+0.27
+0.12
+0.32
+0.22
+0.2 
+0.1 
+0.35
+0.13
+0.13
+0.19
+0.25
+0.15
+0.290.61
+0.074
+0.068
+0.14
+0.1
+0.076
+0.43
+0.084
+0.35
+0.063
+0.12
+0.061
+0.63
+0.077
+0.11
+0.21
+0.21
+0.06
+0.47
+0.053
+0.15
+0.055
+0.069
+0.054
+0.670.59
+0.043
+0.13
+0.13
+0.1
+0.11
+0.51
+0.14
+0.2
+0.036
+0.13
+0.27
+0.44
+0.091
+0.068
+0.27
+0.14
+0.17
+0.35
+0.066
+0.05
+0.057
+0.044
+0.038
+0.810.19
+0.24
+0.11
+0.2
+0.25
+0.27
+0.16
+0.21
+0.23
+0.13
+0.21
+0.22
+0.3
+0.14
+0.13
+0.15
+0.15
+0.11
+0.34
+0.26
+0.22
+0.16
+0.19
+0.18
+0.250.59
+0.18
+0.014
+0.095
+0.13
+0.21
+0.43
+0.025
+0.26
+0.076
+0.049
+0.042
+0.76
+0.071
+0.081
+0.31
+0.097
+0.019
+0.42
+0.16
+0.21
+0.063
+0.11
+0.13
+0.490.47
+0.1
+0.054
+0.21
+0.17
+0.19
+0.23
+0.08
+0.27
+0.24
+0.098
+0.15
+0.5 
+0.096
+0.16
+0.23
+0.082
+0.076
+0.4
+0.21
+0.12
+0.4
+0.13
+0.11
+0.25HyRSM++: Hybrid Relation Guided Temporal Set Matching for Few-shot Action Recognition
+13
+ble 6, we show the performance comparison with self-
+supervised pretraining weights [97]. Results demonstrate
+that our HyRSM++ is still powerful and not limited to the
+specific initialization weights.
+Other relation modeling forms. Previous few-shot image
+classification methods of learning task-specific features have
+also achieved promising results [101, 47]. However, many
+of them use some complex and fixed operations to learn the
+dependencies between images, while our method is straight-
+forward and flexible. Moreover, most previous works only
+use the information within the support set to learn task-
+specific features, ignoring the correlation with query sam-
+ples. In our hybrid relation module, we add the query video
+to the pool of inter-relation modeling to extract relevant in-
+formation suitable for query classification. As illustrated in
+Table 7, we try to remove the query video from the pool
+in HyRSM++, i.e., Support-only, but we can observe that
+after removing the query video, the performance of 1-shot
+and 5-shot on SSv2-Full reduces by 1.3% and 1.0%, respec-
+tively. There are similar conclusions on the Kinetics dataset.
+This evidences that the proposed hybrid relation module is
+reasonable and can effectively extract task-related features,
+thereby promoting query classification accuracy.
+Robustness to noise labels. To demonstrate the robustness
+of HyRSM++ to noise samples, we simulate the presence
+of noise labels in the dataset in Figure 8. From the results,
+we can observe that performance generally decreases as the
+proportion of noise rises. However, our HyRSM++ still ex-
+hibits higher performance than other methods, which illus-
+trates the robustness of our method and its adaptability to
+complex conditions.
+4.4 Comparison with other matching approaches
+Our proposed temporal set matching metric Bi-MHM aims
+to accurately find the corresponding video frames between
+video pairs by relaxing the strict temporal ordering con-
+straints. The following comparative experiments in Table 8
+are carried out under identical experimental setups, i.e., re-
+Table 11 Complexity analysis for 5-way 1-shot SSv2-Full evaluation.
+The experiments are carried out on one Nvidia V100 GPU.
+Method
+Backbone
+Param
+FLOPs
+Latency
+Acc
+HyRSM
+ResNet-18
+13.8M
+3.64G
+36.5ms
+46.6
+HyRSM++
+ResNet-18
+13.8M
+3.64G
+36.5ms
+47.7
+HyRSM
+ResNet-34
+23.9M
+7.34G
+67.5ms
+50.0
+HyRSM++
+ResNet-34
+23.9M
+7.34G
+67.5ms
+50.4
+OTAM [7]
+ResNet-50
+23.5M
+8.17G
+116.6ms
+42.8
+TRX [68]
+ResNet-50
+47.1M
+8.22G
+94.6ms
+42.0
+STRM [84]
+ResNet-50
+73.3M
+8.27G
+113.3ms
+43.1
+HyRSM
+ResNet-50
+65.6M
+8.36G
+83.5ms
+54.3
+HyRSM++
+ResNet-50
+65.6M
+8.36G
+83.5ms
+55.0
+place the OTAM directly with our Bi-MHM while keep-
+ing other settings unchanged. Results show that our Bi-
+MHM performs well and outperforms other temporal align-
+ment methods (e.g., OTAM). We further analyze different
+set matching approaches in Table 9, and the results indi-
+cate that Hausdorff distance is susceptible to noise interfer-
+ence, resulting in the mismatch and relatively poor perfor-
+mance. However, our Bi-MHM shows stability to noise and
+obtains better performance. Furthermore, compared with the
+single directional metric, our proposed bidirectional metric
+is more comprehensive in reflecting the actual distances be-
+tween videos and achieves better performance on few-shot
+tasks. In addition, we observe that the proposed temporal
+set matching metric achieves clear improvement over Bi-
+MHM after incorporating temporal coherence. For instance,
+the temporal set matching metric obtains 0.7%, 1.1% perfor-
+mance gains on 5-way 1-shot, and 5-way 5-shot SSv2-Full
+classification. It indicates the effectiveness of the proposed
+temporal set matching metric.
+4.5 Comparison of temporal coherence manners
+Pioneering work [11, 22, 59] also indicates the important
+role of temporal coherence and shows remarkable results in
+face recognition [59] and unsupervised representation learn-
+ing [22, 27]. However, they also have some limitations as
+noted in Section 3.2, and thus the temporal coherence reg-
+ularization is proposed for smooth video coherence. Ta-
+ble 10 compares the proposed temporal coherence regular-
+ization with existing temporal coherence schemes based on
+OTAM and Bi-MHM. Results show that exploiting tempo-
+ral coherence helps improve the classification accuracy of
+the metrics, which confirms our motivation for consider-
+ing temporal order information during the matching process.
+In addition, our proposed temporal coherence regularization
+achieves more significant improvements than other manners,
+and we attribute this to the smooth property of temporal co-
+herence regularization.
+4.6 Visualization results
+To qualitatively show the discriminative capability of the
+learned task-specific features in our proposed method, we
+visualize the similarities between query and support videos
+with and without the hybrid relation module. As depicted in
+Figure 9, by adding the hybrid relation module, the discrim-
+ination of features is significantly improved, contributing to
+predicting more accurately. Additionally, the matching re-
+sults of the set matching metric are visualized in Figure 10,
+and we can observe that our Bi-MHM is considerably flexi-
+ble in dealing with alignment and misalignment.
+
+14
+Xiang Wang et al.
+Support
+Query
+Support
+Query
+“tipping Sth over” from SSv2-Full
+OTAM
+HyRSM++
+“taking Sth out of Sth” from SSv2-Full
+OTAM
+HyRSM++
+Support
+Query
+“showing Sth next to Sth” from SSv2-Full
+OTAM
+HyRSM++
+Support
+Query
+“riding elephant” from Kinetics
+OTAM
+HyRSM++
+Support
+Query
+“playing trumpet” from Kinetics
+OTAM
+HyRSM++
+Support
+Query
+“filling eyebrows” from Kinetics
+OTAM
+HyRSM++
+Fig. 11 Visualization of activation maps with Grad-CAM [75]. Compared to OTAM [7], HyRSM++ focuses more precisely on classification-
+related regions.
+To further visually evaluate the proposed HyRSM++, we
+compare the activation visualization results of HyRSM++ to
+the competitive OTAM [7]. As shown in Figure 11, the fea-
+tures of OTAM usually contain non-target objects or ignore
+most discriminative parts since it lacks the mechanism of
+learning task-specific embeddings for feature adaptation. In
+contrast, our proposed HyRSM++ processes the query and
+support videos with an adaptive relation modeling operation,
+which allows it to focus on the different target objects. The
+above qualitative experiments illustrate the rationality of our
+model design and the necessity of learning task-related fea-
+tures.
+4.7 Limitations
+In order to further understand HyRSM++, Table 11 il-
+lustrates its differences with OTAM and TRX in terms
+of parameters, computation, and runtime. In the inference
+
+HyRSM++: Hybrid Relation Guided Temporal Set Matching for Few-shot Action Recognition
+15
+Table 12 Comparison to existing semi-supervised few-shot action recognition methods on the meta-testing set of Kinetics and SSv2-Small. The
+experiments are conducted under the 5-way setting, and results are reported as the shot increases from 1 to 5. ”w/o unlabeled data” indicates
+that there is no unlabeled set in a episode, i.e., the traditional few-shot action recognition setting, which can act as the lower bound of the semi-
+supervised counterpart.
+Dataset
+Method
+Backbone
+1-shot
+2-shot
+3-shot
+4-shot
+5-shot
+Kinetics
+OTAM w/o unlabeled data [7]
+Inception-v3
+68.6
+72.7
+74.1
+75.7
+76.9
+DeepCluster CACTUs-MAML [30]
+Inception-v3
+65.1
+72.8
+76.5
+77.9
+79.5
+DeepCluster CACTUs-ProtoNets [30]
+Inception-v3
+66.9
+73.2
+77.0
+78.1
+79.9
+LIM [113]
+Inception-v3
+69.8
+75.9
+78.3
+80.4
+82.6
+HyRSM++ w/o unlabeled data
+Inception-v3
+69.1
+76.0
+78.6
+81.6
+81.9
+HyRSM++
+Inception-v3
+73.7
+79.4
+80.9
+81.8
+83.1
+CMN w/o unlabeled data [112]
+ResNet-50
+60.5
+70.0
+75.6
+77.3
+78.9
+OTAM w/o unlabeled data [7]
+ResNet-50
+73.0
+75.9
+78.7
+81.9
+85.8
+LIM (ensemble) [113]
+ResNet-50, Inception-v3, ResNet-18
+73.3
+78.3
+80.8
+82.4
+84.0
+HyRSM++ w/o unlabeled data
+ResNet-50
+74.0
+80.8
+83.6
+85.3
+86.4
+HyRSM++
+ResNet-50
+79.1
+84.3
+85.4
+86.4
+86.8
+SSv2-Small
+OTAM w/o unlabeled data [112]
+Inception-v3
+36.7
+41.0
+43.6
+44.1
+46.9
+DeepCluster CACTUs-MAML [30]
+Inception-v3
+37.9
+44.5
+45.9
+47.8
+49.9
+DeepCluster CACTUs-ProtoNets [30]
+Inception-v3
+38.4
+44.8
+46.1
+48.0
+50.1
+LIM [113]
+Inception-v3
+41.1
+46.9
+48.0
+51.5
+53.0
+HyRSM++ w/o unlabeled data
+Inception-v3
+41.5
+46.1
+49.5
+52.9
+55.1
+HyRSM++
+Inception-v3
+43.6
+49.5
+51.8
+52.4
+54.5
+CMN w/o unlabeled data [112]
+ResNet-50
+36.2
+42.1
+44.6
+47.0
+48.8
+OTAM w/o unlabeled data [112]
+ResNet-50
+36.4
+42.9
+45.9
+46.8
+48.0
+LIM (ensemble) [113]
+ResNet-50, Inception-v3, ResNet-18
+44.0
+49.8
+51.3
+53.9
+55.1
+HyRSM++ w/o unlabeled data
+ResNet-50
+42.8
+47.1
+52.4
+54.7
+58.0
+HyRSM++
+ResNet-50
+45.4
+51.1
+55.2
+57.4
+58.8
+74.0 
+77.7 
+79.1 
+79.7 
+79.9 
+80.8 
+83.0 
+84.3 
+84.3 
+84.6 
+83.6 
+84.8 
+85.4 
+85.5 
+85.9 
+85.3
+85.8
+86.4
+86.6
+86.8
+86.4
+86.5
+86.7
+86.8
+86.9
+0
+50
+100
+150
+200
+Accuracy (%)
+Kinetics
+1-shot
+2-shot
+3-shot
+4-shot
+5-shot
+73
+75
+79
+77
+85
+81
+87
+83
+89
+Fig. 12 Performance comparison of different amounts of unlabeled
+data for testing in an episode on Kinetics.
+phase, HyRSM++ does not add additional computational
+burden compared to HyRSM because the temporal coher-
+ence regularization is not involved in the calculation. No-
+tably, HyRSM++ introduces extra parameters (i.e., hybrid
+relation module), resulting in increased GPU memory and
+computational consumption. Nevertheless, without complex
+non-parallel classifier heads, the whole inference speed of
+HyRSM++ is faster than OTAM and TRX. We will further
+investigate how to reduce complexity with no loss of perfor-
+mance in the future.
+42.8 
+45.0 
+45.4 
+45.8 
+46.2 
+47.1 
+50.1 
+51.1 
+51.2 
+51.6 
+52.4 
+54.1 
+55.2 
+55.5 
+55.8 
+54.7
+56.8
+57.4
+57.8
+57.9
+58.0 
+58.6 
+58.8 
+59.0 
+59.6 
+0
+50
+100
+150
+200
+Accuracy (%)
+SSv2-Small
+1-shot
+2-shot
+3-shot
+4-shot
+5-shot
+60
+38
+40
+44
+42
+50
+46
+58
+48
+62
+54
+52
+56
+Fig. 13 Performance comparison of different amounts of unlabeled
+data for testing in an episode on the SSv2-Small dataset.
+5 Extension to Semi-supervised Few-shot Action
+Recognition
+In this section, we demonstrate that the proposed HyRSM++
+can be extended to address the more challenging semi-
+supervised few-shot action recognition problem. Follow-
+ing LIM [113], we utilize two common datasets (Kinet-
+ics [8] and SSv2-Small [23]) to perform comparative exper-
+iments. These two datasets are subsets of Kinetics-400 [8]
+and Something-Something-v2 [23], respectively, and the un-
+labeled examples in our experiments are collected from the
+remaining videos of the same category as these subsets. To
+
+16
+Xiang Wang et al.
+Table 13 Comparison to state-of-the-art unsupervised few-shot action
+recognition approaches on UCF101, HMDB51, and Kinetics. ∗ indi-
+cates that the algorithm adopt the same 2D ResNet-50 backbone as
+HyRSM++.
+Method
+Supervision
+UCF101
+HMDB51 Kinetics
+MAML [19]
+Supervised
+-
+-
+54.2
+CMN [112]
+Supervised
+-
+-
+60.5
+TARN [5]
+Supervised
+-
+-
+66.6
+ProtoGAN [43]
+Supervised
+57.8
+34.7
+-
+ARN [105]
+Supervised
+66.3
+45.2
+63.7
+3DRotNet [37]
+Unsupervised
+39.4
+32.4
+27.5
+VCOP [96]
+Unsupervised
+32.9
+27.8
+26.5
+IIC [83]
+Unsupervised
+56.8
+34.7
+37.7
+Pace [87]
+Unsupervised
+25.6
+26.2
+22.4
+MemDPC [83]
+Unsupervised
+49.3
+30.3
+42.0
+CoCLR [26]
+Unsupervised
+52.0
+31.3
+37.6
+MetaUVFS∗ [64]
+Unsupervised
+66.1
+40.0
+50.9
+HyRSM++
+Unsupervised
+68.0
+41.0
+55.0
+64.0 
+64.7 
+66.3 
+66.5 
+68.0 
+66.5 
+66.4 
+50
+75
+100
+125
+150
+175
+200
+Accuracy (%)
+UCF101
+63
+36.1 
+36.1 
+36.9 
+39.2 
+41.0 
+38.9 
+39.7 
+50
+75
+100
+125
+150
+175
+200
+Accuracy (%)
+HMDB51
+51.2 
+52.0 
+52.0 
+54.7 
+55.0 
+54.9 
+54.3 
+50
+75
+100
+125
+150
+175
+200
+Accuracy (%)
+Kinetics
+49
+65
+67
+69
+55
+37
+39
+41
+43
+51
+53
+57
+35
+Fig. 14 Ablation study of different cluster numbers under 5-way 1-
+shot unsupervised few-shot settings.
+conduct the semi-supervised few-shot evaluation, we fol-
+low the mainstream distractor setting [30, 38, 113], where
+the unlabeled set contains other interference classes in each
+episodic task. This setting is more realistic and requires the
+model to be robust to the existence of noisy samples from
+other classes. In our experiments, we fixed the number of
+unlabeled videos in an episodic task to 100.
+Table 12 provides the comparison of our HyRSM++
+against state-of-the-art methods on the two standard semi-
+supervised few-shot benchmarks. We find that HyRSM++
+substantially surpasses the previous approaches, such as
+LIM [113]. Under the semi-supervised 5-way 1-shot sce-
+nario, HyRSM++ produces performance gains of 3.8% and
+2.5% on Kinetics and SSv2-Small than LIM with Inception-
+v3 backbone, respectively. In particular, when using the
+ResNet-50 backbone, our method is even superior to the
+multi-modal fusion method (i.e., LIM), which indicates that
+HyRSM++ enables more accurate pseudo-labels for unla-
+beled data and then can expand the support set to boost the
+classification accuracy of the query videos. In addition, com-
+pared to our supervised counterpart (i.e., HyRSM++ w/o un-
+labeled data), joining unlabeled data is beneficial to allevi-
+ating the data scarcity problem and promotes few-shot clas-
+sification accuracy. We can observe that when ResNet-50
+is adopted as the backbone, the performance of HyRSM++
+with unlabeled data is improved by 5.1% compared to that
+without unlabeled data under the 5-way 1-shot Kinetics
+evaluation.
+To further investigate the effect of unlabeled videos in
+an episode, we conduct comparative experiments with vary-
+ing numbers of unlabeled videos in Figure 12 and Figure 13.
+Experimental results show that as the number of unlabeled
+samples increases, the performance also increases gradually,
+indicating that the introduction of unlabeled data helps gen-
+eralize to unseen categories. Furthermore, we notice that the
+improvement in the 1-shot setting is more significant than
+that in the 5-shot, which shows that under the condition of
+low samples, unlabeled videos can improve the estimation
+of the distribution of new categories more effectively. Mean-
+while, as the amount of unlabeled data increases to a certain
+level, the performance starts to saturate slowly.
+6 Extension to Unsupervised Few-shot Action
+Recognition
+We also extend the proposed HyRSM++ to solve the
+challenging unsupervised few-shot action recognition task
+where labels for training videos are not available. Following
+previous work [38, 36], we adopt the idea of the ”cluster-
+ing first and then meta-learning” paradigm to construct few-
+shot tasks and exploit unlabeled data for training. Our ex-
+periments are based on unsupervised ResNet-50 initializa-
+tion [97], which is self-supervised pre-trained on Kinetics-
+400 [8] without accessing any label information. During the
+clustering process, we utilize the K-means clustering strat-
+egy for each dataset to obtain 150 clusters.
+As presented in Table 13, we compare HyRSM++ with
+current state-of-the-art methods on the UCF101, HMDB51
+and Kinetics datasets under the 5-way 1-shot setting. Note
+that HyRSM++ and MetaUVFS [64] use the same ResNet-
+50 structure as the feature extractor, and our HyRSM++
+shows better performance on each dataset. In particular, we
+observe that our method achieves 68.0% performance on
+the UCF101 dataset, a 1.9% improvement over MetaUVFS,
+and even surpasses the fully supervised ARN. The supe-
+rior performance of HyRSM++ reveals that our approach of
+leveraging relations within and cross videos and the flexible
+metric performs effectively in the low-shot regime. More-
+over, this phenomenon also demonstrates the potential of our
+method to learn a strongly robust few-shot model using only
+unlabeled videos, even though HyRSM++ is not specifically
+
+HyRSM++: Hybrid Relation Guided Temporal Set Matching for Few-shot Action Recognition
+17
+designed for the unsupervised few-shot action recognition
+task.
+In the experiments, one parameter involved in apply-
+ing HyRSM++ to the unsupervised few-shot setting is the
+number of clusters. In Figure 14, we display the perfor-
+mance comparison under different number of clusters. Re-
+sults show that when the number of clusters is 150, the per-
+formance reaches the peak value, which means that if the
+cluster number is too small, it may lead to under-clustering.
+If the number is too large, it may cause over-clustering, dam-
+aging the performance.
+7 Conclusion
+In this work, we have proposed a hybrid relation guided
+temporal set matching (HyRSM++) approach for few-shot
+action recognition. Firstly, we design a hybrid relation mod-
+ule to model the rich semantic relevance within one video
+and cross different videos in an episodic task to generate
+task-specific features. Secondly, built upon the representa-
+tive task-specific features, an efficient set matching metric is
+proposed to be resilient to misalignment and match videos
+accurately. During the matching process, a temporal coher-
+ence regularization is further imposed to exploit temporal
+order information. Furthermore, we extend HyRSM++ to
+solve the more challenging semi-supervised few-shot ac-
+tion recognition and unsupervised few-shot action recog-
+nition problems. Experimental results demonstrate that our
+HyRSM++ achieves the state-of-the-art performance on
+multiple standard benchmarks.
+Acknowledgements This work is supported by the National Natural
+Science Foundation of China under grant 61871435, Fundamental Re-
+search Funds for the Central Universities no.2019kfyXKJC024, 111
+Project on Computational Intelligence and Intelligent Control under
+Grant B18024, and Alibaba Group through Alibaba Research Intern
+Program.
+References
+1. Antoniou A, Storkey A (2019) Assume, augment and learn: Un-
+supervised few-shot meta-learning via random labels and data
+augmentation. arXiv preprint arXiv:190209884 4
+2. Bai Y, Ding H, Sun Y, Wang W (2018) Convolutional set match-
+ing for graph similarity. arXiv preprint arXiv:181010866 3
+3. Bai Y, Ding H, Gu K, Sun Y, Wang W (2020) Learning-based
+efficient graph similarity computation via multi-scale convolu-
+tional set matching. In: AAAI, vol 34, pp 3219–3226 3
+4. Berthelot D, Carlini N, Goodfellow I, Papernot N, Oliver A, Raf-
+fel CA (2019) Mixmatch: A holistic approach to semi-supervised
+learning. In: NeurIPS, vol 32 4
+5. Bishay M, Zoumpourlis G, Patras I (2019) TARN: temporal at-
+tentive relation network for few-shot and zero-shot action recog-
+nition. In: BMVC, p 154 4, 8, 16
+6. Caba Heilbron F, Escorcia V, Ghanem B, Carlos Niebles J (2015)
+Activitynet: A large-scale video benchmark for human activity
+understanding. In: CVPR, pp 961–970 1
+7. Cao K, Ji J, Cao Z, Chang CY, Niebles JC (2020) Few-shot video
+classification via temporal alignment. In: CVPR, pp 10618–
+10627 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15
+8. Carreira J, Zisserman A (2017) Quo vadis, action recognition? a
+new model and the kinetics dataset. In: CVPR, pp 6299–6308 1,
+8, 15, 16
+9. Chen Z, Fu Y, Zhang Y, Jiang YG, Xue X, Sigal L (2019) Multi-
+level semantic feature augmentation for one-shot learning. TIP
+28(9):4594–4605 3
+10. Cho K, Van Merri¨enboer B, Gulcehre C, Bahdanau D, Bougares
+F, Schwenk H, Bengio Y (2014) Learning phrase representa-
+tions using rnn encoder-decoder for statistical machine transla-
+tion. arXiv preprint arXiv:14061078 5
+11. Conners RW, Harlow CA (1980) A theoretical comparison of tex-
+ture algorithms. TPAMI (3):204–222 3, 6, 13
+12. Coskun H, Zia MZ, Tekin B, Bogo F, Navab N, Tombari F, Sawh-
+ney H (2021) Domain-specific priors and meta learning for few-
+shot first-person action recognition. TPAMI 4
+13. Damen D, Doughty H, Farinella G, Fidler S, Furnari A, Kaza-
+kos E, Moltisanti D, Munro J, Perrett T, Price W, et al. (2020)
+The epic-kitchens dataset: Collection, challenges and baselines.
+TPAMI (01):1–1 1, 9
+14. Damen D, Doughty H, Farinella GM, Furnari A, Kazakos E, Ma
+J, Moltisanti D, Munro J, Perrett T, Price W, et al. (2020) Rescal-
+ing egocentric vision. arXiv preprint arXiv:200613256 9
+15. Deng J, Dong W, Socher R, Li LJ, Li K, Fei-Fei L (2009) Ima-
+genet: A large-scale hierarchical image database. In: CVPR, pp
+248–255 9, 12
+16. Dubuisson MP, Jain AK (1994) A modified hausdorff distance
+for object matching. In: ICPR, IEEE, vol 1, pp 566–568 3, 6
+17. Fei-Fei L, Fergus R, Perona P (2006) One-shot learning of object
+categories. TPAMI 28(4):594–611 3
+18. Feichtenhofer C, Fan H, Malik J, He K (2019) Slowfast networks
+for video recognition. In: ICCV, pp 6202–6211 1
+19. Finn C, Abbeel P, Levine S (2017) Model-Agnostic Meta-
+Mearning for Fast Adaptation of Deep Networks. In: ICML 3, 8,
+10, 16
+20. Fu Y, Zhang L, Wang J, Fu Y, Jiang YG (2020) Depth guided
+adaptive meta-fusion network for few-shot video recognition. In:
+ACMMM, pp 1142–1151 4
+21. Gao Y (2003) Efficiently comparing face images using a modi-
+fied hausdorff distance. IEE Proceedings-Vision, Image and Sig-
+nal Processing 150(6):346–350 6
+22. Goroshin R, Bruna J, Tompson J, Eigen D, LeCun Y (2015)
+Unsupervised learning of spatiotemporally coherent metrics. In:
+ICCV, pp 4086–4093 3, 6, 13
+23. Goyal R, Ebrahimi Kahou S, Michalski V, Materzynska J, West-
+phal S, Kim H, Haenel V, Fruend I, Yianilos P, Mueller-Freitag
+M, et al. (2017) The” something something” video database
+for learning and evaluating visual common sense. In: ICCV, pp
+5842–5850 1, 8, 15
+24. Grauman K, Westbury A, Byrne E, Chavis Z, Furnari A, Girdhar
+R, Hamburger J, Jiang H, Liu M, Liu X, et al. (2022) Ego4d:
+Around the world in 3,000 hours of egocentric video. In: CVPR,
+pp 18995–19012 1
+25. Graves A, Mohamed Ar, Hinton G (2013) Speech recognition
+with deep recurrent neural networks. In: ICASSP, pp 6645–6649
+5
+26. Han T, Xie W, Zisserman A (2020) Self-supervised co-training
+for video representation learning. In: NeurIPS, vol 33, pp 5679–
+5690 16
+27. Haresh S, Kumar S, Coskun H, Syed SN, Konin A, Zia Z, Tran
+QH (2021) Learning by aligning videos in time. In: CVPR, pp
+
+18
+Xiang Wang et al.
+5548–5558 3, 13
+28. He K, Zhang X, Ren S, Sun J (2016) Deep residual learning for
+image recognition. In: CVPR, pp 770–778 9
+29. Hou R, Chang H, Ma B, Shan S, Chen X (2019) Cross attention
+network for few-shot classification. In: NeurIPS, pp 4003–4014
+3, 5
+30. Hsu K, Levine S, Finn C (2018) Unsupervised learning via meta-
+learning. In: ICLR 4, 15, 16
+31. Huang H, Zhang J, Zhang J, Wu Q, Xu C (2021) Ptn: A pois-
+son transfer network for semi-supervised few-shot learning. In:
+AAAI, vol 35, pp 1602–1609 4
+32. Huang K, Geng J, Jiang W, Deng X, Xu Z (2021) Pseudo-
+loss confidence metric for semi-supervised few-shot learning. In:
+ICCV, pp 8671–8680 4
+33. Huang Y, Yang L, Sato Y (2022) Compound prototype matching
+for few-shot action recognition. In: ECCV, Springer, pp 351–368
+2, 4, 8, 10
+34. Huttenlocher DP, Klanderman GA, Rucklidge WJ (1993) Com-
+paring images using the hausdorff distance. TPAMI 15(9):850–
+863 3, 6
+35. Jesorsky O, Kirchberg KJ, Frischholz RW (2001) Robust face
+detection using the hausdorff distance. In: AVBPA, Springer, pp
+90–95 3
+36. Ji Z, Zou X, Huang T, Wu S (2019) Unsupervised few-shot learn-
+ing via self-supervised training. arXiv preprint arXiv:191212178
+4, 7, 16
+37. Jing L, Yang X, Liu J, Tian Y (2018) Self-supervised spa-
+tiotemporal feature learning via video rotation prediction. arXiv
+preprint arXiv:181111387 16
+38. Khodadadeh S, Boloni L, Shah M (2019) Unsupervised meta-
+learning for few-shot image classification. In: NeurIPS, vol 32 4,
+7, 16
+39. Kingma DP, Ba J (2014) Adam: A method for stochastic opti-
+mization. arXiv preprint arXiv:14126980 9
+40. Kliper-Gross O, Hassner T, Wolf L (2011) One shot similarity
+metric learning for action recognition. In: SIMBAD, Springer,
+pp 31–45 4
+41. Koizumi Y, Yatabe K, Delcroix M, Masuyama Y, Takeuchi D
+(2020) Speech enhancement using self-adaptation and multi-
+head self-attention. In: ICASSP, pp 181–185 12
+42. Kuehne H, Serre T, Jhuang H, Garrote E, Poggio T, Serre T
+(2011) HMDB: A large video database for human motion recog-
+nition. In: ICCV, DOI 10.1109/ICCV.2011.6126543 9
+43. Kumar Dwivedi S, Gupta V, Mitra R, Ahmed S, Jain A (2019)
+Protogan: Towards few shot learning for action recognition. In:
+ICCVW, pp 0–0 16
+44. Lazarou M, Stathaki T, Avrithis Y (2021) Iterative label clean-
+ing for transductive and semi-supervised few-shot learning. In:
+ICCV, pp 8751–8760 4
+45. Lee DH, et al. (2013) Pseudo-label: The simple and efficient
+semi-supervised learning method for deep neural networks. In:
+ICMLW, vol 3, p 896 7
+46. Li A, Luo T, Xiang T, Huang W, Wang L (2019) Few-shot learn-
+ing with global class representations. In: ICCV, pp 9715–9724
+7
+47. Li H, Eigen D, Dodge S, Zeiler M, Wang X (2019) Finding task-
+relevant features for few-shot learning by category traversal. In:
+CVPR, pp 1–10 2, 3, 5, 13
+48. Li S, Liu H, Qian R, Li Y, See J, Fei M, Yu X, Lin W (2021)
+Ttan: Two-stage temporal alignment network for few-shot action
+recognition. arXiv preprint arXiv:210704782 8, 9, 10
+49. Li X, Sun Q, Liu Y, Zhou Q, Zheng S, Chua TS, Schiele B (2019)
+Learning to self-train for semi-supervised few-shot classification.
+In: NeurIPS, vol 32 4, 7
+50. Li Z, Zhou F, Chen F, Li H (2017) Meta-sgd: Learning to learn
+quickly for few-shot learning. arXiv preprint arXiv:170709835 3
+51. Lin J, Gan C, Wang K, Han S (2020) Tsm: Temporal shift module
+for efficient and scalable video understanding on edge devices.
+TPAMI 1
+52. Liu L, Shao L, Li X, Lu K (2015) Learning spatio-temporal rep-
+resentations for action recognition: A genetic programming ap-
+proach. TCYB 46(1):158–170 1
+53. Liu X, Gao J, He X, Deng L, Duh K, Wang Yy (2015) Repre-
+sentation learning using multi-task deep neural networks for se-
+mantic classification and information retrieval. In: NAACL, pp
+912–921 2
+54. Liu Y, Zhang X, Zhang S, He X (2020) Part-aware prototype
+network for few-shot semantic segmentation. In: ECCV, pp 142–
+158 7
+55. Lu J, Gong P, Ye J, Zhang C (2020) Learning from very few
+samples: A survey. arXiv preprint arXiv:200902653 3
+56. Lu J, Jin S, Liang J, Zhang C (2020) Robust few-shot learning
+for user-provided data. TNNLS 32(4):1433–1447 3
+57. Lu P, Bai T, Langlais P (2019) Sc-lstm: Learning task-specific
+representations in multi-task learning for sequence labeling. In:
+NAACL, pp 2396–2406 2
+58. Mitra A, Biswas S, Bhattacharyya C (2016) Bayesian modeling
+of temporal coherence in videos for entity discovery and summa-
+rization. TPAMI 39(3):430–443 3
+59. Mobahi H, Collobert R, Weston J (2009) Deep learning from
+temporal coherence in video. In: ICML, pp 737–744 3, 6, 13
+60. Mohanaiah P, Sathyanarayana P, GuruKumar L (2013) Image
+texture feature extraction using glcm approach. IJSRP 3(5):1–5
+3
+61. M¨uller M (2007) Dynamic time warping. Information Retrieval
+for Music and Motion pp 69–84 11
+62. Nguyen KD, Tran QH, Nguyen K, Hua BS, Nguyen R (2022) In-
+ductive and transductive few-shot video classification via appear-
+ance and temporal alignments. In: ECCV, Springer, pp 471–487
+2, 4, 8, 10
+63. Nishiyama M, Yuasa M, Shibata T, Wakasugi T, Kawahara T,
+Yamaguchi O (2007) Recognizing faces of moving people by hi-
+erarchical image-set matching. In: CVPR, pp 1–8 3
+64. Patravali J, Mittal G, Yu Y, Li F, Chen M (2021) Unsupervised
+few-shot action recognition via action-appearance aligned meta-
+adaptation. In: ICCV, pp 8484–8494 4, 16
+65. Peng B, Lei J, Fu H, Zhang C, Chua TS, Li X (2018) Unsuper-
+vised video action clustering via motion-scene interaction con-
+straint. TCSVT 30(1):131–144 1
+66. Peng M, Zhang Q, Xing X, Gui T, Fu J, Huang X (2019) Learning
+task-specific representation for novel words in sequence labeling.
+In: IJCAI 2
+67. Perez L, Wang J (2017) The effectiveness of data augmenta-
+tion in image classification using deep learning. arXiv preprint
+arXiv:171204621 3
+68. Perrett T, Masullo A, Burghardt T, Mirmehdi M, Damen D
+(2021) Temporal-relational crosstransformers for few-shot action
+recognition. In: CVPR, pp 475–484 2, 4, 6, 7, 8, 9, 10, 11, 12, 13
+69. Qin T, Li W, Shi Y, Gao Y (2020) Diversity helps: Unsupervised
+few-shot learning via distribution shift-based data augmentation.
+arXiv preprint arXiv:200405805 4
+70. Ratner AJ, Ehrenberg HR, Hussain Z, Dunnmon J, R´e C (2017)
+Learning to compose domain-specific transformations for data
+augmentation. In: NeurIPS, NIH Public Access, vol 30, p 3239 3
+71. Ren M, Triantafillou E, Ravi S, Snell J, Swersky K, Tenenbaum
+JB, Larochelle H, Zemel RS (2018) Meta-learning for semi-
+supervised few-shot classification. In: ICLR 3, 7
+72. Rezaei M, Fr¨anti P (2016) Set matching measures for external
+cluster validity. TKDE 28(8):2173–2186 3
+73. Saito Y, Nakamura T, Hachiya H, Fukumizu K (2020) Exchange-
+able deep neural networks for set-to-set matching and learning.
+In: ECCV, Springer, pp 626–646 3
+
+HyRSM++: Hybrid Relation Guided Temporal Set Matching for Few-shot Action Recognition
+19
+74. Santoro A, Bartunov S, Botvinick M, Wierstra D, Lillicrap T
+(2016) Meta-learning with memory-augmented neural networks.
+In: ICML, PMLR, pp 1842–1850 3
+75. Selvaraju RR, Cogswell M, Das A, Vedantam R, Parikh D, Batra
+D (2017) Grad-cam: Visual explanations from deep networks via
+gradient-based localization. In: ICCV, pp 618–626 14
+76. Snell J, Swersky K, Zemel R (2017) Prototypical networks for
+few-shot learning. In: NeurIPS, vol 30, pp 4077–4087 3, 9, 11
+77. Sohn K, Berthelot D, Carlini N, Zhang Z, Zhang H, Raffel CA,
+Cubuk ED, Kurakin A, Li CL (2020) Fixmatch: Simplifying
+semi-supervised learning with consistency and confidence. In:
+NeurIPS, vol 33, pp 596–608 7
+78. Soomro K, Zamir AR, Shah M (2012) Ucf101: A dataset of 101
+human actions classes from videos in the wild. arXiv preprint
+arXiv:12120402 9
+79. Sudha N, et al. (2007) Robust hausdorff distance measure for
+face recognition. Pattern Recognition 40(2):431–442 3, 6
+80. Sung F, Yang Y, Zhang L, Xiang T, Torr PH, Hospedales TM
+(2018) Learning to compare: Relation network for few-shot
+learning. In: CVPR, pp 1199–1208 3
+81. Szegedy C, Vanhoucke V, Ioffe S, Shlens J, Wojna Z (2016)
+Rethinking the inception architecture for computer vision. In:
+CVPR, pp 2818–2826 11
+82. Takacs B (1998) Comparing face images using the modified
+hausdorff distance. Pattern recognition 31(12):1873–1881 3, 6
+83. Tao L, Wang X, Yamasaki T (2020) Self-supervised video rep-
+resentation learning using inter-intra contrastive framework. In:
+ACMMM, pp 2193–2201 16
+84. Thatipelli A, Narayan S, Khan S, Anwer RM, Khan FS, Ghanem
+B (2022) Spatio-temporal relation modeling for few-shot action
+recognition. In: CVPR 4, 8, 10, 11, 12, 13
+85. Vaswani A, Shazeer N, Parmar N, Uszkoreit J, Jones L, Gomez
+AN, Kaiser Ł, Polosukhin I (2017) Attention is all you need. In:
+NeurIPS, pp 5998–6008 5
+86. Vinyals O, Blundell C, Lillicrap T, Kavukcuoglu K, Wierstra D
+(2016) Matching Networks for One Shot Learning. In: NeurIPS,
+arXiv:1606.04080v2 2, 3, 4, 8, 10
+87. Wang J, Jiao J, Liu YH (2020) Self-supervised video representa-
+tion learning by pace prediction. In: ECCV, Springer, pp 504–521
+16
+88. Wang L, Xiong Y, Wang Z, Qiao Y, Lin D, Tang X, Van Gool
+L (2018) Temporal segment networks for action recognition in
+videos. TPAMI 41(11):2740–2755 1, 4
+89. Wang X, Zhang S, Qing Z, Shao Y, Gao C, Sang N (2021) Self-
+supervised learning for semi-supervised temporal action pro-
+posal. In: CVPR, pp 1905–1914 1
+90. Wang X, Zhang S, Qing Z, Shao Y, Zuo Z, Gao C, Sang N (2021)
+Oadtr: Online action detection with transformers. ICCV 12
+91. Wang X, Zhang S, Qing Z, Tang M, Zuo Z, Gao C, Jin R, Sang N
+(2022) Hybrid relation guided set matching for few-shot action
+recognition. In: CVPR 3, 9
+92. Weng R, Lu J, Hu J, Yang G, Tan YP (2013) Robust feature set
+matching for partial face recognition. In: ICCV, pp 601–608 3
+93. Weng R, Lu J, Tan YP (2016) Robust point set matching for par-
+tial face recognition. TIP 25(3):1163–1176 3
+94. Wu J, Zhang T, Zhang Z, Wu F, Zhang Y (2022) Motion-
+modulated temporal fragment alignment network for few-shot
+action recognition. In: CVPR, pp 9151–9160 2, 4, 8, 10
+95. Xian Y, Korbar B, Douze M, Torresani L, Schiele B, Akata Z
+(2021) Generalized few-shot video classification with video re-
+trieval and feature generation. TPAMI 4
+96. Xu D, Xiao J, Zhao Z, Shao J, Xie D, Zhuang Y (2019) Self-
+supervised spatiotemporal learning via video clip order predic-
+tion. In: CVPR, pp 10334–10343 16
+97. Xu J, Wang X (2021) Rethinking self-supervised correspondence
+learning: A video frame-level similarity perspective. In: ICCV,
+pp 10075–10085 11, 13, 16
+98. Ye HJ, Hu H, Zhan DC, Sha F (2020) Few-shot learning via
+embedding adaptation with set-to-set functions. In: CVPR, pp
+8808–8817 3
+99. Ye HJ, Ming L, Zhan DC, Chao WL (2022) Few-shot learning
+with a strong teacher. TPAMI 3
+100. Yoo S, Bahng H, Chung S, Lee J, Chang J, Choo J (2019) Col-
+oring with limited data: Few-shot colorization via memory aug-
+mented networks. In: CVPR, pp 11283–11292 3
+101. Yoon SW, Seo J, Moon J (2019) Tapnet: Neural network aug-
+mented with task-adaptive projection for few-shot learning. In:
+ICML, pp 7115–7123 2, 3, 13
+102. Yu CB, Qin HF, Cui YZ, Hu XQ (2009) Finger-vein image recog-
+nition combining modified hausdorff distance with minutiae fea-
+ture matching. Interdisciplinary Sciences: Computational Life
+Sciences 1(4):280–289 6
+103. Yu Z, Chen L, Cheng Z, Luo J (2020) Transmatch: A transfer-
+learning scheme for semi-supervised few-shot learning. In:
+CVPR, pp 12856–12864 4
+104. Zhang H, Cisse M, Dauphin YN, Lopez-Paz D (2018) mixup:
+Beyond empirical risk minimization. In: ICLR 7
+105. Zhang H, Zhang L, Qi X, Li H, Torr PH, Koniusz P (2020) Few-
+shot action recognition with permutation-invariant attention. In:
+ECCV, Springer, pp 525–542 1, 4, 8, 9, 10, 16
+106. Zhang S, Zhou J, He X (2021) Learning implicit temporal align-
+ment for few-shot video classification. In: IJCAI 2, 4, 8, 9, 10
+107. Zhao C, Shi W, Deng Y (2005) A new hausdorff distance for
+image matching. Pattern Recognition Letters 26(5):581–586 3
+108. Zheng S, Chen S, Jin Q (2022) Few-shot action recognition
+with hierarchical matching and contrastive learning. In: ECCV,
+Springer, pp 297–313 2, 4, 8, 10
+109. Zhou B, Andonian A, Oliva A, Torralba A (2018) Temporal rela-
+tional reasoning in videos. In: ECCV, pp 803–818 8
+110. Zhou ZH, Li M (2005) Tri-training: Exploiting unlabeled data
+using three classifiers. TKDE 17(11):1529–1541 7
+111. Zhou ZQ, Wang B (2009) A modified hausdorff distance using
+edge gradient for robust object matching. In: IASP, IEEE, pp
+250–254 6
+112. Zhu L, Yang Y (2018) Compound memory networks for few-shot
+video classification. In: ECCV, pp 751–766 1, 2, 4, 8, 9, 10, 15,
+16
+113. Zhu L, Yang Y (2020) Label independent memory for semi-
+supervised few-shot video classification. TPAMI 44(1):273–285,
+DOI 10.1109/TPAMI.2020.3007511 4, 7, 8, 10, 12, 15, 16
+

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+Safer Together: Machine Learning Models Trained on Shared Accident
+Datasets Predict Construction Injuries Better than Company-Specific
+Models
+Submitted to Automation in Construction
+Antoine J.-P. Tixier1a, Matthew R. Hallowella,b
+aSafetyAI R&D
+bUniversity of Colorado at Boulder
+Abstract
+Highlights
+• 9 companies from 3 domains (construction, electric T&D, oil & gas) shared their accident datasets.
+• Machine learning models were trained to predict safety outcomes from fundamental attributes.
+• Models trained on all datasets (full generic models) outperformed the company-specific models
+in 82% of the company-domain-outcome combinations, with large gains in F1 score (+4.4 on
+average and up to +15.3).
+• On average, generic models predicted 2.26 categories more than specific models (up to 7), making
+for more useful forecasts in practice.
+• Per-domain generic models were not always better than full generic models.
+• Combining generic and specific models (data quantity and relevance) was often very beneficial.
+• Generic models give companies devoid of accident datasets access to safety predictions.
+• Generic models address safety cross-organizational learning and dissemination in construction.
+In this study, we capitalized on a collective dataset repository of 57k accidents from 9 companies be-
+longing to 3 domains and tested whether models trained on multiple datasets (generic models) predicted
+safety outcomes better than the company-specific models. We experimented with full generic models
+(trained on all data), per-domain generic models (construction, electric T&D, oil & gas), and with en-
+sembles of generic and specific models. Results are very positive, with generic models outperforming the
+company-specific models in most cases while also generating finer-grained, hence more useful, forecasts.
+Successful generic models remove the needs for training company-specific models, saving a lot of time
+and resources, and give small companies, whose accident datasets are too limited to train their own mod-
+els, access to safety outcome predictions. It may still however be advantageous to train specific models
+to get an extra boost in performance through ensembling with the generic models. Overall, by learning
+lessons from a pool of datasets whose accumulated experience far exceeds that of any single company,
+and making these lessons easily accessible in the form of simple forecasts, generic models tackle the
+holy grail of safety cross-organizational learning and dissemination in the construction industry.
+Keywords: construction safety, artificial intelligence, supervised learning, injury prediction,
+transfer learning, data sharing, collective intelligence
+1antoine.tixier@safetyfunction.com
+arXiv:2301.03567v1  [cs.LG]  9 Jan 2023
+
+1. Introduction
+The SafetyAI council is a community of large organizations from the construction, oil &
+gas, and electric Transmission and Delivery (T&D) domains, that share their safety-related data
+with the SafetyAI Research and Development (R&D) team.
+Before exploiting the data, the R&D team is in charge of standardizing the datasets received
+by each company, which is crucial, as each one features different variables and different category
+names for each variable. Standardization makes sure that all datasets are based on the same
+taxonomy, i.e., speak the same language.
+The SafetyAI community dataset, comprising close to a million events including near misses,
+observations, good catches, etc., is only accessible to the R&D team, a neutral party, which guar-
+antees that it is impossible for companies to see each other’s data, and that the output of all the
+R&D conducted on the collective dataset is made available to the entire community. This is of
+paramount importance, in a very competitive environment.
+In this study, we started by extracting attributes from accident reports. We briefly introduce
+the attribute framework in what follows.
+1.1. Attribute-based framework
+Attributes are basic descriptors of construction work that are observable before accident
+occurrence, and cover means, methods, and environmental conditions [1, 2]. One advantage of
+the attribute-based framework over modeling at the task or work package level is that attributes
+are fundamental and universal. That is, any situation from any site around the world, in any
+industry sector, can be characterized by a set of attributes. Attributes can be recorded on-the-fly
+on site, or can be extracted offline from various mediums such as photos and text reports. For
+instance, four attributes can be extracted from the narrative worker tripped on a cable
+when carrying a 2x4 to his truck: (1) cable, (2) object on the floor, (3) lumber, and
+(4) light vehicle.
+Narratives are particularly well-suited if the goal is to use attributes for predictive modeling.
+Indeed, in incident report databases, narratives are often paired with outcomes such as accident
+type, injury severity, body part impacted, etc. Attributes also completely anonymize narratives,
+which is especially desirable when considering a pool of datasets aggregated from different
+companies. For any given event, everything that remains is a set of attributes and a set of
+standardized safety outcomes.
+However, manually extracting attributes from large amounts of text reports is very costly in
+terms of human resources and pose inter-annotator agreement issues. To solve this problem, we
+developed and validated a Natural Language Processing (NLP) tool based on rules and lexicons
+[3]. We later proved that using the attributes extracted by the tool to predict safety outcomes
+was effective and valid [4, 5]. We also used the attributes extracted by the tool for unsupervised
+learning applications, such as clustering and visualization [6], and risk modeling and simulation
+[7].
+1.2. Differences with our previous research and objective of the current study
+In our original study [4], we provided a proof for the concept of predicting safety outcomes
+from attributes, both extracted with the NLP tool. Then, in [5], we showed that attributes were
+still highly predictive when the safety outcomes were given by independent human annotations,
+which definitely validated the approach. We also used a much larger dataset than in the orig-
+inal study, two new supervised learning algorithms, model stacking, a healthier experimental
+setup with more appropriate performance metrics, and we analyzed per-category attribute im-
+portance scores. We also showed that unlike what we had concluded in [4], injury severity was
+predictable from attributes.
+2
+
+In the present research, we interested ourselves with a new, completely different problem.
+We had access to a pool of accident datasets coming from 9 companies, and our goal was to:
+“Test whether predictive models trained on a generic dataset (i.e., aggregated from the datasets
+of multiple companies) outperformed the models trained on the specific dataset of each com-
+pany.”
+More precisely, we experimented with two types of generic models:
+• Full generic model: one model trained on the datasets of all companies.
+• Per-domain generic models: one model per industry sector, trained only on the datasets
+of the companies involved in that sector (or the parts thereof, as some companies belong
+to multiple domains).
+The potential advantages of generic models are numerous:
+1. Usually with machine learning, the more data, the better, so generic models are expected
+to bring improvements in predictive skill compared to the company-specific models. This
+is not guaranteed however, as one important question is whether (1) more data (generic
+datasets) or (2) more relevant data (specific datasets) is better.
+2. By being trained on larger datasets, the generic models learn to predict a greater variety
+of outcome categories than the specific models, making for more useful forecasts.
+3. Successful generic models would remove the needs for training specific models for each
+company, saving a lot of time and resources.
+4. Alternatively, if company-specific models are already available, combining them with the
+generic models may provide an extra boost in performance.
+5. Last but not least, successful generic models would give small companies -whose accident
+datasets are too limited to train their own specific models- access to high quality safety
+outcome forecasts.
+From a high level, generic models tackle the holy grail of safety cross-organizational learn-
+ing and dissemination in the construction industry. Indeed, generic models (1) learn lessons
+from a pool of datasets whose quantity and diversity2 of accumulated experience far exceeds
+that of any single company, and (2) disseminate these lessons as forecasts, which are clear, di-
+rect, and easily accessible information, via, e.g., a user interface (desktop or mobile) or API
+taking attributes as input and returning probabilities for each category of each outcome.
+Moreover, one should note that in the pool, the individual biases of each dataset, due to
+specific annotators, reporting practices and policies, etc., tend to average out. Consequently, the
+lessons learned by the supervised learning algorithms on the generic datasets are more objective
+and broadly applicable than that learned on the specific datasets.
+2. Background
+The needs to share standardized incident data at the industry level to enable collaborative
+learning have long been recognized in aviation and transportation [8]. Some examples include
+2Diversity of situations, means and methods, environmental conditions, geographical areas...
+3
+
+the NASA-managed Aviation Safety Reporting System (ASRS) database, created in 1976 and
+featuring over a million incidents, or the European Coordination Center for Accident and In-
+cident Reporting Systems (ECCAIRS) database, started in 2004. Such collective repositories
+also exist in the chemical industry, with the Major Accident Reporting System (eMARS) of the
+European Commission, launched in 1982, and the Process Safety Incident Database (PSID) of
+the Center for Chemical Process Safety [9].
+However, the construction industry still lacks comparable initiatives. The needs for data
+storage and access infrastructures for construction safety did start to receive some attention
+recently [10, 11], but most efforts placed themselves at the company or project level. Cross-
+organizational safety data collection is still rare in practice [12, 13]. This is a major issue,
+as collaborative machine learning at the industry level is not possible until a common pool of
+standardized datasets has been put together. This provided the motivation for us to create the
+SafetyAI council in 2020.
+One should note that some consortiums already exist, such as the INGAA Foundation, the
+Edison Electric Institute (EEI), the Construction Safety Research Alliance (CSRA), or the Na-
+tional Safety Council (NSC), but their activities do not revolve around systematic large-scale
+accident data collection and analysis. These initiatives rather involve working groups, com-
+munities of practice, qualitative analyses, and conferences, towards building communications,
+policies, best practices, business intelligence, safety culture and leadership, training material,
+etc. In other words, they are based on “soft” methods for knowledge sharing and collabo-
+rative learning at the human level. They do not primarily conduct “hard” scientific research
+and software development, and do not pool accident datasets for AI applications and automatic
+large-scale learning and dissemination.
+3. Data Description
+As already explained, as part of the SafetyAI initiative, we had access to a pool of safety
+datasets coming from nine large companies from the construction, oil & gas, and electric Trans-
+mission and Delivery (T&D) domains. One company, Company73, also had about 600 corporate
+services (office) events for the severity outcome. We kept these cases as training data for the
+full generic model but did not train a specific model on them.
+Member companies conduct work mostly in North America, and rely on their own teams as
+well as contractors. The collective dataset covers the period 2000 to 2022, with a distribution
+biased towards the last decade and especially more recent years.
+While the entire pool comprises almost a million events including near misses and observa-
+tions, we focused on accident cases only in this effort. As can be seen in Table 1, the sizes of
+the individual datasets ranged from 2k to 20k cases, with an average of 6k per company. There
+were 57262 accident cases in total, recorded over tens of millions of work hours.
+We considered the same outcomes as in [5]: injury severity, body part impacted, injury
+type, and accident type. The columns corresponding to each outcome were selected from the
+company datasets and normalized to use a common, standard set of categories, shown in Table
+2. Not all outcomes were available for every event of every company. From the narrative of
+each report, we extracted with the NLP tool [3] the original set of 80 attributes [3, 5], plus 11
+new items (see Table A.7). We also used the tool to extract a fifth outcome, energy source, that
+was not available in the company datasets.
+3Company names have been anonymized.
+4
+
+Comp.1
+Comp.2
+Comp.3
+Comp.4
+Comp.5
+Comp.6
+Comp.7
+Comp.8
+Comp.9
+Domains
+Constr.,
+elec.
+Oilgas
+Constr.,
+oilgas
+Elec.
+Constr.
+Constr.,
+elec.
+Elec.,
+oilgas,
+corp.
+Oilgas
+Elec.
+Regions
+Canada
+California
+NAM
+NAM
+NAM
+NAM
+NAM,
+Mexico
+World⋆
+Southeast
+USA
+n
+4481
+1965
+4072
+5321
+7245
+4310
+8345
+19298
+2225
+Table 1: Company overview. NAM: North America (Canada + USA). Constr.: construction. Elec: electric T&D.
+Oilgas: oil & gas. ⋆Including ships and rigs. Corp: corporate.
+Injury Severity
+Body Part
+Injury Type
+Accident Type
+Energy Source
+first aid
+38994
+hand
+15782
+cut
+14086
+handling
+6379
+motion
+33958
+report-only
+6993
+head
+10296
+strain
+10069
+fall
+5374
+gravity
+15904
+lost time
+5319
+leg
+6550
+contusion
+8558
+exposure
+3986
+chemical
+2411
+medical
+4913
+arm
+5943
+foreign body
+3348
+struck
+3834
+biological
+2044
+recordable
+1043
+trunk
+5375
+pinch
+1756
+contact
+2269
+thermal
+1691
+foot
+4632
+fracture
+1681
+caught
+1758
+mechanical
+611
+multiple/entire
+942
+burn
+1454
+overexertion
+1523
+pressure
+296
+irritation
+1222
+equipment
+1449
+electricity
+181
+pain
+1194
+PPE
+949
+radiation
+166
+exhaustion
+1054
+transitioning
+578
+bite
+710
+error
+425
+Table 2: Outcome category counts, across all companies and domains. PPE: personal protective equipment.
+4. Experimental Setup
+4.1. Splits
+Train, validation and test splits were created for each of the 51 company-domain-outcome
+combinations for which at least 2 categories with more than 100 observations each were avail-
+able (shown in Table 4), by randomly sampling without replacement 64%, 16%, and 20% of
+cases, respectively. The counts summed over companies are shown in Table 3. Note that the
+proportions we used in our previous work [5] were 81%, 9% and 10%, but in the present re-
+search, we decided to reserve more observations for the validation and test sets to make them
+more representative of the training sets, in order to increase the stability and validity of hyper-
+parameter tuning and evaluation4.
+A specific model was trained on each of the 51 company-domain-outcome combinations for
+which sufficient data were available, except for that one combination involving the corporate
+cases, making for a total of 50 specific models.
+For a given domain and a given outcome, the splits of the per-domain generic model were
+obtained by combining, across all companies, the splits corresponding to that domain and that
+outcome. In total, there was one per-domain generic model for each domain and for each
+outcome, hence a total of 3 × 5 = 15 per-domain generic models.
+For a given outcome, the splits of the full generic model were obtained by combining, across
+all companies and across all domains, the splits corresponding to that outcome. In total, there
+was one full generic model for each outcome, hence a total of 5 full generic models.
+For each of the aforementioned cases, we tried 3 different algorithms, as will be explained
+in subsection 4.3. Hence, a total of (15 + 5) × 3 = 60 generic models were trained.
+4Increasing the sizes of the validation and test sets was a good alternative to k-fold cross-validation, which would
+have taken too much time.
+5
+
+# Companies
+Train
+Val
+Test
+Severity
+Construction
+4
+9980
+2494
+3119
+Electric T&D
+4
+6672
+1669
+2085
+Oil & Gas
+4
+18381
+4595
+5744
+Corporate
+1
+418
+105
+131
+Full
+9
+35451
+8863
+11079
+Body Part
+Construction
+4
+8209
+2052
+2565
+Electric T&D
+4
+6036
+1508
+1885
+Oil & Gas
+3
+15788
+3947
+4933
+Full
+9
+30033
+7507
+9383
+Injury Type
+Construction
+4
+6267
+1566
+1958
+Electric T&D
+4
+4764
+1191
+1489
+Oil & Gas
+3
+14960
+3740
+4675
+Full
+9
+25991
+6497
+8122
+Acc. Type
+Construction
+2
+2740
+685
+856
+Electric T&D
+2
+1600
+400
+500
+Oil & Gas
+3
+2910
+728
+910
+Full
+6
+7250
+1813
+2266
+En. Source
+Construction
+4
+4875
+1218
+1524
+Electric T&D
+3
+2637
+660
+825
+Oil & Gas
+2
+2600
+650
+813
+Full
+8
+10112
+2528
+3162
+Table 3: Split counts for each domain-outcome combination, summed over companies. For # Companies, full ̸=
+total as some companies belong to multiple domains (see Tables 1 and 4).
+Construction
+Electric T&D
+Oil & Gas
+Corp.
+Comp.
+S
+B
+IT
+AT
+E
+S
+B
+IT
+AT
+E
+S
+B
+IT
+AT
+E
+S
+1
+x
+x
+x
+x
+2
+x
+x
+x
+3
+x
+x
+x
+x
+x
+x
+x
+4
+x
+x
+x
+x
+x
+5
+x
+x
+x
+x
+x
+6
+x
+x
+x
+x
+x
+x
+x
+x
+7
+x
+x
+x
+x
+x
+x
+x
+x
+x
+8
+x
+x
+x
+x
+x
+9
+x
+x
+x
+x
+x
+Table 4: The 51 company-domain-outcome combinations associated with at least 2 categories with more than 100
+observations each. S: severity, B: body part, IT: injury type, AT: accident type, E: energy source. Corp.: corportate.
+4.2. Class imbalance
+To address the problem of class imbalance, weights inversely proportional to category
+counts in the training set were computed with the formula max(counts)/counts, like in
+[5]. During training, these weights forced the models to pay more attention to the cases from
+the minority categories. Per-category counts with training weights can be found in Tables B.8
+and B.9 for the 15 domain-outcome combinations.
+4.3. Supervised learning algorithms
+Like in [5], we relied on three popular machine learning models: Random Forest (RF) [14],
+eXtreme Gradient Boosting (XGBoost or XGB) [15], and linear Support Vector Machine (SVM)
+[16]. More precisely, we used the Python’s scikit-learn implementations of Random
+Forest5 and linear SVM6, while, for XGBoost, we used the original Python library7 and in
+5https://scikit-learn.org/stable/modules/generated/sklearn.ensemble.RandomForestClassifier.html
+6https://scikit-learn.org/stable/modules/generated/sklearn.svm.LinearSVC.html
+7https://xgboost.readthedocs.io/en/latest/python/python api.html#module-xgboost.sklearn
+6
+
+particular the GPU-accelerated implementation of the “fast histogram” algorithm (gpu hist)
+as the tree method8.
+For theoretical details about each algorithm, we refer the reader to our paper [5], publicly
+available9.
+4.4. Hyperparameter optimization
+We tuned the models by performing grid searches on the validation sets. Details about the
+parameters searched are available in Appendix C. The final models were trained on the union
+of the training and validation sets with the best parameter values. Both the specific models and
+the generic models were tested on the test sets of the specific models, to ensure fair comparison.
+As already explained, there were 50 such test sets, one for each company-domain-outcome
+combination.
+4.5. Transfer learning by stacking generic and specific models
+As was mentioned in the introduction, one important question is the extent to which (1) more
+data (generic datasets) or (2) more relevant data (specific datasets) is better. In what follows,
+we explore a way to move past this binary choice and have a tradeoff between quantity and
+relevance.
+Inspired by transfer learning, which is very successful in computer vision [17] and NLP
+[18, 19, 20, 21], we experimented with combining the predictions of the generic and specific
+models via an ensemble model.
+Very briefly, in AI, transfer learning refers to a two-step process. First, a model is trained
+at solving a general task on large amounts of data. This phase is called the pretraining phase,
+as it allows the model to acquire generic knowledge (e.g., in NLP, reading and writing), that
+is applicable to a great variety of situations downstream. Second, the pretrained model is fine-
+tuned on a specific task of interest, often associated with a much smaller dataset (e.g. in NLP,
+summarization, classification, question answering, paraphrase detection, etc.).
+In our case, the generic and the specific models have to perform the same task, i.e., predict-
+ing a given safety outcome10, and there is no pretraining phase per se, in that the generic and the
+specific models are two different models. However, our approach is similar in spirit to transfer
+learning, as our goal is to capitalize on generic knowledge gained from large amounts of data to
+improve performance on a specific task associated with a smaller dataset.
+More precisely, for each company-domain-outcome combination, we trained a meta-model
+taking as input the weighted elementwise sum of the probabilistic forecasts of the best generic
+and specific models11. We used a simple logistic regression12 as our meta-model, with the C
+parameter fixed and equal to 0.2, like in [5]. We grid searched the validation set to find the best
+values of coefficients a and b where:
+inputensemble = a × outputgeneric + b × outputspecific
+(1)
+Besides performance considerations, using tunable weights improves interpretability, by
+providing information regarding which of the generic model or the specific model makes the
+most important contribution to predictive skill.
+8https://xgboost.readthedocs.io/en/latest/gpu/
+9https://arxiv.org/pdf/1908.05972.pdf
+10The generic model has to perform a more difficult version of the task, though (more categories to predict).
+11The entries of the specific model vector for the categories that it did not predict were set to zero.
+12https://scikit-learn.org/stable/modules/generated/sklearn.linear model.LogisticRegression.html
+7
+
+We tried values from 0.1 to 1 with 0.1 steps, holding the other parameter equal to 1, and
+conversely. That is, the following 19 pairs: (0.1, 1), (0.2, 1), ... , (1, 1), (1, 0.1), (1, 0.2), ... ,
+(1, 0.9).
+SVM issue. By design, the implementation of the linear SVM model we used, linearSVC,
+only returns discrete predictions, that is, a single label corresponding to the most likely category,
+rather than a probability distribution over all categories. To address this issue, in [5], we tried
+retraining the best SVM using the SVC implementation13 with linear Kernel. However, results
+were not convincing. Therefore, in the present study, we decided simply not to use model
+stacking when one of the two models involved (e.g., best generic or specific model) was a
+SVM.
+4.6. Performance metrics
+Due to the large class imbalance for all outcomes, measuring classification performance
+with accuracy was inadequate. Rather, we computed precision, recall, and F1-score.
+Precision, respectively recall, for category i, is equal to the number of correct predictions
+for category i (number of hits), divided by the number of predictions made for category i (hits
+and false alarms), respectively by the number of observations in category i (hits and misses).
+precision =
+Ci,i
+�K
+j=1 Cj,i
+recall =
+Ci,i
+�K
+j=1 Ci,j
+(2)
+Where the confusion matrix C is a square matrix of dimension K ×K (K being the number
+of categories) and whose (i, j)th element Ci,j indicates how many of the observations known
+to be in category i were predicted to be in category j. Finally, we computed the F1-score, the
+harmonic mean of precision and recall:
+F1 = 2 × precision × recall
+precision + recall
+(3)
+4.7. Configuration
+We relied on a single Ubuntu 20.04.4 machine featuring a 4.9 MHz 12-thread CPU, a 12
+GB Nvidia Titan V GPU, 64 GB of RAM, R version 4.1.3 [22], and Python version 3.8.13 with
+scikit-learn version 1.1.1 [23]. Running all experiments took approximately ten days.
+5. Results
+Each generic model (full and per-domain), as well as ensembles thereof (stacking approach
+described in section 4.5) was tested on the test set of each company-domain-outcome combina-
+tion and compared against the best performing specific model for this combination.
+Results are very positive. As can be seen in Table 5, across all companies, the generic mod-
+els (full or per-domain) outperform the specific models 82% of the time, i.e., for 41 company-
+domain-outcome combinations out of 50. Detailed per-company results can be found in Ap-
+pendix E for the full generic models and Appendix F for the per-domain generic models. At
+the company level, improvements are brought on average for 80.6% of outcomes (across all
+domains), ranging from 33.3% for Company2 to 100% for Company1, Company4, Company6,
+and Company9.
+13https://scikit-learn.org/stable/modules/generated/sklearn.svm.SVC.html
+8
+
+Construction
+Electric T&D
+Oil & Gas
+C
+S
+B
+IT
+AT
+E
+S
+B
+IT
+AT
+E
+S
+B
+IT
+AT
+E
+1
++1.26
++0.2
++2.55
++3.07
+2
+x
++3.15
+x
+3
+x
++6.56
++0.49
+x
++12.47
++0.99
+x
+4
++3.49
++0.47
++3.06
++0.98
++0.63
+5
+x
++2.64
++1.29
++3.14
++2.79
+6
++12.86
++4.39
++1.63
++7.09
++12.87
++5
++11.19
++0.59
+7
+x
++6.69
++15.3
+x
++1.04
++9.54
++2.12
++2.2
+8
++1.11
++0.47
++2.78
+x
++3.13
+9
++1.56
++5.38
++12.16
++5.01
++6.16
+Table 5: Company-level max gains. x: no improvement. S: severity, B: body part, IT: injury type, AT: accident type,
+E: energy source. C: company
+Furthermore, as shown in Fig. 1, gains are high on average (+4.4 in F1 score) and reach im-
+pressive values, e.g., +15.3 for Company7 on electric T&D-injury type, +12.87 for Company6
+on electric T&D-severity, +12.86 for Company6 on construction-severity, +6.56 for Company3
+on construction-body part, etc. And all of that, while predicting more categories.
+Distribution of Company−level Max Gains
+Gain in F1 score over specific models
+0
+5
+10
+15
+0
+2
+4
+6
+8
+10
+12
+14
+Counts
++
++
++++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
+++
++
++
+++
++
++
++
++
++
+Figure 1: Company-level max gains, across all domains and outcomes. n=41, min=0.2, max=15.3, mean=4.4.
+There are only 9 domain-outcome combinations over 50, across 5 companies, on which
+the generic models do not bring any quantitative improvement. However, since their forecasts
+are more informative (more categories predicted), it may still make sense in practice to use the
+generic models in lieu of the specific models, even on these combinations. For instance, for
+Company3-oil & gas-accident type, the specific model only predicts exposure and struck, but
+the generic model also predicts the categories caught, fall, and overexertion.
+The F1 scores averaged over all companies are shown in Table 6. Overall, the generic mod-
+els bring improvement over the specific models for 73.3 % of the domain-outcome combinations
+(11 out of 15). As shown on the right of Fig. 2, maximum gains range from 0.95 (for electric
+T&D-energy source) to 9.98 (for electric T&D-injury type) with an average of 3.37. Also, not
+only do the 11 best generic models outperform their specific counterparts with a comfortable
+margin, but they also generate finer-grained forecasts, which are much more useful in practice.
+More specifically, generic models predict 2.26 additional categories on average, even up to 7
+for construction-injury type (while still providing a gain of 3.48 in F1 score). This is remarkable,
+considering that the more categories to be predicted, the more difficult the task (see Appendix
+D).
+9
+
+Distribution of All Gains
+Gain in F1 score over specific models
+Counts
+0
+2
+4
+6
+8
+10
+0
+5
+10
+15
+20
+25
+30
+Counts
+++
++
++
++
++
++ + +
++
++
++
++
++
++
++
++
++
++
+++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++++
++
++++
+++
++
++
++
++
+Distribution of Max Gains
+Gain in F1 score over specific models
+Frequency
+0
+2
+4
+6
+8
+10
+0
+1
+2
+3
+4
+5
+6
+Counts
++++
++
++
++
++
++
++
++
++
+Figure 2: Gains averaged over companies. Left: n=48, min=0.16, max=9.98, mean=2.22. Right: n=11, min=0.95,
+max=9.98, mean=3.37.
+The construction and oil & gas domains see gains for 3 outcomes out of 5, while on the
+electric T&D domain, we observe improvement for every outcome. Further, for the body part,
+injury type, and energy source outcomes, there is at least one generic model that outperforms
+its specific counterpart, on every domain, while the severity and accident type outcomes see
+improvements only on the electric T&D domain.
+However, it is important to note that even on those 4 domain-outcome combinations on
+which the generic models do not offer gains in predictive performance, it can still be desirable
+to use them in practice over the specific models, as they generate more informative forecasts,
+with 2 additional categories predicted, on average.
+Overall, more than half of all F1 scores recorded for the generic models (79 out of 150,
+or 53%) are greater or within two points of that of the specific models, while predicting 1.83
+more categories on average. And, as shown on the left of Fig. 2, the 48 generic models that
+outperform their specific counterparts bring on average an improvement of 2.22 in F1 score.
+5.1. Body part, injury type, and energy source
+Some of the greatest improvements are observed for injury type, where the best generic
+models provide large average gains of 3.48, 9.98, and 3.07, respectively on the construction,
+electric T&D, and oil & gas domains, while predicting on average 4.25 more categories than
+the company-specific models. This large boost in performance is remarkable considering the
+significant increase in task difficulty.
+Similarly, for energy source, the best generic models provide 4.48, 0.95, and 1.83 improve-
+ments in F1 score, while predicting 1.53 more categories on average; and for body part, the
+gains are 3.38, 3.42, and 1.37, with 0.17 more categories predicted.
+5.2. Severity and accident type
+For severity and accident type, the generic models outperform the company-specific ones
+on the electric T&D domain, with 3.09 and 2.03 gains in F1 scores, while predicting 2 and 0.5
+more categories on average.
+On the construction and oil & gas domains, the best generic models are between 2.4 and 6
+points below the company-specific ones. However, they still offer the benefit of predicting more
+categories (+1.6 on average).
+10
+
+Construction
+Electric T&D
+Oil & Gas
+Full
+Dom.
+Full
+Dom.
+Full
+Dom.
+Severity
+F1
+SVM
+gen
+31.66
+30.29
+35.52
+35.79
+30.92
+28.97
+RF
+gen
+27.26
+31.04
+33.37
+41.28
+23.48
+26.08
+ens
+30.33
+31.82
+41.17†
+43.88†
+28.98
+30.76
+XGB
+gen
+26.98
+28.74
+36.4
+39.69⋆
+24.25
+24.19
+ens
+29.67
+31.81
+40.01⋆
+40.95
+29.14
+31.44
+spec
+35.34†
+40.79
+33.86†
+Count
+# categories
+spec
+3.5
+3
+3.5
+gen
+5
+5
+5
+5
+5
+5
+# datasets
+9
+4
+9
+4
+9
+4
+Body Part
+F1
+SVM
+gen
+25.51
+31.32
+26.68
+25.24
+23.65
+24.66
+RF
+gen
+34.41†
+31.81
+34.94†
+33.05
+30.22†
+28.85†
+ens
+30.33⋆
+30.39⋆
+34.08
+29.25
+26.43
+27.54⋆
+XGB
+gen
+32.86
+32.23†
+33.76
+35.21†
+29.22
+28.59⋆
+ens
+29.05⋆
+29.2⋆
+32.31
+34.71
+26.37
+27.74⋆
+spec
+31.03
+31.79
+28.85
+Count
+# categories
+spec
+6
+5.5
+6
+gen
+6
+6
+6
+6
+6
+6
+# datasets
+9
+4
+9
+4
+9
+3
+Injury Type
+F1
+SVM
+gen
+38.73
+42.78⋆
+53.7†
+42.91⋆
+35.11⋆
+33.89
+RF
+gen
+40.03
+42.11⋆
+42.44⋆
+41.68⋆
+32.15
+32.82
+ens
+41.75
+45.46†
+49.42
+45.10
+38.65†
+38.97
+XGB
+gen
+36.76
+42.55⋆
+41.03
+41.48
+31.33
+32.34
+ens
+47.4†
+45.45
+51.44
+49.28†
+38.05
+39.79†
+spec
+43.92
+43.72
+36.72
+Count
+# categories
+spec
+4
+5.25
+7
+gen
+11
+6
+11
+8
+11
+11
+# datasets
+9
+4
+9
+4
+9
+3
+Accident Type
+F1
+SVM
+gen
+42.39
+42.20
+44.58
+44.84
+60.69
+64.85
+RF
+gen
+42.46
+42.42
+48.58†
+50.2†
+63.14
+64.12
+ens
+44.35
+40.8
+41.29
+39.72
+66.40
+65.74
+XGB
+gen
+48.27
+49.21
+47.80⋆
+49.58
+58.58
+63.04
+ens
+42.02
+43.40
+41.08
+43.53
+64.56
+67.13
+spec
+54.98†
+48.17
+73.16†
+Count
+# categories
+spec
+3.5
+4.5
+2.67
+gen
+5
+5
+5
+5
+5
+4
+# datasets
+6
+2
+6
+2
+6
+3
+Energy Source
+F1
+SVM
+gen
+74.12†
+73.78
+77.64⋆
+78.87†
+69.54⋆
+55.63
+RF
+gen
+72.71
+73.91†
+77.12⋆
+78.61
+71.12
+70.72†
+ens
+69.06⋆
+69.64⋆
+75.83
+76.48⋆
+70.58
+69.12⋆
+XGB
+gen
+73.77
+71.04
+78.83†
+75.61
+72.22†
+70.35⋆
+ens
+69.05⋆
+70.23
+76.47⋆
+75.03
+70.98
+70.38⋆
+spec
+69.64
+77.92
+70.39
+Count
+# categories
+spec
+2.25
+2.67
+3
+gen
+5
+3
+5
+3
+5
+4
+# datasets
+8
+4
+8
+3
+8
+2
+Table 6: Results averaged over companies. †: best of their sub-column. Bold/⋆: better than/within 2 pts of spec. Full: full generic
+model (one per outcome, same across domains). Dom.: per-domain generic model (one per outcome per domain). Gen/spec:
+generic/specific. Ens: ensemble thereof. # datasets: number of company datasets forming the generic dataset. Note: for the same
+outcome, # categories and # datasets are the same for Full across domains, we repeat them only to ease comparison.
+5.3. Full vs. per-domain
+In what follows, we refer to the full and per-domain models and their ensemble versions.
+When considering full generic models, the average improvement in F1-score over the specific
+models is 2.85 and there are 2.44 additional categories predicted (min=0, max=7), while when
+11
+
+considering per-domain generic models, the average improvement is 2.57 and 1.38 additional
+categories are predicted (min=0, max=4). The per-domain models reach a higher max score than
+the full models on 9 combinations out of 15 (60%), and in 5 out of 11 (45%) when the specific
+models are outperformed. The full and per-domain models outperform the specific models on
+the same 11 domain-outcome combinations.
+So, in terms of performance, there is no clear winner. However, since the full generic models
+predict more categories, and are also simpler conceptually (just one model per outcome), full
+models seem like the way to go. This conclusion however will need to be validated when more
+datasets are available for each domain. One thing to note, however, is that specific models may
+still be desirable in the context of model stacking, as covered next.
+5.4. Generic vs. ensemble (generic + specific)
+The transfer learning-like stacking approach, i.e., combining the predictions of the generic
+and specific models, boosts performance over the generic models (both full and per-domain)
+on all domains for the severity and injury type outcomes, in some cases for accident type, and
+nowhere for body part and energy source.
+For severity, the average gains are of 3.93, and range from 0.78 to an impressive 7.8 (for
+electric T&D-full-RF). Results are even more impressive for injury type. Gains range from 1.72
+to 10.64 (for construction-full-XGB), with a high average of 6.17.
+It is interesting to note that for severity and injury type, very few of the generic models
+outperform the specific models in the first place, and it is only by combining their predictions
+with that of the specific models that absolute best performance can be reached, on the electrical
+domain for severity, and on all domains for injury type.
+We also observe that conversely, for body part and energy source, where model stacking
+does not bring additional skill, the generic models are stronger than the specific models in the
+first place.
+All in all, these results may suggest that ensembling only works when the generic models are
+not already better than the specific models. However, this rule does not hold everywhere (e.g.,
+construction-accident type-XGB), so additional data, experiments and results will be necessary
+to draw any general conclusion here.
+5.5. Quantity vs. relevance
+As far as whether more data or more relevant data is best, Fig. 3 shows the distributions of
+the best a and b coefficients as determined on the validation sets. It tends to indicate that, on
+average, the best tradeoff involves anywhere from a little bit to a lot of generic model (anywhere
+in the [0.1,1] range, with peaks towards [0.1,0.2] and [0.9,1]), but almost always a lot of specific
+model (between 0.9 and 1). In other words, data relevance always seems important, while
+the contribution of data quantity fluctuates. However, this is only a general trend. As can be
+seen in the detailed results per company (Appendix E and Appendix F), in some cases, the
+contribution of the generic model is more important than that of the specific model, e.g., (1,0.6)
+for Company6-XGB in the first table of Appendix F.
+5.6. Best model type
+For the full generic models, the best algorithm is RF (6 domain-outcome combinations over
+15), followed by SVM (5/15) and XGB (4/15). When stacked with the specific model, RF
+reaches best performance in 10 out of 15 combinations.
+12
+
+Coefficient Distributions
+Coefficient Value
+Count
+0.2
+0.4
+0.6
+0.8
+1.0
+0
+10
+20
+30
+40
+50
+60
+generic
+specific
+Coefficient Distributions
+Coefficient Value
+Count
+0.2
+0.4
+0.6
+0.8
+1.0
+0
+10
+20
+30
+40
+50
+generic
+specific
+Figure 3: Distributions of the best coefficient values a (generic) and b (specific). Left: full. Right: per-domain.
+When considering the per-domain generic models, SVM obtains the best score 7 times out
+of 15, followed by RF (5/15) and XGB (3/5). However, when used in the ensemble, XGB is the
+best (10/15).
+RF and XGB are better choices than SVM as they consistently top the scores and can be
+used in ensembles. In terms of performance though, there is no clear winner between the two.
+One or the other could be used interchangeably. However, XGBoost is superior in practice as
+far as deployment is concerned, as the Random Forest models take a lot of disk space, even after
+applying some compression tricks.
+6. Conclusion
+We showed that generic models provide consistent and large improvements over company-
+specific models. Moreover, generic models issue finer-grained forecasts that are more useful in
+practice, as they predict more categories of each safety outcome.
+Generic models remove the needs for training company-specific models, saving a lot of time
+and resources, and give small companies, whose accident datasets are too limited to train their
+own models, access to safety outcome predictions.
+Per-domain generic models (trained on data from a specific industry sector) are not always
+better than full generic models (trained on all data). Ensembling generic and specific models is
+often very beneficial. Therefore, it might still be worth training specific models to combine their
+predictions with that of the generic models. If specific models are already in use, combining
+them with the generic models may provide a boost in performance.
+The forecasts are in essence clear and direct information that can be accessed via a user
+interface (as a desktop or mobile webpage or application), or via an API for integration into any
+existing ecosystem. In each case, the only input required is a set of attributes, and the output are
+probabilities for each category of each outcome.
+By learning lessons from a pool of datasets whose accumulated experience far exceeds that
+of any single company, and making these lessons easily accessible, generic models tackle the
+holy grail of safety cross-organizational learning and dissemination in the construction industry.
+7. Acknowledgements
+We thank the Nvidia corporation for donating the Titan V GPU that was used in this re-
+search, as part of their GPU grant program.
+13
+
+8. References
+References
+[1] M. Desvignes, Requisite empirical risk data for integration of safety with advanced technologies and intelligent
+systems, Ph.D. thesis, University of Colorado at Boulder (2014).
+URL https://scholar.colorado.edu/downloads/0r967398g
+[2] M. P. Villanova, Attribute-based risk model for assessing risk to industrial construction tasks, Ph.D. thesis,
+University of Colorado at Boulder (2014).
+URL https://scholar.colorado.edu/downloads/jd472w76t
+[3] A. J.-P. Tixier, M. R. Hallowell, B. Rajagopalan, D. Bowman, Automated content analysis for construction
+safety: a natural language processing system to extract precursors and outcomes from unstructured injury
+reports, Automation in Construction 62 (2016) 45–56. doi:10.1016/j.autcon.2015.11.001.
+[4] A. J.-P. Tixier, M. R. Hallowell, B. Rajagopalan, D. Bowman, Application of machine learning to construction
+injury prediction, Automation in Construction 69 (2016) 102–114. doi:10.1016/j.autcon.2016.05.
+016.
+[5] H. Baker, M. R. Hallowell, A. J.-P. Tixier, Ai-based prediction of independent construction safety outcomes
+from universal attributes, Automation in Construction 118 (2020) 103146.
+[6] A. J.-P. Tixier, M. R. Hallowell, B. Rajagopalan, D. Bowman, Construction safety clash detection: identify-
+ing safety incompatibilities among fundamental attributes using data mining, Automation in Construction 74
+(2017) 39–54.
+[7] A. J.-P. Tixier, M. R. Hallowell, B. Rajagopalan, Construction safety risk modeling and simulation, Risk
+Analysis 37 (10) (2017) 1917–1935. doi:10.1111/risa.12772.
+[8] L. Tanguy, N. Tulechki, A. Urieli, E. Hermann, C. Raynal, Natural language processing for aviation safety
+reports: From classification to interactive analysis, Computers in Industry 78 (2016) 80–95.
+[9] A. L. Sepeda, Lessons learned from process incident databases and the process safety incident database (psid)
+approach sponsored by the center for chemical process safety, Journal of hazardous materials 130 (1-2) (2006)
+9–14.
+[10] Q. T. Le, D. Y. Lee, C. S. Park, A social network system for sharing construction safety and health knowledge,
+Automation in Construction 46 (2014) 30–37.
+[11] A. Pedro, A.-T. Pham-Hang, P. T. Nguyen, H. C. Pham, Data-driven construction safety information sharing
+system based on linked data, ontologies, and knowledge graph technologies, International journal of environ-
+mental research and public health 19 (2) (2022) 794.
+[12] K. W. Edwin, Sharing incident experiences: a roadmap towards collective safety information in the norwegian
+construction industry, International Journal of Occupational Safety and Ergonomics (2022) 1–11.
+[13] K. Wasilkiewicz, Information flow and knowledge transfer of accident investigation results in the norwegian
+construction industry, in: Safety and Reliability–Safe Societies in a Changing World, CRC Press, 2018, pp.
+2855–2862.
+[14] L. Breiman, Random forests, Machine Learning 45 (1) (2001) 5–32.
+[15] T. Chen, C. Guestrin, Xgboost: A scalable tree boosting system, in: Proceedings of the 22nd ACM SIGKDD
+International Conference on Knowledge Discovery and Data Mining, ACM, 2016, pp. 785–794.
+[16] B. E. Boser, I. M. Guyon, V. N. Vapnik, A training algorithm for optimal margin classifiers, in: ACM
+Proceedings of the Fifth Annual Workshop on Computational learning theory, 1992, pp. 144–152. doi:
+10.1145/130385.130401.
+[17] A. Krizhevsky, I. Sutskever, G. E. Hinton, Imagenet classification with deep convolutional neural networks, in:
+Advances in neural information processing systems, 2012, pp. 1097–1105.
+14
+
+[18] A.
+Radford,
+K.
+Narasimhan,
+T.
+Salimans,
+I.
+Sutskever,
+Improving
+language
+under-
+standing
+by
+generative
+pre-training,
+URL
+https://s3-us-west-2.
+amazonaws.
+com/openai-
+assets/researchcovers/languageunsupervised/language understanding paper. pdf.
+[19] J. Devlin, M.-W. Chang, K. Lee, K. Toutanova, Bert: Pre-training of deep bidirectional transformers for lan-
+guage understanding, arXiv preprint arXiv:1810.04805.
+[20] M. Lewis, Y. Liu, N. Goyal, M. Ghazvininejad, A. Mohamed, O. Levy, V. Stoyanov, L. Zettlemoyer, Bart:
+Denoising sequence-to-sequence pre-training for natural language generation, translation, and comprehension,
+arXiv preprint arXiv:1910.13461.
+[21] M. K. Eddine, A. J.-P. Tixier, M. Vazirgiannis, Barthez: a skilled pretrained french sequence-to-sequence
+model, arXiv preprint arXiv:2010.12321.
+[22] R Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Com-
+puting, Vienna, Austria (2022).
+URL https://www.R-project.org/
+[23] F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer,
+R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, E. Duchesnay, Scikit-
+learn: Machine learning in Python, Journal of Machine Learning Research 12 (2011) 2825–2830.
+Appendices
+Appendix A. Attribute List
+adverse low temps
+fuses⋆
+machinery
+spark
+bolt
+grinding
+manlift
+splinter/sliver
+breaker⋆
+grout
+mud
+spool
+cable
+guardrail/handrail
+nail
+stairs
+cable tray
+hammer
+no/improper PPE
+steel/steel sections
+chipping
+hand size pieces
+object at height
+stripping
+cleaning
+hazardous substance
+object on the floor
+stud
+clearance⋆
+heat source/high temps
+piping
+switch/switching⋆
+concrete
+heater⋆
+pole⋆
+tank
+concrete liquid
+heavy material/tool
+pontoon
+transformer⋆
+conduit
+heavy vehicle
+poor housekeeping
+uneven surface
+confined work space
+hose
+poor visibility
+unpowered tool
+congested work space
+improper body position
+powered tool
+unpowered transporter
+crane
+improper procedure/inattention
+rebar
+unstable support/surface
+door
+improper security of materials
+relay⋆
+valve
+drill
+improper security of tools
+repetitive motion
+vault⋆
+dunnage
+insect/animal
+scaffold
+welding
+electricity
+job trailer
+screw
+wind
+exiting
+ladder
+sharp edge
+wire
+fan⋆
+lifting/pulling/manipulating
+slag
+working at height
+fatigued dizzy
+light vehicle
+slippery surface
+working below elev wksp/mat
+forklift
+LOTO/labeling⋆
+small particle
+working overhead
+formwork
+lumber
+soffit
+wrench
+Table A.7: 92 attributes used in this study. LOTO: lockout-tagout. PPE: personal protective equipment. ⋆: eleven
+new attributes added since [4, 5].
+15
+
+Appendix B. Detailed split counts
+Severity
+Train
+w
+Val
+Test
+Construction
+report-only
+917
+8.2
+226
+283
+1st aid
+7486
+1.0
+1876
+2369
+medical
+470
+15.9
+114
+140
+recordable
+147
+50.9
+28
+42
+lost time
+960
+7.8
+250
+285
+total
+9980
+2494
+3119
+Electric T&D
+report-only
+2392
+1.2
+576
+712
+1st aid
+2809
+1.0
+736
+905
+medical
+554
+5.1
+140
+162
+recordable
+310
+9.1
+74
+101
+lost time
+607
+4.6
+143
+205
+total
+6672
+1669
+2085
+Oil & Gas
+report-only
+929
+14.8
+244
+279
+1st aid
+13766
+1.0
+3405
+4279
+medical
+1919
+7.2
+489
+618
+recordable
+152
+90.6
+42
+52
+lost time
+1615
+8.5
+415
+516
+total
+18381
+4595
+5744
+Corporate
+report-only
+97
+3.3
+31
+22
+1st aid
+321
+1.0
+74
+109
+total
+418
+105
+131
+Full
+report-only
+4335
+5.6
+1077
+1296
+1st aid
+24382
+1.0
+6091
+7662
+medical
+2943
+8.3
+743
+920
+recordable
+609
+40.0
+144
+195
+lost time
+3182
+7.7
+808
+1006
+total
+35451
+8863
+11079
+Body Part
+Train
+w
+Val
+Test
+Construction
+arm
+1059
+2.6
+285
+338
+foot
+694
+3.9
+167
+232
+hand
+2732
+1.0
+701
+864
+head
+1682
+1.6
+394
+494
+leg
+958
+2.9
+262
+307
+trunk
+1084
+2.5
+243
+330
+total
+8209
+2052
+2565
+Electric T&D
+arm
+1061
+1.4
+274
+319
+foot
+372
+4.0
+89
+135
+hand
+1473
+1.0
+368
+452
+head
+1246
+1.2
+318
+403
+leg
+1084
+1.4
+251
+307
+trunk
+800
+1.8
+208
+269
+total
+6036
+1508
+1885
+Oil & Gas
+arm
+1445
+3.9
+386
+477
+foot
+1741
+3.2
+421
+568
+hand
+5586
+1.0
+1385
+1740
+head
+3514
+1.6
+887
+1088
+leg
+2053
+2.7
+498
+596
+trunk
+1449
+3.9
+370
+464
+total
+15788
+3947
+4933
+Full
+arm
+3565
+2.7
+945
+1134
+foot
+2807
+3.5
+677
+935
+hand
+9791
+1.0
+2454
+3056
+head
+6442
+1.5
+1599
+1985
+leg
+4095
+2.4
+1011
+1210
+trunk
+3333
+2.9
+821
+1063
+total
+30033
+7507
+9383
+Accident Type
+Train
+w
+Val
+Test
+Construction
+caught
+396
+2.3
+105
+137
+exposure
+119
+7.8
+38
+40
+fall
+803
+1.2
+200
+243
+overexertion
+492
+1.9
+128
+160
+struck
+930
+1.0
+214
+276
+total
+2740
+685
+856
+Electric T&D
+caught
+207
+2.2
+55
+62
+exposure
+454
+1.0
+123
+142
+fall
+403
+1.1
+102
+143
+overexertion
+288
+1.6
+51
+65
+struck
+248
+1.8
+69
+88
+total
+1600
+400
+500
+Oil & Gas
+caught
+198
+7.7
+43
+53
+exposure
+526
+2.9
+127
+184
+fall
+1527
+1.0
+393
+463
+struck
+659
+2.3
+165
+210
+total
+2910
+728
+910
+Full
+caught
+801
+3.4
+203
+252
+exposure
+1099
+2.5
+288
+366
+fall
+2733
+1.0
+695
+849
+overexertion
+780
+3.5
+179
+225
+struck
+1837
+1.5
+448
+574
+total
+7250
+1813
+2266
+Energy Source
+Train
+w
+Val
+Test
+Construction
+chemical
+76
+42.7
+21
+14
+gravity
+1551
+2.1
+405
+479
+motion
+3248
+1.0
+792
+1031
+total
+4875
+1218
+1524
+Electric
+biological
+221
+7.6
+52
+88
+gravity
+733
+2.3
+179
+230
+motion
+1683
+1.0
+429
+507
+total
+2637
+660
+825
+Oil & Gas
+chemical
+70
+21.2
+13
+21
+gravity
+1485
+1.0
+361
+448
+motion
+914
+1.6
+246
+300
+thermal
+131
+11.3
+30
+44
+total
+2600
+650
+813
+Full
+biological
+221
+26.4
+52
+88
+chemical
+146
+40.0
+34
+35
+gravity
+3769
+1.6
+945
+1157
+motion
+5845
+1.0
+1467
+1838
+thermal
+131
+44.6
+30
+44
+total
+10112
+2528
+3162
+Table B.8: Split counts (1/2). w: training weights.
+16
+
+Injury Type
+Train
+w
+Val
+Test
+Construction
+contusion
+728
+3.6
+185
+229
+cut
+2644
+1.0
+682
+795
+fob
+399
+6.6
+84
+118
+fracture
+100
+26.4
+24
+39
+pinch
+267
+9.9
+90
+97
+strain
+2129
+1.2
+501
+680
+total
+6267
+1566
+1958
+Electric T&D
+bite
+129
+12.3
+35
+42
+burn
+75
+21.2
+14
+21
+contusion
+861
+1.8
+216
+277
+cut
+1305
+1.2
+330
+400
+fob
+209
+7.6
+46
+69
+fracture
+176
+9.0
+39
+53
+irritation
+420
+3.8
+101
+141
+strain
+1589
+1.0
+410
+486
+total
+4764
+1191
+1489
+Oil & Gas
+bite
+168
+27.6
+39
+52
+burn
+572
+8.1
+150
+179
+contusion
+3587
+1.30
+848
+1091
+cut
+4638
+1.0
+1160
+1509
+exhaustion
+75
+61.8
+24
+25
+fob
+1440
+3.2
+381
+455
+fracture
+622
+7.5
+160
+199
+irritation
+127
+36.5
+37
+42
+pain
+704
+6.6
+176
+215
+pinch
+720
+6.4
+181
+231
+strain
+2307
+2.0
+584
+677
+total
+14960
+3740
+4675
+Full
+bite
+297
+28.9
+74
+94
+burn
+647
+13.3
+164
+200
+contusion
+5176
+1.7
+1249
+1597
+cut
+8587
+1.0
+2172
+2704
+exhaustion
+75
+114.5
+24
+25
+fob
+2048
+4.2
+511
+642
+fracture
+898
+9.6
+223
+291
+irritation
+547
+15.7
+138
+183
+pain
+704
+12.2
+176
+215
+pinch
+987
+8.7
+271
+328
+strain
+6025
+1.4
+1495
+1843
+total
+25991
+6497
+8122
+Table B.9: Split counts (2/2). w: training weights.
+Appendix C. Hyperparameter Optimization Details
+For Random Forest14, we searched the number of trees (ntree parameter, from 100 to
+1600 with steps of 100), the number of variables to try when making each split (mtry, from 5
+to 45 with steps of 5), and the leaf size (nodesize, 1, 2, 5, 10, 25, and 50).
+For XGBoost15, we searched the maximum depth of a tree in the sequence (max depth,
+from 3 to 6 with steps of 1), the learning rate (learning rate, 0.01, 0.05, and 0.1), the
+minimum leaf size (min child weight, 1, 3, 5, and 10), the percentage of training instances
+to be used in building each tree (subsample, 0.3, 0.5, 0.7, and 1) , and the percentage of
+predictors to be considered in making each split of a given tree (colsample bylevel, 0.3,
+0.5, 0.7, and 1). The number of trees in the sequence (ntrees) was set to 2000. The loss was
+the multinomial one. Finally, for the SVM model, we optimized the C parameter (C, 10x with
+x taking 3000 evenly spaced values in [−9, 9]).
+14https://scikit-learn.org/stable/modules/generated/sklearn.ensemble.RandomForestClassifier.html
+15https://xgboost.readthedocs.io/en/latest/parameter.html
+17
+
+Appendix D. Illustration of Task Difficulty vs. Number of Categories
+To illustrate how the prediction task gets more and more difficult as the number of cate-
+gories increases, we designed a synthetic example in which 105 observations were drawn from
+an increasing number of categories (2 to 12). Class imbalance was simulated by drawing from
+the categories with probabilities following the lognormal distribution (mean=0, sd=2). We con-
+sidered two baselines: a random baseline, that predicts categories uniformly at random, and a
+most frequent baseline, which always returns the most frequent category. Our proxy for diffi-
+culty was one minus the F1 score of the baselines. In other words, the less well the baselines
+are doing, the more difficult the task. We can see on Fig. D.4 that the task difficulty rapidly
+increases with the number of categories, and that going from 2 to 6 categories almost makes the
+task twice as hard.
+2
+4
+6
+8
+10
+12
+0.5
+0.6
+0.7
+0.8
+0.9
+Prediction Task Difficulty vs Number of Categories
+Number of Categories
+1 − F1 score
+Random Baseline
+Most Frequent Baseline
+Figure D.4
+Appendix E. Per-Company Results for the Full Generic Models
+Note: the ensemble (“ens”) rows are left blank whenever the specific model is a SVM, as
+we could not use ensembling in this case (the forecast of the SVM is not probabilistic).
+Appendix E.1. Severity
+Comp.1
+Comp.3
+Comp.5
+Comp.6
+Avg
+spec
+29.51
+32.62
+45.35
+33.9
+35.34†
+SVM
+gen
+20.23
+25.64
+34.01
+46.76
+31.66
+gen
+25.75
+21.54
+29.75
+31.99
+27.26
+RF
+ens
+28.68
+31.62
+30.69
+30.33
+coef.
+(0.4,1)
+(0.8,1)
+(0.4,1)
+gen
+27.58
+23.26
+27.58
+29.48
+26.98
+XGB
+ens
+28.85
+28.34
+31.82
+29.67
+coef.
+(0.1,1)
+(0.3,1)
+(0.5,1)
+#lev. spec
+4
+4
+3
+3
+3.5
+#lev. gen
+5
+5
+5
+5
+5
+Table E.10: Severity, construction. †: best model on average.
+18
+
+Comp.4
+Comp.6
+Comp.7
+Comp.9
+Avg
+spec
+29.48
+45.66
+57.67
+30.34
+40.79
+SVM
+gen
+20.93
+42.56
+46.67
+31.9
+35.52
+gen
+27.46
+38.61
+39.97
+27.42
+33.37
+RF
+ens
+28.73
+53.62
+41.17†
+coef.
+(1,0.9)
+(0.5,1)
+gen
+27.39
+53.02
+39.24
+25.95
+36.4
+XGB
+ens
+28.74
+51.27
+40.01⋆
+coef.
+(0.8,1)
+(0.2,1)
+#lev. spec
+4
+2
+2
+4
+3
+#lev. gen
+5
+5
+5
+5
+5
+Table E.11: Severity, electric T&D. †: best model on average. Bold/⋆: better/within 2pts of the company-specific
+model.
+Comp.2
+Comp.3
+Comp.8
+Comp.7
+Avg
+spec
+42.53
+24.74
+39.72
+28.44
+33.86†
+SVM
+gen
+37.91
+22.53
+38.85
+24.41
+30.92
+gen
+17.96
+17.12
+35.69
+23.14
+23.48
+RF
+ens
+27.87
+24.05
+39.81
+24.2
+28.98
+coef.
+(0.2,1)
+(0.7,1)
+(0.7,1)
+(0.1,1)
+gen
+16.75
+23.25
+35.27
+21.72
+24.25
+XGB
+ens
+27.89
+25.36
+39.61
+23.7
+29.14
+coef.
+(0.2,1)
+(1,0.8)
+(0.3,1)
+(0.1,1)
+#lev. spec
+3
+4
+3
+4
+3.5
+#lev. gen
+5
+5
+5
+5
+5
+Table E.12: Severity, oil & gas. †: best model on average.
+Appendix E.2. Body Part
+Comp.1
+Comp.3
+Comp.5
+Comp.6
+Avg
+spec
+34.14
+26.48
+32.09
+31.39
+31.03
+SVM
+gen
+23.26
+25.09
+27.02
+26.66
+25.51
+gen
+34.14
+33.04
+34.68
+35.78
+34.41†
+RF
+ens
+33.49
+22.43
+32.7
+32.7
+30.33⋆
+coef.
+(0.4,1)
+(0.1,1)
+(0.7,1)
+(0.6,1)
+gen
+31.92
+30.57
+34.73
+34.22
+32.86
+XGB
+ens
+32.44
+20.38
+32.62
+30.77
+29.05⋆
+coef.
+(0.1,1)
+(0.2,1)
+(0.2,1)
+(0.5,1)
+#lev. spec
+6
+6
+6
+6
+6
+#lev. gen
+6
+6
+6
+6
+6
+Table E.13: Body part, construction. †: best model on average. Bold/⋆: better/within 2pts of the company-specific
+model.
+19
+
+Comp.4
+Comp.6
+Comp.7
+Comp.9
+Avg
+spec
+29.25
+27.7
+46.34
+23.86
+31.79
+SVM
+gen
+19.21
+28.86
+38.26
+20.4
+26.68
+gen
+27.96
+32
+51.02
+28.76
+34.94†
+RF
+ens
+27.94
+50.75
+23.56
+34.08
+coef.
+(0.4,1)
+(0.1,1)
+(0.4,1)
+gen
+28.24
+31.12
+46.44
+29.24
+33.76
+XGB
+ens
+27.96
+41.17
+27.81
+32.31
+coef.
+(0.2,1)
+(0.1,1)
+(0.5,1)
+#lev. spec
+6
+6
+4
+6
+5.5
+#lev. gen
+6
+6
+6
+6
+6
+Table E.14: Body part, electric T&D. †: best model on average. Bold: better the company-specific model.
+Comp.2
+Comp.8
+Comp.7
+Avg
+spec
+22.96
+32.41
+31.17
+28.85
+SVM
+gen
+22.66
+26.23
+22.06
+23.65
+gen
+26.11
+32.34
+32.21
+30.22†
+RF
+ens
+20.31
+32.88
+26.09
+26.43
+coef.
+(0.1,1)
+(1,0.1)
+(0.1,1)
+gen
+25.5
+32.36
+29.81
+29.22
+XGB
+ens
+16.26
+32.28
+30.56
+26.37
+coef.
+(0.1,1)
+(1,0.3)
+(0.2,1)
+#lev. spec
+6
+6
+6
+6
+#lev. gen
+6
+6
+6
+6
+Table E.15: Body part, oil & gas. †: best model on average. Bold: better the company-specific model.
+Appendix E.3. Injury Type
+Comp.1
+Comp.3
+Comp.5
+Comp.6
+Avg
+spec
+54
+37.7
+33.91
+50.07
+43.92
+SVM
+gen
+34.67
+36.66
+34.78
+48.81
+38.73
+gen
+47.84
+33.86
+33.11
+45.3
+40.03
+RF
+ens
+47.6
+31.98
+45.67
+41.75
+coef.
+(0.2,1)
+(0.1,1)
+(0.4,1)
+gen
+46.46
+23.99
+31.9
+44.7
+36.76
+XGB
+ens
+56.55
+35.2
+50.46
+47.4†
+coef.
+(0.6,1)
+(0.4,1)
+(0.2,1)
+#lev. spec
+3
+3
+6
+4
+4
+#lev. gen
+11
+11
+11
+11
+11
+Table E.16: Injury type, construction. †: best model on average. Bold: better the company-specific model.
+Comp.4
+Comp.6
+Comp.7
+Comp.9
+Avg
+spec
+39.21
+43.4
+47.28
+44.98
+43.72
+SVM
+gen
+42.27
+54.59
+60.78
+57.14
+53.7†
+gen
+26.44
+42.41
+56.74
+44.16
+42.44⋆
+RF
+ens
+39.33
+59.52
+49.42
+coef.
+(1,0.5)
+(1,0.2)
+gen
+28.57
+41.6
+51.45
+42.49
+41.03
+XGB
+ens
+40.31
+62.58
+51.44
+coef.
+(1,0.8)
+(1,0.1)
+#lev. spec
+5
+6
+4
+6
+5.25
+#lev. gen
+11
+11
+11
+11
+11
+Table E.17: Injury type, electric T&D. †: best model on average. Bold/⋆: better/within 2pts of the company-specific
+model.
+20
+
+Comp.2
+Comp.8
+Comp.7
+Avg
+spec
+35.39
+34.04
+40.72
+36.72
+SVM
+gen
+27.97
+30.67
+46.69
+35.11⋆
+gen
+23.72
+32.22
+40.52
+32.15
+RF
+ens
+36.82
+40.48
+38.65†
+coef.
+(0.5,1)
+(0.7,1)
+gen
+23.69
+31.01
+39.28
+31.33
+XGB
+ens
+35.09
+41
+38.05
+coef.
+(1,0.7)
+(1,0.6)
+#lev. spec
+3
+10
+8
+7
+#lev. gen
+11
+11
+11
+11
+Table E.18: Injury type, oil & gas. †: best model on average. Bold/⋆: better/within 2pts of the company-specific
+model.
+Appendix E.4. Accident Type
+Comp.3
+Comp.5
+Avg
+spec
+68.63
+41.34
+54.98†
+SVM
+gen
+41.87
+42.91
+42.39
+gen
+40.44
+44.48
+42.46
+RF
+ens
+44.35
+44.35
+coef.
+(1,0.7)
+gen
+54.04
+42.51
+48.27
+XGB
+ens
+42.02
+42.02
+coef.
+(1,1)
+#lev. spec
+2
+5
+3.5
+#lev. gen
+5
+5
+5
+Table E.19: Accident type, construction. †: best model on average.
+Comp.4
+Comp.9
+Avg
+spec
+43.15
+53.2
+48.17
+SVM
+gen
+36.46
+52.71
+44.58
+gen
+40.05
+57.11
+48.58†
+RF
+ens
+41.29
+41.29
+coef.
+(0.4,1)
+gen
+38.13
+57.46
+47.8⋆
+XGB
+ens
+41.08
+41.08
+coef.
+(0.4,1)
+#lev. spec
+5
+4
+4.5
+#lev. gen
+5
+5
+5
+Table E.20: Accident type, electric T&D. †: best model on average. Bold/⋆: better/within 2pts of the company-
+specific model.
+21
+
+Comp.3
+Comp.8
+Comp.7
+Avg
+spec
+80.91
+85
+53.58
+73.16†
+SVM
+gen
+58.06
+78.09
+45.92
+60.69
+gen
+61.67
+78.03
+49.71
+63.14
+RF
+ens
+78.46
+54.35
+66.4
+coef.
+(0.1,1)
+(1,0.1)
+gen
+46.65
+76.93
+52.16
+58.58
+XGB
+ens
+73.8
+55.31
+64.56
+coef.
+(1,0.7)
+(1,0.7)
+#lev. spec
+2
+2
+4
+2.67
+#lev. gen
+5
+5
+5
+5
+Table E.21: Accident type, oil & gas. †: best model on average.
+Appendix E.5. Energy Source
+Comp.1
+Comp.3
+Comp.5
+Comp.6
+Avg
+spec
+71.69
+70.97
+68.07
+67.82
+69.64
+SVM
+gen
+74.76
+78.16
+70.86
+72.69
+74.12†
+gen
+70.36
+76.03
+70.14
+74.31
+72.71
+RF
+ens
+71.05
+68.02
+68.1
+69.06⋆
+coef.
+(0.9,1)
+(0.2,1)
+(0.4,1)
+gen
+74.33
+83.44
+64.62
+72.7
+73.77
+XGB
+ens
+71.88
+66.81
+68.47
+69.05⋆
+coef.
+(0.4,1)
+(0.1,1)
+(0.4,1)
+#lev. spec
+2
+2
+3
+2
+2.25
+#lev. gen
+5
+5
+5
+5
+5
+Table E.22: Energy source, construction.
+†: best model on average. Bold/⋆: better/within 2pts of the company-
+specific model.
+Comp.4
+Comp.6
+Comp.9
+Avg
+spec
+79.5
+73.22
+81.05
+77.92
+SVM
+gen
+76.59
+70.61
+85.73
+77.64⋆
+gen
+74.99
+73.06
+83.32
+77.12⋆
+RF
+ens
+77.85
+73.81
+75.83
+coef.
+(0.9,1)
+(0.2,1)
+gen
+76.43
+72.85
+87.21
+78.83†
+XGB
+ens
+79.41
+73.52
+76.47⋆
+coef.
+(0.2,1)
+(0.3,1)
+#lev. spec
+3
+2
+3
+2.67
+#lev. gen
+5
+5
+5
+5
+Table E.23: Energy source, electric T&D. †: best model on average. Bold/⋆: better/within 2pts of the company-
+specific model.
+22
+
+Comp.8
+Comp.7
+Avg
+spec
+68.98
+71.8
+70.39
+SVM
+gen
+68.73
+70.36
+69.54⋆
+gen
+70.43
+71.81
+71.12
+RF
+ens
+68.27
+72.89
+70.58
+coef.
+(0.4,1)
+(1,0.2)
+gen
+70.44
+74
+72.22†
+XGB
+ens
+68.72
+73.25
+70.98
+coef.
+(0.1,1)
+(0.3,1)
+#lev. spec
+4
+2
+3
+#lev. gen
+5
+5
+5
+Table E.24: Energy source, oil & gas. †: best model on average. Bold/⋆: better/within 2pts of the company-specific
+model.
+Appendix F. Per-Company Results for the Per-Domain Generic Models
+Note: the ensemble (‘ens’) rows are left blank whenever the specific model is a SVM, as we
+could not use ensembling in this case (the forecast of the SVM is not probabilistic).
+Appendix F.1. Severity
+Comp.5
+Comp.3
+Comp.6
+Comp.1
+Avg
+spec
+45.35
+32.62
+33.9
+29.51
+35.34†
+SVM
+gen
+39.86
+26.03
+32.61
+22.66
+30.29
+gen
+34.1
+27.7
+32.62
+29.74
+31.04
+RF
+ens
+31.2
+33.5
+30.77
+31.82
+Coeffs
+(0.8,1)
+(0.6,1)
+(1,0.3)
+gen
+30.84
+26.84
+28.33
+28.95
+28.74
+XGB
+ens
+31.3
+34.14
+30
+31.81
+Coeffs
+(0.3,1)
+(1,0.6)
+(0.5,1)
+#categories spec
+3
+4
+3
+4
+3.5
+#categories gen
+5
+5
+5
+5
+5
+Table F.25: Severity, construction. †: best model on average.
+Comp.7
+Comp.4
+Comp.9
+Comp.6
+Avg
+spec
+57.67
+29.48
+30.34
+45.66
+40.79
+SVM
+gen
+36
+30.47
+24.62
+52.06
+35.79
+gen
+47.19
+30.91
+28.5
+58.53
+41.28
+RF
+ens
+54.8
+32.97
+43.88†
+Coeffs
+(1,0.6)
+(1,0.9)
+gen
+44.23
+30.59
+26.49
+57.44
+39.69⋆
+XGB
+ens
+54.95
+26.96
+40.95
+Coeffs
+(0.4,1)
+(0.1,1)
+#categories spec
+2
+4
+4
+2
+3
+#categories gen
+5
+5
+5
+5
+5
+Table F.26: Severity, elec. †: best model on average. Bold/⋆: better/within 2pts of the company-specific model.
+23
+
+Comp.7
+Comp.3
+Comp.8
+Comp.2
+Avg
+spec
+28.44
+24.74
+39.72
+42.53
+33.86†
+SVM
+gen
+26.7
+21.05
+40.83
+27.31
+28.97
+gen
+26.22
+19.82
+35.7
+22.59
+26.08
+RF
+ens
+27.59
+22.38
+40.22
+32.86
+30.76
+Coeffs
+(0.8,1)
+(1,0.9)
+(1,0.8)
+(0.7,1)
+gen
+23.82
+19.7
+33.09
+20.15
+24.19
+XGB
+ens
+27.97
+25.73
+40.1
+31.96
+31.44
+Coeffs
+(0.4,1)
+(1,0.7)
+(1,0.2)
+(0.7,1)
+#categories spec
+4
+4
+3
+3
+3.5
+#categories gen
+5
+5
+5
+5
+5
+Table F.27: Severity, oil & gas. †: best model on average.
+Appendix F.2. Body part
+Comp.5
+Comp.3
+Comp.6
+Comp.1
+Avg
+spec
+32.09
+26.48
+31.39
+34.14
+31.03
+SVM
+gen
+31.08
+28.14
+31.92
+34.13
+31.32
+gen
+32.19
+27.06
+33.64
+34.34
+31.81
+RF
+ens
+31.23
+25.77
+35.14
+29.41
+30.39⋆
+Coeffs
+(0.1,1)
+(0.2,1)
+(0.6,1)
+(0.1,1)
+gen
+33.6
+29.91
+32.48
+32.92
+32.23†
+XGB
+ens
+32.34
+20.72
+32.33
+31.41
+29.2⋆
+Coeffs
+(0.1,1)
+(0.2,1)
+(0.5,1)
+(0.1,1)
+#categories spec
+6
+6
+6
+6
+6
+#categories gen
+6
+6
+6
+6
+6
+Table F.28: Body part, construction. †: best model on average. Bold/⋆: better/within 2pts of the company-specific
+model.
+Comp.7
+Comp.4
+Comp.9
+Comp.6
+Avg
+spec
+46.34
+29.25
+23.86
+27.7
+31.79
+SVM
+gen
+34.02
+18.16
+19.6
+29.17
+25.24
+gen
+48.21
+26.31
+25.71
+31.97
+33.05
+RF
+ens
+39.52
+28.63
+19.59
+29.25
+Coeffs
+(0.1,1)
+(0.1,1)
+(0.1,1)
+gen
+53.03
+28.55
+26.56
+32.7
+35.21†
+XGB
+ens
+49.41
+29.72
+25.01
+34.71
+Coeffs
+(1,1)
+(0.6,1)
+(0.2,1)
+#categories spec
+4
+6
+6
+6
+5.5
+#categories gen
+6
+6
+6
+6
+6
+Table F.29: Body part, elec. †: best model on average. Bold/⋆: better/within 2pts of the company-specific model.
+24
+
+Comp.7
+Comp.8
+Comp.2
+Avg
+spec
+31.17
+32.41
+22.96
+28.85†
+SVM
+gen
+27.22
+25.91
+20.84
+24.66
+gen
+29.64
+31.8
+25.12
+28.85†
+RF
+ens
+30.15
+31.66
+20.81
+27.54⋆
+Coeffs
+(0.1,1)
+(0.1,1)
+(0.4,1)
+gen
+29.69
+32.36
+23.72
+28.59⋆
+XGB
+ens
+31.84
+32.1
+19.28
+27.74⋆
+Coeffs
+(1,0.5)
+(1,0.1)
+(0.2,1)
+#categories spec
+6
+6
+6
+6
+#categories gen
+6
+6
+6
+6
+Table F.30: Body part, oil & gas.
+†: best model on average. Bold/⋆: better/within 2pts of the company-specific
+model.
+Appendix F.3. Injury type
+Comp.5
+Comp.3
+Comp.6
+Comp.1
+Avg
+spec
+33.91
+37.7
+50.07
+54
+43.92
+SVM
+gen
+34.16
+36.31
+51.7
+48.97
+42.78⋆
+gen
+33.56
+33.91
+49.34
+51.64
+42.11⋆
+RF
+ens
+33.38
+48.57
+54.42
+45.46†
+Coeffs
+(0.1,1)
+(0.1,1)
+(1,0.2)
+gen
+33.3
+38.19
+48.17
+50.54
+42.55⋆
+XGB
+ens
+34.74
+47.08
+54.53
+45.45
+Coeffs
+(0.3,1)
+(0.1,1)
+(0.5,1)
+#categories spec
+6
+3
+4
+3
+4
+#categories gen
+6
+6
+6
+6
+6
+Table F.31: Injury type, construction. †: best model on average. Bold/⋆: better/within 2pts of the company-specific
+model.
+Comp.7
+Comp.4
+Comp.9
+Comp.6
+Avg
+spec
+47.28
+39.21
+44.98
+43.4
+43.72
+SVM
+gen
+57.28
+28.12
+44.54
+41.7
+42.91⋆
+gen
+53.99
+29.2
+43.07
+40.47
+41.68⋆
+RF
+ens
+51.33
+38.87
+45.1
+Coeffs
+(0.8,1)
+(0.4,1)
+gen
+49.62
+29.63
+40.26
+46.42
+41.48
+XGB
+ens
+59.48
+39.09
+49.28†
+Coeffs
+(1,0.3)
+(1,0.3)
+#categories spec
+4
+5
+6
+6
+5.25
+#categories gen
+8
+8
+8
+8
+8
+Table F.32: Injury type, elec. †: best model on average. Bold/⋆: better/within 2pts of the company-specific model.
+25
+
+Comp.7
+Comp.8
+Comp.2
+Avg
+spec
+40.72
+34.04
+35.39
+36.72
+SVM
+gen
+50.26
+33.24
+18.18
+33.89
+gen
+39.57
+33.87
+25.02
+32.82
+RF
+ens
+41.69
+36.25
+38.97
+Coeffs
+(1,0.7)
+(0.8,1)
+gen
+38.32
+32.64
+26.07
+32.34
+XGB
+ens
+42.88
+36.7
+39.79†
+Coeffs
+(1,0.7)
+(1,0.1)
+#categories spec
+8
+10
+3
+7
+#categories gen
+11
+11
+11
+11
+Table F.33: Injury type, oil & gas. †: best model on average. Bold/⋆: better/within 2pts of the company-specific
+model.
+Appendix F.4. Accident type
+Comp.5
+Comp.3
+Avg
+spec
+41.34
+68.63
+54.98†
+SVM
+gen
+44.25
+40.15
+42.2
+gen
+41.37
+43.48
+42.42
+RF
+ens
+40.8
+40.8
+Coeffs
+(0.1,1)
+gen
+43.21
+55.21
+49.21
+XGB
+ens
+43.4
+43.4
+Coeffs
+(1,0.1)
+#categories spec
+5
+2
+3.5
+#categories gen
+5
+5
+5
+Table F.34: Accident type, construction. †: best model on average.
+Comp.4
+Comp.9
+Avg
+spec
+43.15
+53.2
+48.17
+SVM
+gen
+39.45
+50.22
+44.84
+gen
+44.13
+56.28
+50.2†
+RF
+ens
+39.72
+39.72
+Coeffs
+(0.3,1)
+gen
+40.96
+58.21
+49.58
+XGB
+ens
+43.53
+43.53
+Coeffs
+(0.4,1)
+#categories spec
+5
+4
+4.5
+#categories gen
+5
+5
+5
+Table F.35: Accident type, elec. †: best model on average. Bold/⋆: better/within 2pts of the company-specific model.
+Comp.7
+Comp.3
+Comp.8
+Avg
+spec
+53.58
+80.91
+85
+73.16†
+SVM
+gen
+55.04
+59.76
+79.75
+64.85
+gen
+51.77
+58.06
+82.53
+64.12
+RF
+ens
+53.02
+78.46
+65.74
+Coeffs
+(1,0.1)
+(0.1,1)
+gen
+49.03
+62.16
+77.93
+63.04
+XGB
+ens
+55.7
+78.56
+67.13
+Coeffs
+(1,0.9)
+(0.1,1)
+#categories spec
+4
+2
+2
+2.67
+#categories gen
+4
+4
+4
+4
+Table F.36: Accident type, oil & gas. †: best model on average.
+26
+
+Appendix F.5. Energy source
+Comp.5
+Comp.3
+Comp.6
+Comp.1
+Avg
+spec
+68.07
+70.97
+67.82
+71.69
+69.64
+SVM
+gen
+70.3
+76.99
+73.32
+74.5
+73.78
+gen
+68.17
+79.98
+74.28
+73.21
+73.91†
+RF
+ens
+67.85
+69.62
+71.45
+69.64⋆
+Coeffs
+(0.1,1)
+(0.9,1)
+(0.4,1)
+gen
+62.88
+73.28
+74.91
+73.09
+71.04
+XGB
+ens
+68.05
+70.75
+71.9
+70.23
+Coeffs
+(0.1,1)
+(0.7,1)
+(0.5,1)
+#categories spec
+3
+2
+2
+2
+2.25
+#categories gen
+3
+3
+3
+3
+3
+Table F.37: Energy source, construction.
+†: best model on average. Bold/⋆: better/within 2pts of the company-
+specific model.
+Comp.4
+Comp.9
+Comp.6
+Avg
+spec
+79.5
+81.05
+73.22
+77.92
+SVM
+gen
+78.31
+85.83
+72.46
+78.87†
+gen
+80.13
+83.73
+71.96
+78.61
+RF
+ens
+79.75
+73.22
+76.48⋆
+Coeffs
+(0.5,1)
+(0.1,1)
+gen
+75.63
+82.34
+68.86
+75.61
+XGB
+ens
+77.15
+72.91
+75.03
+Coeffs
+(1,0.8)
+(0.1,1)
+#categories spec
+3
+3
+2
+2.67
+#categories gen
+3
+3
+3
+3
+Table F.38: Energy source, elec.
+†: best model on average. Bold/⋆: better/within 2pts of the company-specific
+model.
+Comp.7
+Comp.8
+Avg
+spec
+71.8
+68.98
+70.39
+SVM
+gen
+49.94
+61.33
+55.63
+gen
+69.34
+72.11
+70.72†
+RF
+ens
+70.44
+67.8
+69.12⋆
+Coeffs
+(1,0.5)
+(0.4,1)
+gen
+72.58
+68.12
+70.35⋆
+XGB
+ens
+72.06
+68.69
+70.38⋆
+Coeffs
+(0.2,1)
+(0.1,1)
+#categories spec
+2
+4
+3
+#categories gen
+4
+4
+4
+Table F.39: Energy source, oil & gas. †: best model on average. Bold/⋆: better/within 2pts of the company-specific
+model.
+27
+

EtAzT4oBgHgl3EQfw_5J/content/tmp_files/2301.01730v1.pdf.txt

@@ -0,0 +1,925 @@
+arXiv:2301.01730v1  [quant-ph]  4 Jan 2023
+Multitime Quantum Communication: Interesting But
+Not Counterfactual
+Robert B. Griffiths∗
+Department of Physics
+Carnegie Mellon University
+Pittsburgh, PA 15213
+Version of 4 January 2023
+Abstract
+A protocol for transmission of information between two parties introduced by Salih
+et al., Phys. Rev. Lett. 110 (2013) 170502 (hereafter SLAZ), involves sending quan-
+tum amplitude back and forth through a quantum channel in a series of steps, rather
+than simply sending a signal in one direction. The authors claimed that their protocol
+was “counterfactual” in the sense that while a quantum channel is needed to connect
+the parties, its actual usage becomes vanishingly small in the asymptotic limit as the
+number of steps tends to infinity. Here we show that this claim is incorrect because it
+uses probabilistic reasoning that is not valid at intermediate times in the presence of
+quantum interference. When ill-defined probabilities are replaced with a well-defined
+measure of channel usage here called “Cost”, equal to the absolute square of the am-
+plitude sent through the channel, the total Cost does not go to zero in the asymptotic
+limit of a large number of steps, but is bounded below by a rigorous inequality. A
+detailed analysis shows that this bound is satisfied in the SLAZ protocol. The analysis
+leading to the bound uses the fact that the Gram matrix formed by inner products of
+a collection of pure quantum states is additive over Hilbert subspaces and invariant
+under unitary time transformations. Its off-diagonal elements, which in general are not
+positive, play a significant role in the formal argument as well as providing a somewhat
+strange way of visualizing the transfer of information.
+Contents
+I
+Introduction
+2
+II
+One-Way Protocols
+3
+II.1
+Multiple Channels in Parallel . . . . . . . . . . . . . . . . . . . . . . . . . .
+3
+II.2
+One Channel Used Multiple Times . . . . . . . . . . . . . . . . . . . . . . .
+5
+∗Electronic address: rgrif@cmu.edu
+1
+
+III Two-way Protocols
+6
+III.1 Gram Matrices
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+6
+III.2 Basic Two-Way Protocol
+. . . . . . . . . . . . . . . . . . . . . . . . . . . .
+6
+III.3 Sending One Classical Bit
+. . . . . . . . . . . . . . . . . . . . . . . . . . .
+7
+III.4 Lower Bound on Costs
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+10
+IV The SLAZ Protocol
+11
+IV.1 Description of the Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . .
+11
+IV.2 Calculation of Costs and Overlap
+. . . . . . . . . . . . . . . . . . . . . . .
+13
+IV.3 Discussion of Costs and Probabilities . . . . . . . . . . . . . . . . . . . . . .
+14
+V
+Conclusion
+15
+I
+Introduction
+The motivation for this paper is a scheme for the transmission of quantum informatiom
+introduced by Salih et. al [1] with the title “Protocol for direct counterfactual quantum com-
+munication”, and referred to hereafter as SLAZ, the initials of the authors. One ordinarily
+thinks of the transmission of information as sending a signal through a channel from sender
+to receiver. However the idea in SLAZ is that information can be sent from Bob to Alice if
+the quantum particle used to carry the information starts off in Alice’s domain, and a part
+of its quantum amplitude is sent to Bob through a quantum channel. Bob modifies this is
+some way before sending (or possibly not sending) it back to Alice, depending on the signal
+he wants to send. Alice then employs what Bob has returned to begin a second round of
+sending amplitude to Bob, who again modifies it before returning it, and so forth. This
+back-and-forth motion can continue for a large number of rounds until the information that
+Bob is sending has arrived in Alice’s domain, where she can carry out a measurement or
+perhaps perform additional processing. A key feature of protocols of this type is that all the
+intermediate steps can be represented by purely unitary time evolution, with intermediate
+time measurements, if any replaced by unitaries—a process of purification.
+The use of amplitude rather than particle in the previous paragraph is intentional, be-
+cause the state of the photon or other particle is in general a coherent superposition of parts
+associated with different spatial locations: Alice’s domain, Bob’s domain, and the channel
+connecting them. One generally thinks of a particle as something with a spatial location,
+but in quantum mechanics one cannot simultaneously ascribe particle and wave properties
+to the same entity at the same time because of wave-particle duality. In Hilbert-space quan-
+tum mechanics physical properties, such as location in space, are represented by projectors
+(Sec. III.5 of [2]), and when a projector representing a wave, think of |ψ⟩⟨ψ|, does not com-
+mute with a projector specifying a spatial location, ignoring this fact can rapidly lead to
+paradoxes. The double-slit paradox is an example: when a coherent wave passes through
+the slit system one cannot say through which slit the particle passed.
+The term “counterfactual” in the original SLAZ paper has the following significance. A
+quantum channel connecting the communicating parties is essential: this is not a case of
+mysterious nonlocal influences of the sort which are sometimes invoked to explain quantum
+violations of Bell inequalities.
+However, if the number of steps in an SLAZ protocol is
+2
+
+sufficiently large, the magnitude of the amplitude sent through the channel in each step can
+be made very small, and vanishes in the limit as the number of steps tends to infinity.
+A similar claim of counterfactuality has been made in much of the rather substantial
+literature motivated by the original SLAZ publication, which contains various modifications
+and extensions of the original protocol. There have also been criticisms of these counter-
+factual claims, and (of course) replies to criticisms. The Conclusion, Sec. V, of the present
+paper contains a few remarks about how its results apply to some of these publications, but
+a review, much less a detailed discussion, lies far outside its scope. The interested reader is
+referred to the extensive bibliographies found in [3,4].
+The aim of the present paper is to study the use of quantum channels in protocols of the
+SLAZ type, in particular the sense in which this usage is or is not counterfactual. To this end
+the concept of the Cost of using a quantum channel, the absolute square of the amplitude
+passing through it in a particular step in the protocol, is suggested, for reasons discussed
+in Sec. II, as a useful substitute for “probabiilities”, which in a quantum context are often
+ill-defined. The example of multiple channels in parallel, which few would want to claim are
+“counterfactual”, serves as an introduction to how information can be sent through a single
+channel in a single direction at multiple times, in a process in which all of the intermediate
+steps are represented by unitary maps.
+The main mathematical results of this paper are in Sec. III: Gram matrices and some of
+their properties are discssed in Sec. III.1, while Sec. III.2 gives the basic structure of simple
+two-way multiple time protocols. Section III.3 considers simple schemes for transmitting one
+classical bit, while the rigorous lower bound that undermines various counterfactual claims
+is the topic of Sec. III.4.
+The original SLAZ protocol is studied in detail in Sec. IV. In particular the total Cost
+of transmitting a classical bit λ = 0, in which Bob reflects the amplitude back to Alice, and
+for transmitting λ = 1, in which he absorbs rather than returns it, are evaluated explicitly.
+It turns out that in the asymptotic limit the λ = 1 Cost is miniscule, but that for λ = 0
+is enormous, while the product of the two remains finite and satisfies the rigourous bound
+in Sec. III.4. The mistaken claim that the SLAZ protocol is counterfactual results from two
+errors: a concept of channel use which would be questionable even for a classical stochastic
+process, and an improper use of probabilities in a way that violates quantum principles.
+The concluding Sec. V has a summary of the main results of this paper, a few comments
+on some parts of the literature related to SLAZ, and some suggestions for future directions
+of research.
+This author believes that protocols of the SLAZ type are quite interesting,
+deserve further exploration, and might contribute to useful ways of studying multipartite
+and multitime transmission of quantum information, as in quantum networks. And that such
+studies would prove more fruitful in the absence of misleading claims of counterfactuality.
+II
+One-Way Protocols
+II.1
+Multiple Channels in Parallel
+Think of quantum information as the information carried by a photon as it passes through
+a quantum channel, such as an optical fiber.
+The information could be encoded in its
+polarization. Rather than using a single channel, one could imagine sending the photon as a
+3
+
+coherent state through a set of N channels in parallel, using a collection of beamsplitters to
+divide up the initial amplitude among the different channels, and a corresponding collection
+to later recombine them. Let us suppose that the normalized |Φ⟩ that represents the photon
+at some intermediate time is a superposition of amplitudes
+|Φ⟩ =
+N
+�
+n=1
+cn|φn⟩
+(1)
+associated with the individual channels, labeled by n. Define the Cost qn associated with
+the use of channel n, and the total Cost Q for the channel system as:
+qn := |cn|2,
+Q :=
+N
+�
+n=1
+qn.
+(2)
+If the |φn⟩ and |Φ⟩ are normalized, Q is equal to 1, so one might identify qn with the prob-
+ability that the photon is in channel n. But what does that mean? In standard (textbook)
+quantum mechanics probability refers to the outcome of a measurement, but a measurement
+carried out at an intermediate time, when the quantum state is a coherent superposition
+over various locations, can alter what occurs later, and hence it is dangerous to associate
+such a probability with a situation in which a measurement does not take place.
+Another way of viewing this difficulty is to recall that von Neumann (Sec. III.5 of [2])
+identified quantum physical properties—which in classical physics are asociated with sets
+of points in the classical phase space—with projectors, self-adjoint idempotent operators,
+P = P † = P 2, on the quantum Hilbert space. For example, in the case of a spin-half particle
+the projectors
+P = (I − σz)/2
+R = (I + σx)/2,
+(3)
+where I is the identity and σz and σx are Pauli operators, represent the properties Sz =
+−¯h/2 and Sx = +¯h/2, respectively. In general, if two projectors P and R commute their
+product PR = RP represents the property P AND R. But if they do not commute, neither
+PR nor RP is a projector, and so neither represents a quantum property. In some sense
+noncommutation is the very essence of quantum mechanics; it is what distinguishes it from
+classical physics. The use of standard (Kolmogorov) probabilities requires a sample space of
+mutually-exclusive possibilities, one and only one of which occurs in a particular run of an
+experiment. In quantum theory such a sample space is a collection of mutually orthogonal
+projectors that sum to the identity, a projective decomposition of the identity. For example, R
+and I −R in (3) in the case of spin half; see (7) below for the general definition. In quantum
+mechanics there are often many possible sample spaces that one might be interested in,
+and carelessly combining incompatible spaces—some projectors in one do not commute with
+projectors in the other—inevitably leads to paradoxes rather than physical understanding.
+In the present context the dyad |Φ⟩⟨Φ| is a projector that does not commute with any
+of the projectos |φn⟩⟨φn| for which cn is nonzero, and thus it is meaningless to assign a
+probability to the latter in a situation where the coherent state |Φ⟩ will later be transformed
+by the final beamsplitters into the original state that entered the channel system.
+4
+
+II.2
+One Channel Used Multiple Times
+The possible advantanges, if any, of using many channels in parallel can also be realized
+by employing a single channel and sending quantum amplitude through it at a succession of
+times; this is what makes protocols of the SLAZ type of some interest. Let us suppose that
+information is being sent from Bob to Alice. One can think of the photon at a particular time
+as being in a coherent superposition of amplitudes in three different physical locations: Alice’s
+domain A, Bob’s domain B, and the channel C connecting them. The same symbols can be
+used for the Hilbert-space projectors associated with these locations, thus operators which are
+self-adjoint and idempotent, A = A† = A2, and mutually orthogonal, AB = BC = AC = 0.
+They sum to the identity
+A + C + B = I
+(4)
+and hence form a projective decomposition of the identity—see the general definition in
+(7) below. A projective decomposition of the identity is the quantum counterpart of the
+sample space of mutually exclusive possibilities essential for using standard (Kolmogorov)
+probability theory in the case of a quantum system. Note that A, B, and C are subspaces of a
+single Hilbert space, not subsystems represented by a tensor product. If the quantum particle
+possesses other degrees of freedom, these projectors are to be understood using the usual
+convention as including the identity operator on these additional degrees of freedom. Thus
+for a photon, A means that it is located in Alice’s domain, whatever may be its polarization.
+Bob can send a particular type of information λ to Alice by starting with a normalized
+reference state |ψ0⟩ = B|ψ0⟩, the particle is somewhere in his domain B, and using a unitary
+transformation Bλ acting on the subspace B + C to place it in a state
+|ψλ
+1⟩ = C|ψλ
+1⟩ = Bλ|ψ0⟩,
+(5)
+in the channel, at which point it travels through the channel to Alice. As the channel has no
+effect except to transmit the particle from one end to the other, we simplify the discussion
+(here and later) by using the same symbol for the ket that arrives at Alice’s end. She then
+applies a unitary A that does not depend on λ, for she does not know what Bob is sending,
+to empty the channel and arrive at a final state
+|ψλ
+2⟩ = A|ψλ
+2⟩ = A|ψλ
+1⟩,
+(6)
+which she can then measure or subject to further processing.
+This single-round transmission process can be carried out in a number of rounds in which
+during the n’th round Bob employs a unitary Bλ
+n acting on the B + C subspace to map an
+amplitude cn|ψ0⟩ into C, which is initially empty, and which travels to Alice, who uses a
+unitary An acting on A + C to remove it from the channel, which is then empty and ready
+for the next round. One way to visualize this is that Bob has a domain B of high dimension,
+and at the outset splits up the initial amplitude |ψ0⟩ into pieces placed in different subspaces
+of B with the help of a suitable set of beamsplitters. At round n the unitary Bλ
+n interchanges
+the appropriate subspace of B with the empty C. Alice’s A is also large, and her An maps
+whatever Bob has sent into an empty subspace reserved for this purpose. When the run is
+completed Alice can then combine the amplitudes in these different subspaces into a smaller
+space—e.g., using beamsplitters—or she can do a similar combination at the end of each
+5
+
+round. Of course Alice’s and Bob’s unitaries cannot be chosen independently; the two must
+work together to design the protocol. What is unknown to Alice is Bob’s choice of λ for a
+particular run; this is the information that she can extract at the end.
+Some multiple-time protocols employ measurements by Alice at intermediate times. In
+cases such as the original SLAZ scheme, discussed below in Sec. IV, it is possible to store the
+amplitude that could have triggered the measuring device in an empty subspace in Alice’s
+domain and put off the measurement until the protocol is finished. Of course, amplitudes
+that correspond to several measurements in succession can be combined, just as in the case
+of simultaneous transmission through several channels in parallel, as discussed earlier.
+III
+Two-way Protocols
+III.1
+Gram Matrices
+Let {Pj} be a projective decomposition of the Hilbert space identity I:
+I =
+�
+j
+Pj,
+Pj = P †
+j ,
+PjPk = δjkPj,
+(7)
+and let {|ψµ⟩}, µ = 0, 1, . . ., be a collection of kets on the same Hilbert space. The Gram
+matrix
+Gµν = ⟨ψµ|ψν⟩ =
+�
+j
+Gµν(Pj) =
+�
+j
+⟨ψµ|Pj|ψν⟩
+(8)
+is additive in that it is a sum over contributions from the different subspaces. In addition,
+Gµν is invariant (or conserved) under a unitary operation U that acts on every ket in the
+collection {|ψµ⟩}. Also, if this unitary acts on only some of the subspaces, say P1 and P2, and
+is the identity operator on the others, then while both Gµν(P1) and Gµν(P2) may change,
+their sum Gµν(P1) + Gµν(P2) remains unchanged. That Gram matrices are additive and
+conserved plays an important role in what follows.
+We shall refer to the diagonal elements Gµµ(Pj), which are non-negative, as weights.
+As these are rather like probabilities, their additivity and conservation is not surprising.
+However, that the same is true of the nondiagonal elements Gµν(Pj) with µ ̸= ν, hereafter
+referred to as overlaps, comes as something of a surprise, especially since |ψµ⟩ and |ψν⟩ may
+refer to two different runs of an experiment, one on Friday and one on Monday. Nonetheless,
+overlaps play a key role in the following analysis, not only as part of the mathematics but
+also in a surprising but useful “intuitive” way of thinking about what is going on. The
+absolute value of an overlap corresponds to a notion of fidelity in quantum information, but
+in general an overlap is a complex number, and the fact that it can be negative as well as
+positive is a key element in what follows.
+III.2
+Basic Two-Way Protocol
+In the following discussion the projective decomposition of the identity (7) that will
+concern us is {A, C, B}, where A means that the photon or other quantum particle is in
+Alice’s domain, B that it is Bob’s domain, and C in the channel connecting them. At the
+6
+
+beginning of a two-way protocol of the SLAZ type in which Bob is sending information to
+Alice of the photon amplitude is in Alice’s domain A. She initiates the run by sending some
+amplitude to Bob through the channel. He then modifies it and returns some or all of it to
+Alice, in a manner that depends on the information λ he wishes to transmit. Alice processes
+what Bob has returned, and begins the second round by again sending amplitude to Bob,
+who again returns it, etc. This can go on for N rounds, following which Alice makes a
+measurement to determine the value of λ.
+In further detail: At the beginning of round n, Alice uses a unitary An1 acting on A + C
+to map some of the amplitude in A into an empty channel C. This amplitude then flows
+through the channel to Bob, where he empties the channel into B, does some processing,
+and then maps some amplitude back into C. This flows to Alice, who empties C into A
+using a unitary An2. We assume that “flow through the channel” does not change anything,
+and hence it is convenient not to think of C as divided into close-to-Alice, close-to-Bob, and
+in-between subspaces, but simply imagine that Alice and then Bob and then Alice are acting
+on a single C subspace. Alice uses unitaries that act on A + C and are independent of λ,
+while Bob uses unitaries Bλ
+n, that depend on the information λ he wants to transmit, which
+act on C + B. Both the Alice and Bob unitaries will in general depend upon the round n,
+but Alice’s do not depend upon λ. In addition we impose the restriction that Bob’s actions
+are passive in the sense that that the magnitude of the amplitude he sends back to Alice
+in round n cannot be greater than what he has just received. This last condition clearly
+differentiates these two-way protocols from the one-way protocols of Sec. II.2.
+The requirement that Alice and Bob only employ unitary operations simplifies the anal-
+ysis. It is true that various published protocols of this type, including the original SLAZ
+version to be discussed in Sec. IV, employ nonunitary measurements at intermediate times.
+In the cases of interest to us these measurements can be replaced by unitary operations
+which allow the measurements to be put off until the end of the run, in a manner indicated
+in Sec. II.2 and employed in the discussion in Sec. IV.
+To quantify the channel usage for these protocols we use the notions of Cost, equal to the
+absolute square of the amplitude for a single use of the channel, and total Cost for the sum of
+the Costs involved in a single experimental run, as in Sec. II.1, see (2). An important issue
+connected to claims that these protocols are counterfactual has to do with the difference
+between Cost and probability, as will be discussed later for the SLAZ protocol in Sec. IV—
+the importance of this has already been noted in Sec. II.1. In particular we will be interested
+in identifying protocols that minimize the overall Cost, as in the example discussed next.
+III.3
+Sending One Classical Bit
+In the simplest SLAZ protocol Bob wants to send a single classical bit, λ = 0 or 1, to
+Alice. At the start all of the amplitude is in A for both a λ = 0 and a λ = 1 run, so all four
+of the initial Gram matrix elements Gµν
+0 (A), µ and ν equal to 0 or 1, are equal to 1. The
+goal is that after N rounds the result will be
+Gµν
+N (A) = δµν,
+(9)
+so that Alice can determine the value of λ Bob has sent by making a measurement in an
+appropriate basis. Thus the desired change is that during the course of the run the overlaps,
+7
+
+the off diagonal elements G01(A) and G10(A), decrease from 1 to 0, while the weights G00(A)
+and G11(A), remain equal to 1.
+At this point it is worth noting that if both weights are not maintained—for example if
+at the end G00(A) = 1 while G11(A) = G01(A) = 0, Alice can still extract the value of λ by
+measuring whether or not the photon is in the state |ψ0⟩. Let us call this, for want of a better
+term, a partial protocol in contrast to a full protocol that results in (9). A partial protocol
+can be used for one-way transmission, and the obvious advantage is that it costs nothing to
+transmit λ = 1. A possible disadvantage is that when Alice’s measurement reveals nothing
+it could be because of some failure in the channel or in the measuring device. In the present
+discussion we focus on full protocols.
+A very simple way to implement such a protocol is that on the very first step Alice sends
+the entire amplitude to Bob, with a Cost of 1 for this use of the channel. Bob then simply
+modifies this using the unitary Bλ and sends it back to Alice, either in one round or several
+rounds, with Alice sending nothing back. The Cost for using the channel in the Bob-to-Alice
+direction is also 1, see the discussion in Sec. II.2. Hence a total Cost of 2 for the protocol
+as a whole. Notice that since there is no restriction on Bλ this rather trivial protocol can be
+used to send “quantum” information. From the perspective of Cost, two-way protocols of
+the kind under discussion are interesting because a classical bit, λ = 0 or 1, can be sent at
+a total Cost of 1 rather than 2. And as shown below in Sec. III.4, the product of the Costs
+for λ = 0 and 1 cannot be less than 1.
+To discuss the successive steps in protocols that optimize the Cost, we need an appropriate
+notation. We will represent kets as row vectors as in the following example
+|ψ⟩ = |a; c; b⟩ = |a1, a2, a3; c1, c2; b1, b2⟩
+(10)
+where the dimensions of the A, B, and C subspaces are d(A) = 3, d(B) = 2 and d(C) = 2.
+Note that we are dealing with a direct sum of subspaces, A ⊕ B ⊕ C, not a tensor product
+of subsystems. In much of what follows, B is empty or can be ignored, so |a; c⟩ will suffice;
+this and other minor variants in notation should be self-explanatory.
+Let us start with an extremely simple one-round full protocol with d(A) = 2, d(C) = 1.
+It consists of the following steps:
+|a1, a2; c⟩ = |1, 0; 0⟩ → |1/
+√
+2, 0; 1/
+√
+2⟩
+⇒ |1/
+√
+2, 0; (−1)λ/
+√
+2⟩ → |1/
+√
+2, (−1)λ/
+√
+2; 0⟩,
+(11)
+where 0 means this amplitude is equal to zero; do not confuse it with the label 0 for one of
+the two orthogonal states of a qubit. Here → indicates the action of a unitary on A + C
+carried out by Alice, and ⇒ a λ-dependent unitary on C carried out by Bob. The action by
+Bob could involve intermediate steps requiring the B subspace, but its net effect is only to
+change the contents of C, so there is no need to include B in the discussion.
+In words: At the outset all of the amplitude is in Alice’s A, a1 = 1. She maps half (in
+the sense of the absolute square) of it into C and sends it to Bob, who either sends it back
+unchanged in order to transmit λ = 0, or with the opposite phase to send λ = 1. Alice then
+empties the channel into the a2 position, using a unitary on A + C that is independent of λ,
+as it simply requires interchanging two subspaces. A final measurement by Alice determines
+which of the two orthogonal states is present in A, and thus which bit Bob was sending.
+8
+
+Next consider what is happening to the Gram matrices Gµν(A) and Gµν(C) during the
+successive steps. In particular, the overlap G01(A) is equal to 1 at the outset, and the first
+step reduces it to 1/2 by placing 1/2 in C. Bob’s action changes G01(C) from +1/2 to −1/2,
+and this negative contribution to the overlap moves back into A when Alice empties the
+channel, leading to the desired G01(A) = 0. On the other hand, whereas the weight G00(A)
+is reduced to 1/2 during the first step, Bob’s action does not change the sign of G00(C), so in
+the final step Alice moves this weight back to its initial value of 1, and similarly for G11(A).
+Thus the goals of a full protocol have been achieved.
+The Costs of using the channel are easily evaluated: 1/2 for the Alice-to-Bob step and
+the same for Bob-to-Alice, for a total Cost of Qλ = 1, the same for λ = 0 and 1. These
+satisfy the rigorous lower bound worked out below in Sec. III.4, so this protocol is optimal
+if one uses total Cost as an appropriate measure of channel usage.
+This protocol is easily extended to an equally efficient version involving N rounds, N any
+positive integer. Let
+ǫ = 1/2N,
+(12)
+and for the first, n = 1, round replace (11) with
+|1, 0; 0⟩ → |
+√
+1 − ǫ, 0; √ǫ ⟩ ⇒ |
+√
+1 − ǫ, 0; (−1)λ√ǫ ⟩ → |
+√
+1 − ǫ, (−1)λ√ǫ; 0⟩,
+(13)
+while for round n + 1,
+|
+√
+1 − nǫ, (−1)λ√nǫ; 0⟩ → |
+�
+1 − (n + 1)ǫ, (−1)λ√nǫ; √ǫ⟩
+⇒ |
+�
+1 − (n + 1)ǫ, (−1)λ√nǫ; (−1)λ√ǫ ⟩ → |
+�
+1 − (n + 1)ǫ, (−1)λ�
+(n + 1)ǫ; 0⟩,
+(14)
+where it is straightforward to show that there exists a λ-independent unitary for the last
+step. The final result at the end of round N is the same as in (11), the case in which N = 1,
+and again the total Cost is Q0 = Q1 = 1, independent of λ. One can also let ǫ depend on n,
+thus ǫn > 0 for round n, subject to the condition
+�
+n
+ǫn = 1/2,
+(15)
+and the Cost is again equal to 1.
+There are other protocols with larger Costs which may have some practical advantage.
+Thus rather than a scalar amplitude, Alice might use photon polarization, say horizontal
+H, which Bob could return as H to send λ = 0 or rotate to vertical V to send λ = 1.
+In this case the Costs are Q0 = Q1 = 2, so twice that for an optimal one-way protocol.
+However, there is now no need to maintain a particular phase relation between what is in
+Alice’s domain and what is available to Bob during each round. If polarization is easier to
+maintain than phase—one leaves that up to the experts—one could imagine the added Cost
+being worthwhile if Alice has a large apparatus capable of generating single photons, while
+Bob, off on a trip to spy on Eve, needs only something easily carried in a suitcase.
+The protocol used in SLAZ, in which Bob returns the amplitude for λ = 0, but absorbs
+it or feeds it to a measuring apparatus for λ = 1, looks less promising. Because the λ = 1
+weight only moves from Alice to Bob it is difficult to have G11(A) = 1 at the end of the
+protocol. In fact SLAZ, discussed in Sec. IV, employs a clever trick (“Zeno effect”) to get
+around this problem, albeit at the cost of a large number of rounds to keep the probability
+of failure small, and a large channel usage Cost for one of the bits.
+9
+
+III.4
+Lower Bound on Costs
+The additivity and conservation properties of the Gram matrix Gµν introduced in Sec. III.1
+will now be used to obtain lower bounds on the total Cost of two-way protocols of the sort
+exemplified by, but not limited to, the case of 1 classical bit discussed above in Sec. III.3.
+Using the |a; c⟩ notation of (10)—the b entry is not needed in the following discussion—round
+n of an N round protocol consists of the following steps carried out on A + C:
+|aµ; 0⟩n → |¯aµ; cµ⟩n ⇒ |¯aµ; ˆcµ⟩n → |aµ; 0⟩n+1.
+(16)
+Here µ labels the bit which Bob is transmitting during this run. Thus after Alice uses a
+unitary An1 on A + C to move some amplitude, |cµ⟩n into an initially empty channel. Bob
+applies a unitary Bµ
+n to C + B, leading to an amplitude |ˆcµ⟩n—note the hat added to c—in
+the channel. If Bob’s action is passive, as assumed in Sec. III.3 (and in the later discussion
+of SLAZ in Sec. IV), one would have
+∥ˆcµ∥n ≤ ∥cµ∥n,
+(17)
+but this conditions is actually not needed to obtain the general results and inequalities given
+below, which thus apply equally to one-way multi-time transmission. As a final step Alice
+employs a unitary An2 on A + C to empty the channel by placing its amplitude into A. It is
+important that Alice’s unitaries An1 and An2, unlike Bob’s Bµ
+n, do not depend upon µ, which
+can be different in different runs of the experiment.
+The change in the Gram matrix associated with A during round n is given by
+Gµν
+n+1(A) − Gµν
+n (A) = ⟨aµ|aν⟩n+1 − ⟨aµ|aν⟩n = ⟨ˆcµ|ˆcν⟩n − ⟨cµ|cν⟩n,
+(18)
+where ⟨aµ|aν⟩n is the inner product of |aµ⟩n and |aν⟩n. The equality follows from the fact that
+Gµν(A + C) is invariant under An1 and An2, and additive: Gµν(A + C) = Gµν(A) + Gµν(C).
+To discuss the total change during N rounds, n = 1, 2, . . . N, it is convenient to define
+|Cµ⟩ := {|cµ⟩1, |cµ⟩2, . . . |cµ⟩N},
+| ˆCµ⟩ := {|ˆcµ⟩1, |ˆcµ⟩2, . . . |ˆcµ⟩N}
+(19)
+with inner products
+⟨Cµ|Cν⟩ =
+N
+�
+n=1
+⟨cµ|cν⟩n,
+⟨ ˆCµ| ˆCν⟩ =
+N
+�
+n=1
+⟨ˆcµ|ˆcν⟩n.
+(20)
+Summing (18) over N rounds yields the following formula
+∆Gµν(A) = Gµν
+N (A) − Gµν
+0 (A) = ⟨ ˆCµ| ˆCν⟩ − ⟨Cµ|Cν⟩,
+(21)
+for the total change in the A portion of the Gram matrix during the full protocol. This
+quantity is bounded by
+|∆Gµν(A)| ≤ |⟨ ˆCµ| ˆCν⟩| + |⟨Cµ|Cν⟩| ≤ ∥ ˆCµ∥ · ∥ ˆCν∥ + ∥Cµ∥ · ∥Cν∥
+(22)
+using the norm ⟨Cµ|Cµ⟩ = ∥Cµ∥2.
+10
+
+Next define the total Cost Kµ for Alice-to-Bob and ˆKµ for Bob-to-Alice uses of the
+channel, with Qµ their sum:
+Kµ = ⟨Cµ|Cµ⟩ = ∥Cµ∥2,
+ˆKµ = ⟨Cµ|Cµ⟩ = ∥ ˆCµ∥2,
+Qµ = Kµ + ˆKµ.
+(23)
+Combining (22) and (23) gives
+|∆Gµν(A)| ≤
+√
+KµKν +
+�
+ˆKµ ˆKν ≤
+�
+QµQν.
+(24)
+This yields an upper bound
+∆Gµµ(A) ≤ Qµ
+(25)
+for a non-negative diagonal weight, and for the off-diagonal overlap:
+|∆Gµν(A)| ≤
+�
+QµQν.
+(26)
+In the particular case of the 1-bit two-way protocol, Sec. III.3, the aim is to reduce G01(A)
+from its initial value of 1 to 0 after N rounds. Setting µ = 0 and ν = 1 in (26), we see that
+to achieve this result it is necessarily the case that the Costs Q0 and Q1 for sending bits
+λ = 0 and λ = 1 must satisfy the condition
+Q0Q1 ≥ 1.
+(27)
+This is satisfied as an equality with Q0 = Q1 = 1 for the specific protocols discussed in
+Sec. III.3, which shows that they are optimal if total Cost is used as a measure. For more
+general protocols there is no reason to expect that the two Costs will be equal, and in that
+case if, say, the Cost for λ = 1 is made very small, that for λ = 0 must be very large. This
+is in fact the case for the original SLAZ protocol, as discussed below in Sec. IV, which thus
+provides an interesting illustration of such a tradeoff.
+IV
+The SLAZ Protocol
+IV.1
+Description of the Protocol
+The original SLAZ protocol differs from the simpler situation discussed in Sec. III.3 in
+two respects. First, it has a hierarchical structure: there are a large number M of outer
+rounds or cycles, each of which consists of a large number N of inner rounds or cycles, and
+the protocol will succeed with high probability provided
+1 ≪ M ≪ N.
+(28)
+Second, while Bob sends a bit λ = 0 by reflecting the amplitude sent by Alice back into
+the channel, for λ = 1 he simply empties the channel, which can be described as a unitary
+operation in which the C amplitude is placed in Bob’s subspace B. In addition, the original
+SLAZ protocol and some of its modifications involve measurements at intermediate times,
+and these will be replaced in the discussion below by unitary operations in the manner
+suggested at the end of Sec. II.2.
+11
+
+We use a notation
+|ψ⟩ = |a1, a2, a3, a4; c; b⟩
+(29)
+of the form introduced in (10), where the aj are scalar amplitudes in Alice’s domain A =
+A1+A2+A3+A4, c is the amplitude the channel C, and b is in Bob’s domain B. Here capital
+letters are used to denote subspaces and the corresponding projectors, while lower case letters
+indicate (in general complex) scalar amplitudes. While A4 and B are one-dimensional, one
+can also make these larger spaces for reasons that will appear during the discussion. An
+abbreviated notation is often convenient: |a2, a3⟩ in the case of a unitary acting on A2 + A3
+while all the other amplitudes remain unchanged.
+Central to the discussion are unitary operators that represent a rotation by an angle θ
+on a 2-dimensional space:
+R(θ)|α, β⟩ = |α cos θ − β sin θ, α sin θ + β cos θ⟩.
+(30)
+In particular, RM and RN, defined in terms of small angles, play a central role:
+RM := R(θM),
+θM := π/(2M),
+RN := R(θN),
+θN := π/(2N).
+(31)
+Note in particular that
+(RM)M = (RN)N = R(π/2);
+R(π/2) |α, β⟩ = | − β, α⟩.
+(32)
+In view of the fact that θN is a small angle, the following approximations turn out to be
+userful:
+cos θN ≈ exp[−θ2
+N/2] = exp[−π2/(8N2)] ≈ 1 − π2/(8N2),
+(cos θN)N ≈ exp[−π2/(8N)] ≈ 1 − π2/(8N) ≈ 1,
+(33)
+and similarly if N is replaced by M.
+These approximations are useful for understanding the overall structure of the protocol,
+which is the following. At the beginning of outer round m, 1 ≤ m ≤ M, RM is applied to
+A1 + A2 to yield,
+|a1, a2⟩λ = RM|¯a1, ¯a2⟩λ,
+(34)
+where ¯a1 and ¯a2 are the values of these amplitudes at the end of the previous outer round. In
+general they depend upon which bit λ = 0 or 1 is being transmitted, whence the superscript
+label, even though Alice’s operations do not depend upon λ. The very first outer round
+m = 1 begins by applying (34) to the starting state (29) with a1 = 1 and all the other
+amplitudes equal to zero.
+The initial step (34) of outer round m is followed by a sequance of N inner rounds, each
+involving the following steps, here displayed using the type of notation employed in Sec. III.3,
+but now with reference to the subspace A2 + A3 + C.
+|a2, a3; c = 0⟩ → |a′
+2, a′
+3; c = 0⟩ → |a′
+2, 0; a′
+3⟩
+⇒ |a′
+2, 0; (1 − λ)a′
+3⟩ → |a′
+2, (1 − λ)a′
+3; 0⟩,
+(35)
+where
+|a′
+2, a′
+3⟩ = RN|a2, a3⟩.
+(36)
+12
+
+In words, Alice applies the unitary rotation RN, (31), to A2 +A3, and then maps A3 into the
+empty channel. Next comes Bob’s action, indicated by ⇒, to either reflect the amplitude a′
+3
+back into C if he is sending λ = 0, or shift it into his domain B, leaving the channel empty
+if sending λ = 1. Alice, who does not know the value of λ, maps whatever is in the channel
+back into A3 by a unitary that simply exchanges the contents of A3 and C, and then begins
+the next inner round. The result of N inner rounds in succession is
+|a2, a3⟩ →
+�
+|0, a2⟩ for λ = 0,
+|(cos θN)Na2, 0⟩ ≈ |a2, 0⟩ for λ = 1.
+(37)
+where the λ = 1 approximation is justified when N is very large, see (33).
+Following the N inner rounds Alice completes this outer round by applying a unitary to
+A3 + A4 that empties the contents of A3 into A4. For λ = 1, a3 = 0, (37), so this emptying
+step is trivial, while for λ = 0 it is nontrivial, and plays a signficant role in understanding
+the true Costs of the protocol. In the original SLAZ protocol this emptying step is replaced
+by a measurement, but instead of a measurement one can just as well let the amplitudes
+accumulate in A4, which is the perspective used here.
+At the end of the protocol after
+completing M outer rounds the final result is
+λ = 0 : |a1 = 1 − r1, a2 = 0, a3 = 0, a4 = r4, c = 0, b = 0⟩
+λ = 1 : |a1 = s1, a2 = 1 − s2, a3 = 0, a4 = 0, c = 0, b = sb⟩,
+(38)
+where the quantities denoted by rj and sk are small corrections, of order 1/M or M/N
+If these are ignored, all the amplitude is in A1 for λ = 0 or A2 for λ = 1, and a simple
+measurement allows Alice to determine which bit Bob sent.
+IV.2
+Calculation of Costs and Overlap
+It is fairly straightforward to work out the Costs for the SLAZ protocol using approxi-
+mations justified by 1 ≪ M ≪ N, and the results are summarized in Sec. IV.3 below. We
+begin with the case λ = 1. If one ignores small quantities, the nonzero components of |ψ⟩m
+at the beginning and at the end of outer round m are
+a1 = cos(mθM),
+a2 = sin(mθM),
+(39)
+and since MθM = π/2, at the end of outer round M the result is the λ = 1 line in (38).
+The probability that the photon arrives in B during outer round m—the probability that
+Bob will detect it if he uses a measuring device—is the sum of the absolute squares of the
+amplitudes in the channel C in the N inner rounds, as this is an incoherent process:
+N(sin(mθM))2(sin(θN))2 ≈ (π2/4)(sin(mθ/M))2/N.
+(40)
+Summing over m gives the total probability
+K1 = Q1 = (π2/8)(M/N).
+(41)
+that the photon will end up in Bob’s domain by the end of the protocol, which is the same
+as the total Cost for λ = 1.
+13
+
+In the case λ = 0, any amplitude placed by Alice in C is immediately returned by Bob,
+and at the end of each outer round is emptied into a4, so that at the end of outer round m
+the state is
+|ψ⟩m = |a1 = (cos θM)m, a2 = 0, a3 = 0, a4, c = 0, b = 0⟩.
+(42)
+For m = M this is (38) with r1 = (π2/8M). Thus at the end of the protocol a2, a3, c and
+b are strictly zero. The Cost associated with inner round n—note that the channel is used
+twice—is
+2[sin θM · sin(nπ/2N)]2.
+(43)
+Summing over n gives a total of (π2/4)(N/M2) for each outer round, and hence for M outer
+rounds a total Cost of
+Q0 = (π2/4)(N/M2).
+(44)
+To compute the total change in overlap ∆G01(A), note that since for λ = 1 Bob does not
+return an amplitude, only the ⟨Cµ|Cν⟩ term in (21) contributes. The contribution for inner
+round n of outer round m is the product of the factors
+[sin θM sin(nθN)] · [sin(mθM) sin θN]
+(45)
+corresponding to λ = 0 and 1. Summing them yields
+(sin θM sin θN)
+M,N
+�
+m,n
+sin(mθM) sin(nθN) = (π2/4MN)(4MN/π2) = 1,
+(46)
+and hence
+∆G01(A) = −1,
+(47)
+as expected.
+IV.3
+Discussion of Costs and Probabilities
+To summarize the results of Sec. IV.2: The total Costs Q0 and Q1 for λ = 0 and 1 are:
+Q0 = (π2/4)(N/M),
+Q1 = (π2/8)(M/N),
+Q0Q1 ≈ 3.044.
+Q0/Q1 = 2N2/M2.
+(48)
+Given that M ≪ N, Q1 is miniscule, Q0 is enormous, while their product is of order 1, and
+satisfies the rigorous bound (27). The case λ = 1 is the easiest to understand. Since Bob
+does not return the amplitude put into the channel by Alice, the Bob-to-Alice Cost ˆK1 is
+zero. The Alice-to-Bob Cost is |sb|2 in (38), i.e., the probability that at the very end the
+photon is in Bob’s domain. The physical reason for this is that the process by which the
+amplitude gets there is incoherent, no quantum interference, since no amplitude goes back
+through the channel. Bob could either accumulate these amplitudes until the end of the
+protocol and then measure to see if the photon is in B, or carry out a measurement at the
+end of each inner round; in either case the probabilility of his detecting the photon is |sb|2 in
+(38). The situation is analogous to the use of intermediate time measurements in a one-way
+protocol as discussed at the end of Sec. II.2.
+The enormous Cost Q0 for λ = 0 comes about because Bob repeatedly returns the
+amplitude sent by Alice in a coherent process. While the amplitude bouncing back and
+14
+
+forth through the channel is relatively small, of order 1/M, multiplying its absolute square
+by 2N, the number of times this amplitude is is in the channel during each outer round,
+leads to a Cost of order N/M2 for each outer round, and hence a total of order N/M for the
+complete process.
+Clearly the large value of Q0 means the claim that protocol is counterfactual cannot
+be maintained if Cost is used as a criterion for channel use, so it is worth discussing how
+the authors of SLAZ reached a different conclusion. In essence their reasoning was based
+on the small value of the amplitude in A3 at the end of an outer round just before it is
+transferred to A4, as per the discussion in Sec. IV.1. The absolute square of this amplitude
+is the probability that the corresponding detector D3 in Fig. 2(b) in the SLAZ paper will
+be triggered. This amplitude was earlier oscillating back and forth inside the subspace with
+projector S = A2+A3+C, and hence it is reasonable to assume that if this detector triggers,
+the photon was earlier in S during all N inner rounds making up this particular outer round1.
+As this probability is of order 1/M2, the probability that one of the D3 detectors triggers
+during the M outer rounds that make up a given run is of order 1/M, and hence small.
+There are two serious objections to using this small probability to justify the claim that
+the protocol is counterfactual: one classical and the other quantum. Let us start with the
+former. During a particular outer round the photon amplitude in a λ = 0 run rattles back
+and forth inside S a total of N times, and in particular it is in C a total of 2N times.
+Consider a stochastic classical protocol for transmitting information in which most of the
+time Alice and Bob exchange no information at all. However, with a small probability ǫ
+Alice sends a little white ball into the channel leading to Bob, who colors it green or red
+and sends it back to Alice to convey one bit of information. She records the color, paints
+the ball white, and returns it to Bob who again colors it to send a second bit, and so forth,
+for a total of N rounds. The average rate of transmitting information is Nǫ bits, and one
+cannot simply throw away the factor of N and claim that this protocol is in some sense
+‘counterfactual’.
+The quantum difficulty has to do with what can be inferred from the probability that
+the photon was in S = A2 + A3 + C during the inner rounds that make up a particular outer
+round. One may be tempted to use classical reasoning and assume that the probabilities of
+being in each of the mutually exclusive regions, A2, A3, and C, that combine to make up S
+are well-defined and sum to the probability of being in S. But in the presence of quantum
+interference this sort of reasoning is invalid and leads to paradoxes. See the discussion of
+parallel channels in Sec. II.1.
+V
+Conclusion
+The original SLAZ proposal has motivated a large number of papers; see the extensive
+bibliographies in [3, 4]. Merely trying to summarize them, much less provide a detailed re-
+view, lies outside the scope of the present paper. Broadly speaking, this literature consists
+of modifications, extensions, or improvements of the original SLAZ scheme; along with crit-
+icisms of the claim that these protocols are counterfactual and replies to such criticisms. It
+is hoped that the following rather brief comments will provide some orientation.
+1This assumption can be justified using Consistent Histories; see the discussion of measurements in [5,6]
+15
+
+Significant extensions of the original SLAZ scheme by the last three members of the
+original collaboration include: the use of a phase change rather than absorption to transmit
+the λ = 1 bit [7]; a scheme to transmit quantum states by multiple iterations of the original
+SLAZ scheme [8]; using many photons in place of a single photon to transmit a classical
+bit [4]. These and others are certainly interesting ideas from the perspective of transmitting
+quantum information, and worth further exploration.
+On the other hand, in these and all other extensions or modifications of SLAZ this author
+has examined, the claim that the protocol is “counterfactual,” in the sense that the total use
+of a quantum channel is negligible in the asymptotic limit, is subject to the same objections
+discussed in Sec. IV.3: An improper use of probabilistic reasoning in a situation where
+quantum interference means probabilities cannot be defined, and where even in a classical
+situation Cost would be better than probability as a measure of channel usage. The total
+Cost remains finite in the asymptotic limit of a very large number of steps, which means that
+counterfactual claims should be dropped. Doing so will aid, not hinder, the serious study of
+these interesting quantum schemes for transmitting information.
+Shortly after the original SLAZ publication, Vaidman published a Comment [9] claiming
+that in the λ = 0 case in which Bob reflects the amplitude rather than absorbing it, the
+photon which was later (with high probability) detected by Alice must at an earlier time have
+been in the channel C. In their Reply [10] the SLAZ authors pointed out this way of reasoning
+about events at an intermediate time in the presence of quantum interference was invalid,
+and leads to paradoxes, a position supported by the analysis in Sec. IV.2 above. However,
+they then repeated their original counterfactual claim which itself is based on a defective
+understanding of probabilities at an intermediate time. A later and much more extended
+criticism of counterfactuality claims by Vaidman [11] suffers from the same difficulty as his
+earlier Comment.
+Some years later Aharonov and Vaidman [12] claimed to have found a scheme of the
+general SLAZ type which is genuinely counterfactual.
+However, when measurements or
+absortion of a photon at intermediate times are replaced by unitary processes—mapping
+amplitude into an empty subspace reserved for this purpose, as discussed in Sec. IV.1—the
+inequality in Sec. III.4 applies to this case and undermines the counterfactual claim. The
+fundamental difficulty with such claims is that the Hilbert space projector which identifies the
+position of a particle at some intermediate time does not commute with the one representing
+the quantum state evolving unitarily in time.
+The most significant contributions of the present paper to the analysis of SLAZ-type
+protocols is the use of Cost as a measure of channel usage, and the use of Gram matrices for
+discussing information transfer at intermediate times in the presence of quantum interference.
+In particular, the fact that these Gram matrices are additive over subspaces and invariant
+(“conserved”) under unitary time transformations, plays a key part in the discussions in
+Sec. III. A rather surprising feature is the role of off-diagonal elements, “overlaps”, as a type
+of information measure which, unlike most such measures, is not in general positive. That it
+can be negative plays a very signficant part in understanding its intuitive role in information
+transfer. That its total change on Alice’s side must be −1 during the course of a successful
+protocol is confirmed for the SLAZ protocol in Sec. IV.2.
+This use of Gram matrices requires that the intermediate time steps be unitary. In the
+case of SLAZ, measurements at intermediate times can be eliminated by mapping photon
+16
+
+amplitude into empty subspaces, and this can be achieved in certain other cases, e.g., the
+Aharonov and Vaidman protocol [12]. However, it is less clear whether something similar
+could be done in a case in which, for example, Alice uses measurements at intermediate times
+to change later steps in the protocol in hopes of reducing the total Cost. This author believes
+that such an improvement is impossible, because measurements themselves are quantum
+processes whose description simply requires a large enough Hilbert space in Alice’s domain
+[13]. But this has not yet been demonstrated.
+And what is special about classical information? Sending an arbitrary one-qubit quantum
+state from Alice to Bob using the 2-way protocol of Sec. III.3 could be done with a Cost of 2,
+which is to say twice that of simply using a 1-way protocol from Bob to Alice. That this is
+the minimum seems likely, but has not been demonstrated. What about a two-way protocol
+with all the amplitude starting on Alice’s side, with the aim of a perfect transmission of each
+of two specified nonorthogonal states from Bob to Alice—what would be the minimum total
+Cost?
+An interesting feature of the original SLAZ protocol is the enormous ratio 2N2/M2, see
+(48), of the Costs to transmit λ = 0 and 1, in contrast to the relatively simple protocols
+discussed in Sec. III.3 for which the ratio is 1. Because the success of SLAZ depends upon N
+being much larger than M, this large ratio presumably has something to do with Bob’s not
+sending anything back through the channel when λ = 1. Might there be some interesting
+physical principles, in addition to the Zeno effect, hiding here and waiting to be explored?
+In conclusion it is hoped that the thinking and tools employed in this paper will be useful
+for studying other problems of quantum information at intermediate times in situations
+where the careless use of ill-defined probabilities generates paradoxes rather than physical
+understanding. In particular, information transfer among three or more parties, of current
+interest in the study of quantum networks, might benefit from the sort of analysis used here.
+Acknowledgements
+The author expresses his appreciation to Carnegie-Mellon University and its Physics
+Department for continuing support of his activities as an emeritus faculty member.
+References
+[1] Hatim Salih, Zheng-Hong Li, M. Al-Amri, and M. Suhail Zubairy. Protocol for di-
+rect counterfactual quantum communication.
+Phys. Rev. Lett., 110:170502, 2013.
+arXiv:1206.2042.
+[2] Johann von Neumann. Mathematische Grundlagen der Quantenmechanik. Springer-
+Verlag, Berlin, 1932. English translation by R. T. Beyer: Mathematical Foundations of
+Quantum Mechanics, Princeton University Press, Princeton, New Jersey (1955).
+[3] Jonte R. Hance, James Ladyman, and John Rarity. How quantum is quantum counter-
+factual communication? Found. Phys., 51:12, 2021. arXiv:1909.07530.
+17
+
+[4] Zheng-Hong Li, Shang-Yue Feng, M. Al-Amri, and M. Suhail Zubairy. Direct counter-
+factual quantum communication protocol beyond a single photon source. Phys. Rev. A,
+106:032610, 2022. arXiv:2202.03935.
+[5] Robert B. Griffiths. What quantum measurements measure. Phys. Rev. A, 96:032110,
+2017. arXiv:1704.08725.
+[6] Robert B. Griffiths.
+The Consistent Histories Approach to Quantum Mechanics.
+Stanford Encyclopedia of Philosophy, 2019.
+https://plato.stanford.edu/entries/qm-
+consistent-histories/.
+[7] Zheng-Hong Li, M. Al-Amri, and M. Suhail Zubairy. Direct quantum communication
+with almost invisible photons. Phys. Rev. A, 89:052334, 2014.
+[8] Zheng-Hong Li, M. Al-Amri, and M. Suhail Zubairy. Direct counterfactual transmission
+of a quantum state. Phys. Rev. A, 92:052315, 2015.
+[9] Lev Vaidman. Tracing the past of a quantum particle. Phys. Rev. A, 89:024102, 2014.
+arXiv:1312.7566.
+[10] Hatim Salih, Zheng-Hong Li, M. Al-Amri, and M. Suhail Zubairy. Salih et al. reply.
+Phys. Rev. Lett., 112:208902, 2014. arXiv:1404.5392.
+[11] L. Vaidman.
+Counterfactuality of ‘counterfactual’ communication.
+J. Phys. A,
+48:465303, 2015. arXiv:1410.2723.
+[12] Yakir Aharonov and Lev Vaidman.
+Modification of counterfactual communication
+protocols that eliminates weak particle traces.
+Phys. Rev. A, 99:010103, 2019.
+arXiv:1805.10634.
+[13] For a consistent quantum-mechanical description of the measuring process, see [5], the
+relevant sections of [6], and Chs. 17 and 18 of [14].
+[14] Robert B. Griffiths. Consistent Quantum Theory. Cambridge University Press, Cam-
+bridge, U.K., 2002. http://quantum.phys.cmu.edu/CQT/.
+18
+

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+arXiv:2301.05112v1  [gr-qc]  12 Jan 2023
+GWitchHunters: Machine Learning and citizen science
+to improve the performance of Gravitational Wave
+detector
+Massimiliano Razzanoa,b,1,∗, Francesco Di Renzoa,b, Francesco Fidecaroa,b,
+Gary Hemmingc, Stavros Katsanevasc
+aDepartment of Physics, University of Pisa, Largo B. Pontecorvo 3, Pisa, I-56127
+bINFN Section of Pisa, Largo B. Pontecorvo 3, Pisa, I-56127
+cEuropean Gravitational Observatory (EGO),Via E. Amaldi, 5, Cascina,I-56021
+Abstract
+The Gravitational waves have opened a new window on the Universe and
+paved the way to a new era of multimessenger observations of cosmic sources.
+Second-generation ground-based detectors such as Advanced LIGO and Ad-
+vanced Virgo have been extremely successful in detecting gravitational wave
+signals from coalescence of black holes and/or neutron stars. However, in
+order to reach the required sensitivities, the background noise must be inves-
+tigated and removed. In particular, transient noise events called “glitches”
+can affect data quality and mimic real astrophysical signals, and it is there-
+fore of paramount importance to characterize them and find their origin,
+a task that will support the activities of detector characterization of Virgo
+and other interferometers. Machine learning is one of the most promising
+approaches to characterize and remove noise glitches in real time, thus im-
+proving the sensitivity of interferometers. A key input to the preparation of
+a training dataset for these machine learning algorithms can originate from
+citizen science initiatives, where volunteers contribute to classify and analyze
+signals collected by detectors. We will present GWitchHunters, a new citi-
+zen science project focused on the study of gravitational wave noise, that has
+been developed within the REINFORCE project (a ”Science With And For
+Society” project funded under the EU’s H2020 program). We will present
+∗Corresponding author
+Email address: massimiliano.razzano@unipi.it (Massimiliano Razzano)
+1on behalf of the REINFORCE Consortium
+Preprint submitted to Nuclear Instruments and Methods in Physics Research AJanuary 13, 2023
+
+the project, its development and the key tasks that citizens are participating
+in, as well as its impact on the study of noise in the Advanced Virgo detector.
+Keywords:
+gravitational waves, machine learning, citizen science
+PACS: 04.20.–q, 04.30.Tv,
+2000 MSC: 83C35,
+1. Introduction
+Gravitational wave physics is opening an entire new window on the Uni-
+verse. Since their discovery in 2015 [1], the Advanced LIGO [2] and Advanced
+Virgo [3] detectors have carried on three observing runs (O1, O2, O3) and
+unveiled 90 signals produced by the coalescence of compact objects, mostly
+binary black hole with a small fraction of neutron star and/or black hole
+binaries [4].
+Advanced LIGO and Virgo are second-generation laser interferometers with
+Fabry-Perot cavities hosted in km perpendicular arms, that are capable of
+detecting the tiny deformations induced in the fabric of spacetime by the
+passage of gravitational waves. In order to improve the sensitivity of the
+detectors, there is a continuous effort to reduce the background noise due to
+local disturbances. In particular, at low frequencies the noise is dominated by
+seismic and Newtonian noise, while at mid frequencies the main component
+is related to the thermal noise and at high frequencies the noise is mostly
+related to quantum effects.
+The activity of detector characterization and
+noise hunting in gravitational wave detectors is focused on the investigation
+of stationary and non stationary noise sources. In particular, non station-
+ary transient noise events called glitches are of particular importance in the
+noise studies. In fact, glitches can affect data quality and stability and mimic
+real astrophysical signals, thus reducing the effective duty cycle of interfer-
+ometers. The classification and characterization of glitches is therefore key
+to understand the origin of noise in detector. However, glitches have com-
+plex temporal signatures, that make difficult to classify them using standard
+methods. Various works have shown that Machine Learning methods can be
+promising for the classification of glitches [5, 7]. In particular, images built
+from the time-frequency spectrograms of glitches are very effective in show-
+ing the complex morphology of glitches and can be easily given in input to
+machine learning algorithms, including deep convolutional neural networks
+[6]. A possible approach to this problem is based on supervised learning,
+2
+
+that requires large number of labeled glitch samples, that could be produced
+by dedicated citizen science initiatives, where volunteers look at images and
+clssify them.
+A successful example of this method is provided by Gravi-
+tySpy2, a citizen science project focused on the classification of glitches in
+LIGO and Virgo[8]. Here we present GWitchHunters3, a new citizen science
+project complementary to GravitySpy and aimed at improving sensitivity of
+gravitational wave detectors combining citizen science and machine learning.
+2. The REINFORCE Project
+GwitchHunters has been developed within the Research Infrastructures
+FOR Citizens in Europe (REINFORCE) project4. REINFORCE is a Re-
+search & Innovation Project, supported by the EU H2020 SWAFS “Science
+with and for Society” work programme and aimed at creating a series of
+cutting-edge citizen science projects on Frontier Physics research, with the
+goal of engaging >100,000 citizens. REINFORCE is based on four citizen
+science demonstrators focused Gravitational Waves (GWitchHunters), Astro-
+physical neutrinos (Deep Sea Explorers), High Energy Physics (New Particle
+Search at CERN) and muon-based tomography (Cosmic Muon Images). All
+demonstrators are hosted on Zooniverse [9], the world leading platform for
+citizen science projects.
+3. Overview of GWitchHunters
+GwitchHunters has been officially launched on Zooniverse in November
+2021 after a dedicated review phase and offers to citizens a set of different
+tasks of increasing difficulty. Data are presented as spectrograms and come
+from the Virgo O3 run. A Playground task is specifically devoted to learning
+the basics of glitch morphology and its classification. Three other levels offer
+(1) the possibility to classify glitches among a larger set of classes, (2) localize
+the glitches in the time-frequency space, and (3) compare the spectrogram
+in the main channel of Virgo with that produced by auxiliary sensors. This
+last task is particularly innovative, since it offer the possibility of linking the
+glitches observed in the main channel to local disturbancies in the detector,
+2http://https://gravityspy.org/
+3https://www.zooniverse.org/projects/reinforce/gwitchhunters
+4https://www.reinforceeu.eu/
+3
+
+Figure 1: Example of a GWitchHunters spectrogram showing two glitches, as well as the
+rectagle drawn by citizens to locate them
+thus suggesting a possible hint to the origin of each glitch. These tasks can
+be carried both on a personal computer and on mobile devices. The project
+also features a set of tutorials and examples to teach the volunteers how to
+perform the different tasks, as well as a ”Field Guide” containing information
+on the Advanced Virgo detector, the various glitch classes and the auxiliary
+channels used in the project.
+4. First Results and Conclusions
+Since its official launch in November 2021, ∼2800 volunteers have sub-
+scribed to the project, although another significant amount have contributed
+without officially registering. This collective effort has produced more than
+∼400000 classifications of ∼ 40000 data samples so far. In order to pro-
+mote the project and engage citizens, the REINFORCE consortium has or-
+ganized many initiatives, including workshops, press activities, online chal-
+4
+
+Virgo strain channel
+Frequency [Hz]
+Normalizedenergy
+100
+0.8
+0.6
+-0.4
+-0.2
+0:2
+0.4
+0.6
+0.8
+Time [s]lenges5 and training school6. A monitoring of the project website has been
+carried on, showing that these initiatives successfully attracted more volun-
+teers to GWitchHunters, leading to peaks of ∼5000 classifications per day.
+The results of the volunteers analysis are used for training a machine learning
+algorithm that automatically analyze the glitch data. In particular, we fo-
+cused on a 2D convolutional neural network architecture, that has been also
+tested on simulations [6, 10] reaching an accuracy greater than 99%. These
+first tests show how the GWitchHunters project could be successfully used
+to join citizen science and machine learning with the goal of contributing to
+increase the sensitivity of gravitational wave detectors.
+Acknowledgements
+REINFORCE has received funding from the European Union’s Horizon
+2020 research and innovation program, under Grant Agreement no. 872859.
+References
+[1] Abbott, B. P. et al. 2016, Physical Review Letters, 116, 061102.
+doi:10.1103/PhysRevLett.116.061102
+[2] Aasi, J. et al. 2015, Classical and Quantum Gravity, 32, 074001.
+doi:10.1088/0264-9381/32/7/074001
+[3] Acernese, F. et al. 2015, Classical and Quantum Gravity, 32, 024001.
+doi:10.1088/0264-9381/32/2/024001
+[4] Abbott, B. et al. 2021, arXiv:2111.03606
+[5] George, D., Shen, H., & Huerta, E. A. 2017, arXiv:1711.07468
+[6] Razzano, M. & Cuoco, E. 2018, Classical and Quantum Gravity, 35,
+095016. doi:10.1088/1361-6382/aab793
+[7] Powell, J. et al. 2015, Classical and Quantum Gravity, 32, 215012.
+doi:10.1088/0264-9381/32/21/215012
+5e.g. https://www.reinforceeu.eu/winter-challenge-2022
+6e.g. https://reinforce.ea.gr/international-training-course/
+5
+
+[8] Zevin, M. et al. 2017, Classical and Quantum Gravity, 34, 064003.
+doi:10.1088/1361-6382/aa5cea
+[9] Lintott, C. J. et al. 2008, MNRAS, 389, 1179. doi:10.1111/j.1365-
+2966.2008.13689.x
+[10] Cuoco E., et al 2021 Mach. Learn.: Sci. Technol. 2 011002
+6
+

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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='05112v1  [gr-qc]  12 Jan 2023  GWitchHunters: Machine Learning and citizen science  to improve the performance of Gravitational Wave  detector  Massimiliano Razzanoa,b,1,∗, Francesco Di Renzoa,b, Francesco Fidecaroa,b,  Gary Hemmingc, Stavros Katsanevasc  aDepartment of Physics, University of Pisa, Largo B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' Pontecorvo 3, Pisa, I-56127  bINFN Section of Pisa, Largo B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' Pontecorvo 3, Pisa, I-56127  cEuropean Gravitational Observatory (EGO),Via E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' Amaldi, 5, Cascina,I-56021  Abstract  The Gravitational waves have opened a new window on the Universe and  paved the way to a new era of multimessenger observations of cosmic sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='  Second-generation ground-based detectors such as Advanced LIGO and Ad-  vanced Virgo have been extremely successful in detecting gravitational wave  signals from coalescence of black holes and/or neutron stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' However, in  order to reach the required sensitivities, the background noise must be inves-  tigated and removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' In particular, transient noise events called “glitches”  can affect data quality and mimic real astrophysical signals, and it is there-  fore of paramount importance to characterize them and find their origin,  a task that will support the activities of detector characterization of Virgo  and other interferometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' Machine learning is one of the most promising  approaches to characterize and remove noise glitches in real time, thus im-  proving the sensitivity of interferometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' A key input to the preparation of  a training dataset for these machine learning algorithms can originate from  citizen science initiatives, where volunteers contribute to classify and analyze  signals collected by detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' We will present GWitchHunters, a new citi-  zen science project focused on the study of gravitational wave noise, that has  been developed within the REINFORCE project (a ”Science With And For  Society” project funded under the EU’s H2020 program).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' We will present  ∗Corresponding author  Email address: massimiliano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='razzano@unipi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='it (Massimiliano Razzano)  1on behalf of the REINFORCE Consortium  Preprint submitted to Nuclear Instruments and Methods in Physics Research AJanuary 13, 2023  the project, its development and the key tasks that citizens are participating  in, as well as its impact on the study of noise in the Advanced Virgo detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='  Keywords:  gravitational waves, machine learning, citizen science  PACS: 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='–q, 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='Tv,  2000 MSC: 83C35,  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' Introduction  Gravitational wave physics is opening an entire new window on the Uni-  verse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' Since their discovery in 2015 [1], the Advanced LIGO [2] and Advanced  Virgo [3] detectors have carried on three observing runs (O1, O2, O3) and  unveiled 90 signals produced by the coalescence of compact objects, mostly  binary black hole with a small fraction of neutron star and/or black hole  binaries [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='  Advanced LIGO and Virgo are second-generation laser interferometers with  Fabry-Perot cavities hosted in km perpendicular arms, that are capable of  detecting the tiny deformations induced in the fabric of spacetime by the  passage of gravitational waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' In order to improve the sensitivity of the  detectors, there is a continuous effort to reduce the background noise due to  local disturbances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' In particular, at low frequencies the noise is dominated by  seismic and Newtonian noise, while at mid frequencies the main component  is related to the thermal noise and at high frequencies the noise is mostly  related to quantum effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='  The activity of detector characterization and  noise hunting in gravitational wave detectors is focused on the investigation  of stationary and non stationary noise sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' In particular, non station-  ary transient noise events called glitches are of particular importance in the  noise studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' In fact, glitches can affect data quality and stability and mimic  real astrophysical signals, thus reducing the effective duty cycle of interfer-  ometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' The classification and characterization of glitches is therefore key  to understand the origin of noise in detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' However, glitches have com-  plex temporal signatures, that make difficult to classify them using standard  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' Various works have shown that Machine Learning methods can be  promising for the classification of glitches [5, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' In particular, images built  from the time-frequency spectrograms of glitches are very effective in show-  ing the complex morphology of glitches and can be easily given in input to  machine learning algorithms, including deep convolutional neural networks  [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' A possible approach to this problem is based on supervised learning,  2  that requires large number of labeled glitch samples, that could be produced  by dedicated citizen science initiatives, where volunteers look at images and  clssify them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='  A successful example of this method is provided by Gravi-  tySpy2, a citizen science project focused on the classification of glitches in  LIGO and Virgo[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' Here we present GWitchHunters3, a new citizen science  project complementary to GravitySpy and aimed at improving sensitivity of  gravitational wave detectors combining citizen science and machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' The REINFORCE Project  GwitchHunters has been developed within the Research Infrastructures  FOR Citizens in Europe (REINFORCE) project4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' REINFORCE is a Re-  search & Innovation Project, supported by the EU H2020 SWAFS “Science  with and for Society” work programme and aimed at creating a series of  cutting-edge citizen science projects on Frontier Physics research, with the  goal of engaging >100,000 citizens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' REINFORCE is based on four citizen  science demonstrators focused Gravitational Waves (GWitchHunters), Astro-  physical neutrinos (Deep Sea Explorers), High Energy Physics (New Particle  Search at CERN) and muon-based tomography (Cosmic Muon Images).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' All  demonstrators are hosted on Zooniverse [9], the world leading platform for  citizen science projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' Overview of GWitchHunters  GwitchHunters has been officially launched on Zooniverse in November  2021 after a dedicated review phase and offers to citizens a set of different  tasks of increasing difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' Data are presented as spectrograms and come  from the Virgo O3 run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' A Playground task is specifically devoted to learning  the basics of glitch morphology and its classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' Three other levels offer  (1) the possibility to classify glitches among a larger set of classes, (2) localize  the glitches in the time-frequency space, and (3) compare the spectrogram  in the main channel of Virgo with that produced by auxiliary sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' This  last task is particularly innovative, since it offer the possibility of linking the  glitches observed in the main channel to local disturbancies in the detector,  2http://https://gravityspy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='org/  3https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='zooniverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='org/projects/reinforce/gwitchhunters  4https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='reinforceeu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='eu/  3  Figure 1: Example of a GWitchHunters spectrogram showing two glitches, as well as the  rectagle drawn by citizens to locate them  thus suggesting a possible hint to the origin of each glitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' These tasks can  be carried both on a personal computer and on mobile devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' The project  also features a set of tutorials and examples to teach the volunteers how to  perform the different tasks, as well as a ”Field Guide” containing information  on the Advanced Virgo detector, the various glitch classes and the auxiliary  channels used in the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' First Results and Conclusions  Since its official launch in November 2021, ∼2800 volunteers have sub-  scribed to the project, although another significant amount have contributed  without officially registering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' This collective effort has produced more than  ∼400000 classifications of ∼ 40000 data samples so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' In order to pro-  mote the project and engage citizens, the REINFORCE consortium has or-  ganized many initiatives, including workshops, press activities, online chal-  4  Virgo strain channel  Frequency [Hz]  Normalizedenergy  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='2  0:2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='8  Time [s]lenges5 and training school6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' A monitoring of the project website has been  carried on, showing that these initiatives successfully attracted more volun-  teers to GWitchHunters, leading to peaks of ∼5000 classifications per day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='  The results of the volunteers analysis are used for training a machine learning  algorithm that automatically analyze the glitch data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' In particular, we fo-  cused on a 2D convolutional neural network architecture, that has been also  tested on simulations [6, 10] reaching an accuracy greater than 99%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' These  first tests show how the GWitchHunters project could be successfully used  to join citizen science and machine learning with the goal of contributing to  increase the sensitivity of gravitational wave detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='  Acknowledgements  REINFORCE has received funding from the European Union’s Horizon  2020 research and innovation program, under Grant Agreement no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' 872859.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='  References  [1] Abbott, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' 2016, Physical Review Letters, 116, 061102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='061102  [2] Aasi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' 2015, Classical and Quantum Gravity, 32, 074001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='1088/0264-9381/32/7/074001  [3] Acernese, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' 2015, Classical and Quantum Gravity, 32, 024001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='1088/0264-9381/32/2/024001  [4] Abbott, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' 2021, arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='03606  [5] George, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=', Shen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=', & Huerta, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' 2017, arXiv:1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='07468  [6] Razzano, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' & Cuoco, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' 2018, Classical and Quantum Gravity, 35,  095016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='1088/1361-6382/aab793  [7] Powell, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' 2015, Classical and Quantum Gravity, 32, 215012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='1088/0264-9381/32/21/215012  5e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='reinforceeu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='eu/winter-challenge-2022  6e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' https://reinforce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='ea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='gr/international-training-course/  5  [8] Zevin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' 2017, Classical and Quantum Gravity, 34, 064003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='1088/1361-6382/aa5cea  [9] Lintott, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' 2008, MNRAS, 389, 1179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
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+page_content='1365-  2966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='13689.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
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+page_content=', et al 2021 Mach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
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+page_content='CO]  6 Jan 2023  Vertex-Critical (P5, chair)-Free Graphs  Shenwei Huang*†  Zeyu Li‡§  January 4, 2022  Abstract  Given two graphs H1 and H2, a graph G is (H1, H2)-free if it contains no  induced subgraph isomorphic to H1 or H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' A Pt is the path on t vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' A chair  is a P4 with an additional vertex adjacent to one of the middle vertices of the P4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' A  graph G is k-vertex-critical if G has chromatic number k but every proper induced  subgraph of G has chromatic number less than k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' In this paper, we prove that there  are finitely many 5-vertex-critical (P5, chair)-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Graph coloring;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' k-vertex-critical graphs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' forbidden induced subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  1  Introduction  All graphs in this paper are finite and simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' We say that a graph G contains a graph  H if H is isomorphic to an induced subgraph of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' A graph G is H-free if it does  not contain H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' For a family of graphs H, G is H-free if G is H-free for every H ∈  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' When H consists of two graphs, we write (H1, H2)-free instead of {H1, H2}-  free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' As usual, Pt and Cs denote the path on t vertices and the cycle on s vertices,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' A clique (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' independent set) in a graph is a set of pairwise adjacent  (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' nonadjacent) vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' The complete graph on n vertices is denoted by Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' The  graph K3 is also referred to as the triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' The clique number of G, denoted by ω(G),  is the size of a largest clique in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' For two graphs G and H, we use G + H to denote  the disjoint union of G and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If a graph G can be partitioned into k independent sets  S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' , Sk such that there is an edge between every vertex in Si and every vertex in Sj  for all 1 ≤ i < j ≤ k, G is called a complete k-partite graph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' each Si is called a part  of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If we do not specify the number of parts in G, we simply say that G is a complete  multipartite graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' We denote by Kn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=',nk the complete k-partite graph such that the  ith part Si has size ni, for each 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  A q-coloring of a graph G is a function φ : V (G) −→ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' , q} such that φ(u) ̸=  φ(v) whenever u and v are adjacent in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' And a q-coloring of G is also a partition of  V (G) into q independent sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' A graph is q-colorable if it admits a q-coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' The  College  of  Computer  Science,  Nankai  University,  Tianjin  300350,  China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Email:  shenweihuang@nankai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Supported by Natural Science Foundation of Tianjin (20JCY-  BJC01190).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  †Tianjin Key Laboratory of Network and Data Security Technology, Nankai University, Tianjin 300071,  China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  ‡College of Computer Science, Nankai University, Tianjin 300350, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  §Tianjin Key Laboratory of Network and Data Security Technology, Nankai University, Tianjin 300071,  China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  1  Figure 1: The graph chair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  chromatic number of a graph G, denoted by χ(G), is the minimum number q for which  G is q-colorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' We call a graph G is k-chromatic when χ(G) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  A graph G is k-critical if it is k-chromatic and χ(G − e) < χ(G) for any edge  e ∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' We call a graph is critical if it is k-critical for some integer k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' A graph  G is k-vertex-critical if χ(G) = k and χ(G−v) < k for any v ∈ V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' For a set H of  graphs and a graph G, we say that G is k-vertex-critical H-free if it is k-vertex-critical  and H-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Our research is mainly motivated by the following theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Theorem 1 ([7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' For any fixed k ≥ 5, there are infinitely many k-vertex-critical P5-  free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Thus, it is natural to consider which subclasses of P5-free graphs have finitely many  k-vertex-critical graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' The reason for finiteness is that if we know there are only  finitely many k-vertex-critical graphs, then there is a polynomial-time algorithm for  (k − 1)-coloring graphs in that class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' In 2021, Kameron, Goedgebeur, Huang and Shi  [4] obtained the following dichotomy result for k-vertex-critical (P5, H)-free graphs  when |H| = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Theorem 2 ([4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let H be a graph of order 4 and k ≥ 5 be a fixed integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Then  there are infinitely many k-vertex-critical (P5, H)-free graphs if and only if H is 2P2  or P1 + K3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  In [4], it was also asked which five-vertex graphs H can lead to finitely many  k-vertex-critical (P5, H)-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' It is known that there are finitely many 5-vertex-  critical (P5,banner)-free graphs [3, 9], and finitely many k-vertex-critical (P5, P5)-  free graphs for every fixed k [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Hell and Huang proved that there are finitely many  k-vertex-critical (P6, C4)-free graphs [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' This was later generalized to (Pt, Kr,s)-  free graphs in the context of H-coloring [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' This gives an affirmative answer for  H = K2,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Recently, it was also shown that the answer to the above question is  positive if H is gem or P2 + P3 [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Moreover, it was proved that there are finitely  many 5-vertex-critical (P5, bull)-free graphs [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  In this article, we continue such a study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' A chair is a P4 with an additional vertex  adjacent to one of the middle vertices of the P4 (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' In particular, we prove  the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' There are finitely many 5-vertex-critical (P5, chair)-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  2  Preliminaries  For general graph theory notation we follow [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let G = (V, E) be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If uv ∈  E, we say that u and v are neighbors or adjacent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' otherwise u and v are nonneighbors  2  or nonadjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' We use u ∼ v to mean that u and v are neighbors and u ≁ v to mean  that u and v are nonneighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' The neighborhood of a vertex v, denoted by NG(v), is  the set of neighbors of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' For a set X ⊆ V (G), let NG(X) = �  v∈X NG(v) \\ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' We  shall omit the subscript whenever the context is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' For X, Y ⊆ V , we say that X is  complete (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' anticomplete) to Y if every vertex in X is adjacent (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' nonadjacent)  to every vertex in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If X = {x}, we write “x is complete (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' anticomplete) to  Y ” instead of “{x} is complete (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' anticomplete) to Y ”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If a vertex v is neither  complete nor anticomplete to a set S, we say that v is mixed on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If a vertex v is  neither complete nor anticomplete to two ends of an edge, we say that v is distinguish  the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' We say that H is a homogeneous set if no vertex in V − H is mixed on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  More generally, we say that H is homogeneous with respect to a subset S ⊆ V if no  vertex in S can be mixed on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' For S ⊆ V , the subgraph induced by S, is denoted by  G[S].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  A pair of comparable vertices of G is pairwise nonadjacent vertices u, v such that  N(v) ⊆ N(u) or N(u) ⊆ N(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' It is well-known that k-vertex-critical graphs cannot  contain comparable vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' We shall use the following generalization in later proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Lemma 1 ([4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let G be a k-vertex-critical graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Then G has no two nonempty  disjoint subsets X and Y of V (G) that satisfy all the following conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  X and Y are anticomplete to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  χ(G[X]) ≤ χ(G[Y ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Y is complete to N(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  3  New Results  In this section, we prove our new results: there are finitely many 5-vertex-critical  (P5, chair)-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' To prove Theorem 3, we prove the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let G be a 5-vertex-critical (P5, chair)-free graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If G contains a C5,  then G has finite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof of Theorem 3 assuming Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let G be a 5-vertex-critical(P5, chair)-free  graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If G contains C5, then G has finite order by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If G is C5-free, then G  has finite order by a result in [7] that there are only thirteen 5-vertex-critical (P5, C5)-  free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' In either case, G has finite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Next we prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='1  Structure Around C5  In this subsection, we discuss some structural properties of (P5, chair)-free graphs  containing a C5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let G be a connected (P5, chair)-free graph containing an induced  C5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let C = v1, v2, v3, v4, v5 be an induced C5 with vivi+1 being an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' We divide  V \\V (C) as follows, where all indices are modulo 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  S0 = {v ∈ V \\V (C) : NC(v) = ∅},  S1(i) = {v ∈ V \\V (C) : NC(v) = {vi}},  S1  2(i) = {v ∈ V \\V (C) : NC(v) = {vi, vi+1}},  3  S2  2(i) = {v ∈ V \\V (C) : NC(v) = {vi, vi+2}},  S1  3(i) = {v ∈ V \\V (C) : NC(v) = {vi−1, vi, vi+1}},  S2  3(i) = {v ∈ V \\V (C) : NC(v) = {vi−2, vi, vi+2}},  S4(i) = {v ∈ V \\V (C) : NC(v) = {vi−2, vi−1, vi+1, vi+2}},  S5 = {v ∈ V \\V (C) : NC(v) = V (C)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  We use Sm  3 (i ± 1) to denote Sm  3 (i + 1) ∪ Sm  3 (i − 1) for m = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' The notations  Sm  3 (i±2), S4(i±1) and S4(i±2) are defined similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' We now prove some properties  about these sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' S1(i) ∪ S1  2(i) ∪ S2  2(i) = ∅, for all 1 ≤ i ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let u, v be arbitrary two vertices such that v ∈ S1(i) ∪ S1  2(i),  u ∈ S2  2(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Then {v, vi, vi−1, vi−2, vi−3} induces a P5, and {u, vi, vi−1, vi−2} and  {vi+1} induce a chair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' S0 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' We will first show that N(S0) ⊆ S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Since G is connected,  there is a pair of vertices u and v such that u ∈ S0, v ∈ V (G)\\S0 and u ∼ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If  v ∈ S1  3(i) for any i, then {u, v, vi+1, vi+2, vi−2} induces a P5, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If  v ∈ S2  3(i) ∪ S4(i + 1) for any i, then {vi+1, vi, v, vi−2} and {u} induce a chair, a  contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Thus, v can only belong to S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Then, two nonempty disjoint subsets S0  and C of V (G) satisfy the three conditions of Lemma 1, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Therefore,  S0 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' S1  3(i) is clique, for all 1 ≤ i ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' We assume that there are two vertices u, v ∈ S1  3(i) with u ≁ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Then {v, vi+1, vi+2, vi−2} and {u} induce a chair in G, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Each vertex in S4(i) ∪ S5 is either complete or anticomplete to a component  of S2  3(i), for all 1 ≤ i ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' We assume that there is an edge uv of S2  3(i) can be distinguished by vertex  s ∈ S4(i) ∪ S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Without loss of generality, let s ∼ u, s ≁ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Then {vi−1, s, u, v} and  {vi+1} induce a chair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Each vertex in V (G)−(S2  3(i)∪S4(i)∪S5) is either complete or anticomplete  to S2  3(i), for all 1 ≤ i ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' By symmetry, it suffices to prove the claim for i, i+ 1 and i+ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let v ∈ S2  3(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  If v is adjacent to s1 ∈ S1  3(i + 1), then {vi−1, vi−2, v, s1, vi+1} is an induced P5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If v  is not adjacent to s2 ∈ S1  3(i) ∪ S2  3(i + 1) ∪ S4(i + 2), then {vi−1, s2, vi+1, vi+2, v} is  an induced P5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If v is not adjacent to s3 ∈ S2  3(i + 2) ∪ S4(i + 1), then {vi−1, s3, vi+2, v}  and {vi+1} induce a chair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If v is not adjacent to s4 ∈ S1  3(i + 2), then {vi−1, vi, vi+1, v}  and {s4} induce a chair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Every component of S2  3(i) is a homogeneous set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' By Claim 4 and Claim 5, there is no vertex of G\\S2  3(i) that can distinguish an  edge of S2  3(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  4  Let Ti = S1  3(i ± 2) ∪ S2  3(i ± 1) ∪ S2  3(i ± 2) for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' S4(i) is complete to Ti, for all 1 ≤ i ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' By the symmetry, it suffers to prove the claim for S1  3(i+2)∪S2  3(i+1)∪S2  3(i+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Let v ∈ S4(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If v is not adjacent to s1 ∈ S1  3(i + 2), then {vi, vi−1, v, vi+2, s1} in-  duces a P5, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If v is not adjacent to s2 ∈ S2  3(i + 1), then {vi, vi−1, v, vi+2}  and {s2} induce a chair, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If v is not adjacent to s3 ∈ S2  3(i + 2), then  {s3, vi, vi+1, v, vi−2} induces a P5, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' For each s ∈ S1  3(i) ∪ S4(i ± 2), u, v ∈ S4(i) with uv /∈ E, s cannot mix on  {u, v}, for all 1 ≤ i ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' By the symmetry, it suffers to prove the claim for S1  3(i) ∪ S4(i + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let  s ∈ S1  3(i) ∪ S4(i + 2) with s ∼ u, s ≁ v , then {vi, s, u, vi+2, v} induces a P5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Let Ri = S1  3(i ± 1) ∪ S2  3(i) ∪ S4(i ± 1) ∪ S5, for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' For each s ∈ Ri, u, v ∈ S4(i) with uv /∈ E, s is adjacent to at least one of  {u, v}, for all 1 ≤ i ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' By the symmetry, it suffers to prove the claim for S1  3(i + 1) ∪ S2  3(i) ∪ S4(i +  1) ∪ S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let s1 ∈ S1  3(i + 1) ∪ S2  3(i) ∪ S4(i − 1), if s1 is nonadjacent to both {u, v},  then {v, vi−1, vi, s1} and {u} induce a chair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let s2 ∈ S5, if s2 is nonadjacent to both  {u, v}, then {vi, s2, vi−2, v} and {u} induce a chair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Every vertex in S4(i ± 2) is complete to x, y ∈ S4(i) with xy /∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' By symmetry, let v ∈ S4(i + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' v can not mix on x, y by Claim 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If v ≁ x  and v ≁ y, {vi, v, vi−2, x} and {y} induce a chair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Then v is complete to {x, y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='2  Proof of Theorem 4  Let graph family F = {K5, W, P, Q1, Q2, Q3} (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' The adjacency lists of  F are given in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' It is routine to verify that every graph in F is a 5-vertex-  critical (P5, chair)-free graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let G be a 5-vertex-critical (P5, chair)-free graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If G contains  a induced F ∈ F, then G is isomorphic to F since G is 5-vertex-critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Therefore,  we may assume that G is F-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  By Claim 1 and Claim 2, G has a finite order if and only if S3 ∪ S4 ∪ S5 has finite  size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' |S1  3(i)| ≤ 2, for all 1 ≤ i ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If |S1  3(i)| ≥ 3, then S1  3(i) ∪ {vi, vi+1} contains a K5 by Claim 3, a contradic-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' χ(S2  3(i) ∪ S4(i) ∪ S5) ≤ 2, for all 1 ≤ i ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If χ(S2  3(i) ∪ S4(i) ∪ S5) ≥ 3, then the proper subgraph S2  3(i) ∪ S4(i) ∪ S5 ∪  {vi−2, vi+2} has chromatic number at least 5, contradicting that G is 5-vertex-critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' S5 is an independent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  5  0  1  2  3  4  K5  0  1  2  3  4  5  6  W  0  1  2  3  4  5  6  7  8  P  0  1  2  3  4  5  6  7  8  Q1  0  1  2  3  4  5  6  7  8  Q2  0  1  2  3  4  5  6  7  8  Q3  Figure 2: Graph Family F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If there are two adjacent vertices u, v ∈ S5, then G contains a W ∈ F, a  contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Every homogeneous component of S2  3(i) or S4(i) is isomorphic to K1 or  K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let K be a component of S2  3(i) or S4(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Since G has no K5 or W, K has  no triangles or C5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Since G is P5-free, G is bipartite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So χ(K) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Clearly, if  χ(K) = 1, then K is isomorphic to K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Now assume that χ(K) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let X and  Y be the bipartition of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let x ∈ X and y ∈ Y with xy ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Suppose that  (X ∪ Y ) \\ {x, y} ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Since G is 5-vertex-critical, G − ((X ∪ Y ) \\ {x, y}) has a  4-coloring φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Without loss of generality, we may assume that φ(x) = 1 and φ(y) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Now if we color every vertex in X with color 1 and color every vertex in Y with color  2, the resulting coloring is a 4-coloring of G by Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' This contradicts that G is  5-vertex-critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So K is isomorphic to K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' |S2  3(i)| ≤ 3, for all 1 ≤ i ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let K be a component of S2  3(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' We say that K is of type i if χ(K) = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' We  show that there is at most one component of type i for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Take two components  K, K′ of the same type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let k ∈ K and k′ ∈ K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' By Lemma 1, there are vertices u, v  such that u ∈ N(K) \\ N(K′) and v ∈ N(K′) \\ N(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' By Claim 6, uk ∈ E, vk′ ∈ E  and uk′, vk /∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Any vertex in V (G)−(S2  3(i)∪S4(i)∪S5) can’t mix on two vertices  of S2  3(i) by Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So u, v ∈ S4(i) ∪ S5 by our assumption about k, k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If u ≁ v,  {k, u, vi+1, v, k′} induces a P5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Therefore, u ∼ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' By Claim 13, u, v cannot be in  S5 at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' It is easy to see that C ∪ {k, k′, u, v} contains an induced P, a  contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  As a result, |S2  3(i)| ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' S4(i) is a star, or S4(i) is complete to S4(i + 2) ∪ S4(i − 2), for all 1 ≤  i ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If S4(i) is disconnected, S4(i) is complete to S4(i+2)∪S4(i − 2) by Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  If S4(i) is connected, then S4(i) is a bipartite graph by Claim 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If χ(S4(i)) = 1,  S4(i) is isomorphic to K1 and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Now assume that |S4(i)| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let X, Y  be the bipartition of S4(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If |X| ≥ 2 and |Y | ≥ 2, then every vertex in S4(i ± 2) is  6  complete to X ∪ Y by Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Thus, S4(i) is complete to S4(i ± 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Therefore, we  may assume that |X| = 1 and so S4(i) is a star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Recall that Ri = S1  3(i ± 1) ∪ S2  3(i) ∪ S4(i ± 1) ∪ S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If S4(i) is a star, then |S4(i)| ≤ 2 for all 1 ≤ i ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Suppose that S4(i) = X ∪ Y with Y  = {y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  We show that |X| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let x1, x2 ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' By Lemma 1, there exist a ∈ N(x1)\\N(x2) and  b ∈ N(x2)\\N(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Note that any vertex of G − Ri can’t mix on two nonadjacent  vertices of X by Claim 7 - Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So a, b ∈ Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If a ≁ b, {x1, a, vi, b, x2} induces  a P5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So a ∼ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' It is not hard to check that G contains one of Q1, Q2 and Q3, a  contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Thus, there are at most two vertices in X, and so |S4(i)| ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' For each i, when S4(i) is complete to S4(i ± 2) and Ri is not empty, then  |S4(i)| ≤ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' When S4(i) is (P1 + P2)-free, S4(i) is a complete bipartite graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let (X, Y )  be a partition of S4(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' We show that |X|, |Y | ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let x1, x2, x3, x4 be  vertices in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' By Lemma 1, there vertices a1 ∈ N(x1)\\N(x2), a2 ∈ N(x2)\\N(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Notice that a1, a2 ∈ Ri by Claim 7 - Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If a1 ≁ a2, G contains an induced  P5 = {x1, a1, vi, a2, x2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So a1 ∼ a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Then a1 ∈ S1  3(i − 1) ∪ S4(i + 1) and a2 ∈  S1  3(i + 1) ∪ S4(i − 1), otherwise, it is easy to check that G contains one of Q1 and  Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Similarly, there exists a3 ∈ N(x3)\\N(x4), a4 ∈ N(x4)\\N(x3) and a3, a4 ∈  Ri, a3 ∼ a4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Thus {x3, x4} is complete to {a1, a2}, and {x1, x2} is complete to  {a3, a4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' This shows that a1, a2, a3, a4 are pairwise different vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Then a3 ∈  S1  3(i − 1) ∪ S4(i + 1), a4 ∈ S1  3(i + 1) ∪ S4(i − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Recall that S1  3(i − 1) or S1  3(i + 1)  is a clique by Claim 3, and S1  3(i − 1) is complete to S4(i + 1), S1  3(i + 1) is complete  to S4(i − 1) by Claim 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If a1 ≁ a3 and a2 ≁ a4, then a1, a3 ∈ S4(i + 1) and a2, a4 ∈  S4(i − 1), then {vi−2, vi+2, x3, a1, a2} is an induced K5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Otherwise, if a1 ∼ a3,  {vi−1, vi−2, x3, a1, a3} induces K5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So a2 ∼ a4, then {vi+1, vi+2, x3, a2, a4} induces  a K5, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So |S4(i)| ≤ 6 if S4(i) is (P1 + P2)-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Now suppose that S4(i) contains a P1 + P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let P1 + P2 = {a, b, c : a ≁ b, a ≁  c, b ∼ c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' We first prove some useful facts about P1 + P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  S1  3(i) is anticomplete to P1 + P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  (1)  Every x ∈ S1  3(i) is either complete or anticomplete to {a, b, c} by Claim 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If x is  complete to {a, b, c}, then G contains an induced W, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So x is anticom-  plete to {a, b, c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' This completes the proof of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  For any y ∈ Ri, {y, a, b, c} induces either a P4 or a 2P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  (2)  Let y ∈ Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Note that {y} ∪ S4(i) is triangle-free or else G contains a K5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If y is  not adjacent to a, then y ∼ b, y ∼ c by Claim 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Now G induces a K5, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  So y ∼ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If y ≁ b, y ≁ c, then {y, a, b, c} induces a 2P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If y is adjacent to exact one  vertex of {b, c}, we assume by symmetry that y ∼ b, y ≁ c and so {a, y, b, c} induces  a P4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' This completes the proof of (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Next we discuss about S4(i)\\{a, b, c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let x ∈ S1  3(i), z ∈ S4(i)\\{a, b, c}, and  we define Y1 = {y1 ∈ Ri : {y1, a, b, c} induces a P4}, and Y2 = {y2 ∈ Ri :  7  {y2, a, b, c} induces a 2P2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  S1  3(i) is anticomplete to S4(i)\\{a, b, c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  (3)  If z ∼ x, then z is complete to {a, b, c} by (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Now G contains an induced W, a  contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So z ≁ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' This completes the proof of (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  So S1  3(i) is anticomplete to S4(i) by (1) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  For any y1 ∈ Y1, z1 ∈ S4(i)\\{a, b, c}, z1y1, z1c ∈ E, and z1a, z1b /∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  (4)  If z1 ≁ y1, then z1 ∼ c by y1c /∈ E and Claim 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So z1 ≁ b by Claim 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If z1 ≁ a,  {y1, a, b, c, z} induces a P5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So z1 ∼ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Then there is an induced C5 = {a, y1, b, c, z1},  contradicting Claim 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So z1 ∼ y1, then z1 ≁ a and z1 ≁ b since S4(i) is triangle-  free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If z1 ≁ c, {a, y1, b, c} and {z1} induce a chair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So z1 ∼ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' This completes the  proof (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  For any y2 ∈ Y2, z2 ∈ S4(i)\\{a, b, c}, z2y2 ∈ E, and z2a, z2b, z2c /∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  (5)  If z2 ≁ y2, then z2 ∼ b and z2 ∼ c by y2b, y2c /∈ E and Claim 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Then {z2, b, c}  induces a triangle, contradicting Claim 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So z2 ∼ y2 and then z2 ≁ a by the fact that  {y2} ∪ S4(i) is triangle-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If z2 is adjacent to exact one of b, c, then {z2, y2, a, b, c}  induces a P5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So z2 ≁ b and z2 ≁ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' This completes the proof (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  We can infer that any vertex in Ri is complete to S4(i)\\{a, b, c} by (4) and (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Suppose that there exist two vertices z, z′ ∈ S4(i)\\{a, b, c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If Y1 ̸= ∅ and Y2 ̸= ∅,  z is adjacent to c by (4) and is nonadjacent to c by (5), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So Ri =  Y1 or Ri = Y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Note that any vertex in Ri is complete to two ends of an edge of  C5 ∩ N(S4(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Since G is K5-free, z ≁ z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Then N(z) = N(z′) by Claim 7,  contradicting to Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So |S4(i)\\{a, b, c}| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Then |S4(i)| ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' For each i, when S4(i) is complete to S4(i±2) and Ri is empty, |S4(i)| ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If S4(i) is disconnected, then there are two components K1, K2 of S4(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Every  vertex of S1  3(i) is either complete or anticomplete to K1 ∪ K2 by Claim 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So K1  and K2 are homogeneous components by Claim 7 - Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Moreover, N(K1) =  N(K2) ⊆ Ti ∪ S1  3(i) ∪ S4(i ± 2) ∪ C5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' This contradicts Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Therefore, S4(i)  is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Recall that χ(S4(i)) ≤ 2 by Claim 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If χ(S4(i)) = 1, then |S4(i)| = |K1| = 1  and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' When χ(S4(i)) = 2, S4(i) is a bipartite graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let (X, Y ) be the  bipartition of S4(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Every vertex s ∈ S1  3(i) is either complete or anticomplete to  X(resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Y ) by Claim 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So X(resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Y ) is homogeneous with respect to G − Y (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  G − X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If there are x ∈ X, y ∈ Y with x ≁ y, then every vertex s ∈ S1  3(i) cannot  mix on S4(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Then S4(i) is a homogeneous set, and |S4(i)| = |K2| = 2 by Claim 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  If X is complete to Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Then X is a homogeneous set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' For any pairwise vertices  x1, x2 ∈ X, we have N(x1) = N(x2), contradicting Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So |X| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' In the  same way, |Y | = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Therefore, |S4(i)| ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' |S4(i)| ≤ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' It follows from Claim 17 to Claim 19 that |S4(i)| ≤ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  8  Claim 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' |S5| ≤ 255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Suppose that |S5| > 255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' We know any two vertices in S5 are nonadjacent  by Claim 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' By the pigeonhole principle, there are two vertices u, v ∈ S5 such that  N(u) = N(v), contradicting Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So |S5| ≤ 25(|S1  3(i)∪S2  3(i)∪S4(i)|)  ≤ 25(2+3+6) = 255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  The lemma follows from Claim 11, Claim 15, Claim 20 and Claim 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  4  Appendix  Below we give the adjacency lists of graphs in F other than K5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Graph W: {0: 1 4 5 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' 1: 0 2 5 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' 2: 1 3 5 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' 3: 2 4 5 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' 4: 0 3 5 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' 5: 0 1 2 3 4  6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' 6: 0 1 2 3 4 5}  Graph P: {0: 1 4 5 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' 1: 0 2 7 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' 2: 1 3 5 6 7 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' 3: 2 4 5 6 7 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' 4: 0 3 7 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' 5: 0 2  3 7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' 6: 0 2 3 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' 7: 1 2 3 4 5 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' 8: 1 2 3 4 6 7}  Graph Q1: {0: 1 4 5 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' 1: 0 2 5 6 7 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' 2: 1 3 5 6 7 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' 3: 2 4 7 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' 4: 0 3 7 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' 5: 0 1  2 6 7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' 6: 0 1 2 5 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' 7: 1 2 3 4 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' 8: 1 2 3 4 6}  Graph Q2: {0: 1 4 5 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' 1: 0 2 5 6 7 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' 2: 1 3 5 6 7 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' 3: 2 4 5 6 7 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' 4: 0 3 7 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' 5:  0 2 3 6 7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' 6: 0 2 3 5 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' 7: 1 2 3 4 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' 8: 1 2 3 4 6}  Graph Q3: {0: 1 4 5 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
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+page_content=' 8: 1 2 3 4 6 7}  References  [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Bondy and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Murty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Graph Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Springer, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  [2] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Cai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Goedgebeur, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Some results on k-critical P5-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  arXiv:2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='05492 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='CO], 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  [3] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Cai, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Huang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Li, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Shi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Vertex-critical (P5, banner)-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  In Yijia Chen, Xiaotie Deng, and Mei Lu, editors, Frontiers in Algorithmics -  13th International Workshop, FAW 2019, Sanya, China, April 29–May 3, 2019,  Proceedings, volume 11458 of Lecture Notes in Computer Science, pages 111–  120, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  [4] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Cameron, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Goedgebeur, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Huang, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Shi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' k-critical graphs in P5-free  graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Theoretical Computer Science, 864:80–91, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  [5] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Dhaliwal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Hamel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Ho`ang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Maffray, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' McConnell, and  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Panait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' On color-critical (P5, co-P5)-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Discrete Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Mathemat-  ics, 216:142–148, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  [6] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Hell and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Complexity of coloring graphs without paths and cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Discrete Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Mathematics, 216:211–232, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  [7] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Ho`ang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Moore, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Recoskiez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Sawada, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Vatshelle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Constructions  of k-critical P5-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Discrete Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=', 182:91–98, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  9  [8] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Huang and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Critical (P5,bull)-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='04179 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='CO],  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  [9] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Huang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Li, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Shi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Critical (P6, banner)-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Discrete Applied  Mathematics, 258:143–151, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  [10] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Kami´nski and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Pstrucha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Certifying coloring algorithms for graphs without  long induced paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Discrete Applied Mathematics, 261:258–267, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
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