The Gyro-Structure of Some Matrix Manifolds
Authors: Xuan Son Nguyen
Venue: NeurIPS 2022 Accept (2022)
Keywords: manifold learning
representation learning
deep learning
gyrovector spaces

Abstract: In this paper, we study the gyrovector space structure (gyro-structure) of matrix manifolds. Our work is motivated by the success of hyperbolic neural networks (HNNs) that have demonstrated impressive performance in a variety of applications. At the heart of HNNs is the theory of gyrovector spaces that provides a powerful tool for studying hyperbolic geometry. Here we focus on two matrix manifolds, i.e., Symmetric Positive Definite (SPD) and Grassmann manifolds, and consider connecting the Riemannian geometry of these manifolds with the basic operations, i.e., the binary operation and scalar multiplication on gyrovector spaces. Our work reveals some interesting facts about SPD and Grassmann manifolds. First, SPD matrices with an Affine-Invariant (AI) or a Log-Euclidean (LE) geometry have rich structure with strong connection to hyperbolic geometry. Second, linear subspaces, when equipped with our proposed basic operations, form what we call gyrocommutative and gyrononreductive gyrogroups. Furthermore, they share remarkable analogies with gyrovector spaces. We demonstrate the applicability of our approach for human activity understanding and question answering.

The Gyro-Structure of Some Matrix Manifolds
Xuan Son Nguyen ETIS, UMR 8051, CY Cergy Paris Université, ENSEA, CNRS, Cergy, France xuan-son. www..nneeeyyeennn.....mmm fr
Abstract
In this paper, we study the gyrovector space structure (gyro-structurre of matrix manifolds. Our work is motivated by the success of hyperbolic neural networks (HNNs) that have demonstrated impressive performance in variety of applications. At the heart HNNs is the theory of gyrovector spaces that provides a powerful tool for studying hyperbolic geometry. Here we focus i.e., Symmetric Positive Definite (SPD) and Grassmann manifolds, and consider connecting the Riemannian geometry of these manifolds with the basic operations, i.e., the binary operation and scalar multiplication on gyrovector spaces. Our work reveals some interesting facts about SPD and Grassmann manifolds. First, SPD matrices with the Affine-Invarian (Al) and Log-Euclidean (LE) geometries have rich structure with strong connection to hyperbolic geometry. Second, linear subspaces, when equipped with our proposed basic operations, form what we call gyrocommutative and gyrononreductive gyrogroups. Furthermore, they share remarkable analogies with gyrovector spaces. We demonstrate the applicability of our approach for human activity understanding and question answering.
matrix manifolds,
Introduction
Data lying on matrix manifolds are commonly encountered in various applied areas such as medical imaging [3, 37], shape analysis [41], drone classification [6], image recognition [11], and human behavior analysis [10, 15, 16, 20, 21, 22, 31, 40]. These data arise from constraint sets of the problem, for which there a natural representation of elements in the form of matrix arrays [2]. Due to the non-Euclidean nature of these data, traditional optimization algorithms usually fail to obtain good results the matrix manifold setting. While a large body works [6, 7, 8, 10, 20, 21, 22, 31, 33, 48] has been developed to generalize traditional optimization algorithms to this setting, there is still a lack of works that translate the language of differential geometry to basic operations on matrix manifolds that they can be used in computational building blocks of neural network models on these manifolds just as basic operations on Euclidean spaces, e.g., matrix-matrix matrix addition and malar-matrin multiplication are used in deep neural networks (DNNs).
To address the above issue, we propose novel framework based on the theory of gyrovector spaces [44, 45, 46] that has been successfully applied in the context of HNNs [12, 39]. Our aim is to uncover hidden analogies between the target manifolds and Euclidean spaces in the same way that we uncover hidden analogies between hyperbolic and Euclidean spaces [12, 39, 44, 45, 46]. Although there are some works [1, 17, 24, 25, 26, 27, 29] showing the gyro-structure SPD manifolds with the AI geometry, none of them provides rigorous mathematical formulation for the connection between the basic operations of [1, 17, 18, 24, 25, 26, 27] and the Al geometry SPD manifolds. In this paper, show how the basic operations can be constructed from the Riemannian geometry of matrix manifolds, and derive their compact expressions for SPD and Grassmann manifolds. In the case of SPD manifolds with the LE geometry [3], recovers precisely the operations of [3] that give these manifolds a vector space structure. In the case of Grassmann manifolds, we obtain gyrocommutative and gyrononreductive gyrogroups that share remarkable analogies with gyrovector spaces. To the
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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best of our knowledge, our work is the first that studies the structure of Grassmann manifolds under the framework of gyrovector spaces. Our main contributions are (1) We propose method for constructing some basic operations, i.e., matrix-matrix addition and scalar-matrix multiplication on SPD and Grassmann manifolds; (2) We derive compact expressions of these operations for the considered manifolds; (3) We verify the gyro-structure of SPD manifolds, and some axioms of gyrovector spaces for Grassmann manifolds; (4) We showcase our approach on the tasks human activity understanding and question answering.
2
Background
2.1
Gyrovector Spaces
Gyrovector spaces form the setting for hyperbolic geometry in the same way that vector spaces form the setting for Euclidean geometry [44, 45, 46]. We first recap the definitions gyrogroups and gyrocommutative gyrogroups proposed in [44, 45, 46]. For greater mathematical detail and in-depth discussion, we refer the interested reader to these papers.
Definition 2.1 (Gyrogroups [46]). pair (G, + is a groupoid in the sense that it a nonempty set, G. with binary operation, @. A groupoid (G, @) is gyrogroup if its binary operation satisfies the following axioms for a, b, G:
(G1) There at least one element e E called a left identity such that e D a = a.
(G2) There is an element Ga E G called a left inverse of such that Go @ e. G for each a, b E G such that a (a b) @ gyrla, blc (Left Gyroassociative Law).
(G3) There is automorphism gyrla, b)
The antomorphism gyrla, b) is called the syroastomorphism, or the gyration of G generated by b. (G4) gyr(a, b] gyrla la b] (Left Reduction Property).
Definition 2.2 (Gyrocommutative Gyrogroups [46]). A gyrogroup (G, @) is gyrocommutative if satisfies
+ gyr(a, b] @ a) (Gyrocommutative Law).
The following definition of gyrovector spaces is slightly different from Definition 3.2 in [46]. Definition 2.3 (Gyrovector Spaces). A gyrocommutaative gyrogroup (G, equipped with scalar multiplication
(1,z) + is called a gyrovector space it satisfies the following axioms s, E R and a, c. G: and (-1) ④ Oa.
(=)
(V3) (0
(V4) gyr|a (| o gyrla,
(V5) gyr(s 3 a a) 1d. where Id is the identity map.
Note that the axioms of gyrovector spaces considered in our work are more strict than those proposed in [24, 25, 26, 27J. Thus many results proved in [24, 25, 26, 27] can be applied our case, which gives rise to interesting applications.
3 Proposed Approach
For simplicity of exposition, we will concentrate on real matrices. Denote by Mn the space of n m matrices, Sym the space of n x n SPD matrices, Sym, the space of n symmetric matrices, O, the space of n x orthogonal matrices, Gra.p the y-dimensional subspaces of R Let M be Riemannian homogeneous space, TpM be the tangent space of M at P M. Denote by ePp(P) and (og(P) the usual matrix exponential and logarithm of P. Expp (W) the exponential

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map at P that associates to tangent vector W E TpM point of M, Logp(Q) the logarithmic map Q E M at Let (p-q(W) be the parallel transport W from to Q along geodesics connecting P and Q. We will use superscripts for the exponential and logarithmic maps, and the parallel transport to indicate their associated Riemannian metric (in the case of SPD manifolds) or the target manifolds (in the case of Grassmann manifolds). The following definitions generalize those of [12] to the matrix manifold setting.
Definition 3.1. Let r be positive real number. The binary operation P & Q where P. Q E M is obtained by projecting Q' in the tangent space the identity element I M with the logarithmic map, computing the parallel transport of this projection from I to P along geodesics connecting I and Pr, and then projecting back on the manifold with the exponential map, i.e.,
p Q (Expp- (To--p-Logg(Q)
Definition 3.2. The scalar multiplication @ P where t E R and P E EM is obtained by projecting in the tangent space at the identity element M with the logarithmic map. multiplying this projection by the scalar in T,M, and then projecting it back on the manifold with the exponential map, i.e.,
1 8 P Expr( Logr
In addition to the basic operations defined above, we need to determine an automorphism in order verify the gyro-structure of given matrix manifold. In the following, we will provide such automorphisms and derive compact expressions of the basic operations for SPD and Grassmann manifolds. The obtained expressions will ease the task of verifying the axioms of gyrovector spaces. Also, they often lead to simple and efficient implementations of neural networks on the considered manifolds. Furthermore, they enable effective generalizations of some operations on Euclidean spaces, e.g., matrix scaling to these manifolds (see Sections 4.1.2 and 4.2.2).
3.1 Gyrovector Spaces of SPD Matrices
We investigate the gyro-structure of SPD manifolds with the Al and LE geometries in Sections 3.1.1 and 3.1.2, respectively. The Al and LE frameworks are reviewed the supplemental material. We refer the interested reader to that document for theoretical results that reveal hidden analogies between SPD manifolds with the Al and LE geometries and Euclidean spaces.
3.1.1 AI Gyrovector Spaces
We first examine SPD manifolds with the Al geometry. Lemma 3.3 gives a compact expression of the binary operation (matrix-matrix addition). Lemma 3.3. For P, Q E Sym the binary operation P th Q is given as
P P Q - QP))
Proof See the supplemental material.
An implicit assumption in Eq. (3) is that positive definite matrices are taken from the computations related matrix powers. This assumption used all computations in Sections 3.1.1 and 3.1.2.
The identity element Sym is the n x n identity matrix I,. Then, from Eq. (3), the inverse of P is given by
P-1
Lemma 3.4. For P E Sym and E R, the scalar multiplication t @at P is given
E.e P
(4)
Proof See the supplemental material.
Definition 3.5 (AI Gyrovector Spaces). Define a binary operation Cº and a sealar multiplication Tai by Eqs. (3) and (4), respectively. Define a gyroautomorphism generated by P and Q as
gyr P. Q (F,, (P
(5)
where (P, (P (P QP PP P
Theorem 3.6. Gyrogroups (Sym st.) with the scalar multiplication Bas form gyrovector spaces
(Sym @% Rai).

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Proof See the supplemental material.
The expressions of the binary operation, scalar multiplication, and gyroautomorphism given in Eqs. (3), (4), and (5) have appeared in [1, 17, 18]. However, these works concern with algebraic structures referred to generalized gyrovector spaces where their considered set of axioms is different from the one in Section 2. The expressions given in Eq. (5) with r = 1 and r = 2 have also appeared in [24, 25, 26, 27]. However, no general form of the gyroautomorphism is given in these works.
3.1.2 LE Gyrovector Spaces
The LE metrics have proven yield similar results as the Al metries in practice, but with much simpler and faster computations [3]. This motivates us to study the gyro-structure of SPD manifolds with the LE geometry. As far we know, this topic has not been investigated in previous works. The result given in the following lemma is not trivial as closed formulae do not exist for the parallel transport associated with the LE geometry.
Lemma 3.7. For P, Q E Sym,, the binary operation P Øie Q is given
P P Q (exp(log(P")) + log Q*)) $
(6)
Proof See the supplemental material.
From Lemma 3.7, the inverse of P is given by
P
Similarly to the scalar multiplication Cai. the scalar multiplication S1e is constructed from Eq. (2).
Lemma 3.8 gives an expression of this operation that is straightforward. Lemma 3.8. For P E Sym and E R, the scalar multiplication t &it P is given by
1B P
Definition 3.9 (LE Gyrovector Spaces). Define a binary operation @he and a scalar multiplication Sic by Eqs. (6) and (7), respectively. Define gyroautomorphism generated by P and Q as
gy Q. =dd
Theorem 3.10. Gyrogroups (Sym (%,) with the scalar multiplication form gyrovector spaces (Sym o% 0te).
Proof See the supplemental material.
The conclusion of Theorem 3.10 agrees with [3] which shows that the space of SPD matrices with the LE geometry has vector space structure. This vector space structure given by the operations defined in [3] that turn out be the binary operation and scalar multiplication gyrovector spaces the specific case where r 1. Indeed, it can shown that the mapping exp (Sym,, +, (Sym Di. @te) is vector space isomorphism. Thus, for any the operations defined above also turn the space of SPD matrices with the LE geometry into a vector space. This generalizes the result of [3].
LE
3.2 Grassmann Gyrocommutative and Gyrononreductive Gyrogroups
In the previous sections, we concern with the non-positively curved spaces of SPD matrices. this section, we focus on another family of matrix manifolds, i.e., the non-negatively curved Grassmann manifolds. We adopt the following definition for Grassmann manifolds:
Gro.p (P E Max p = P2 = PP, rank(P) p)-
The Riemannian geometry of Grassmann manifolds is reviewed the supplemental material.
I
Let In,p
be the identity element of Gra,p- For the sake of convenience, we denote (P). For P, Q GIn,p, assuming that Ln, and P are not in each other's cut locus (see
Log P = Logi

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the supplemental material for definition of cut locus of Grassmann manifolds), and In and Q are each other's cut locus, the binary operation is defined
P Bgg Q Expp (P (Log (Q))).
Lemma 3.11. For P, E Grn.g the binary operation Bor Q is given as
@gr = (xp((P.In.))) (xpp)))eppppPPpppp Io.))
(8)
where [...] denotes the matrix commutator.
Proof See the supplemental material.
Lemma 3.12 gives a closed-form expression the binary operation terms of and Q.
Lemma 3.12. For P. Q E GIn.z the binary operation P Egr Q is given
Bgr Q = exp { log (II, 2P) - pI, Q exp ( (- log (II, - 2P) 2Im)
Proof See the supplemental material.
For P E Gra.p such that In.p and P are not in each other's cut locus, the inverse Ogr P of P is
defined as
gr P Expí (-Logi gr (P)).
The scalar multiplication is defined as in Eq. (2) (subject the assumption stated above).
Lemma 3.13. For Grn.p and E TR, the scalar multiplication &g P given by
t gr P exp([tP, In.p)) exp(-ItP,
Proof See the supplemental material.
Lemma 3.14 gives closed-form expression of the scalar multiplication in terms of and P.
Lemma 3.14. For E Grn.p and E IR, the scalar multiplication 1. &8gr P is given by
1ggr P = exp (5 log (IIa 2I, 2I, In.p exp log (II, 2P) (In
2))
Proof See the supplemental material.
Grassmann manifolds, when equipped with the above operations, do not form gyrovector spaces. However, as we will show later, they still form spaces that verify most of the axioms of gyrovector spaces. Note that since these operations can only be defined under the assumptions stated the beginning of this section, all the axioms in Section 2.1 must be considered under these assumptions. We thus make them implicitly in the following. In order to specify the structure of Grassmann manifolds, we need to define some new algebraic structures that are generalizations of groups (Definitions 3.15 and 3.16) and vector spaces (Definition 3.17).
Definition 3.15 (Nonreductive Gyrogroups). groupoid (G, ()) is nonreductive gyrogroup if its binary operation satisfies axioms (GI), (G2), and (G3).
Definition 3.16 (Gyrocommutative and Gyrononreductive Gyrogroups). A nonreductive gy- rogroup (G, +) is gyrocommutative and gyrononreductive if it satisfies the Gyrocommutative Law. Definition 3.17 (Nonreductive Gyrovector Spaces). gyrocommutative and gyrononreductive gyrogroup (G, θ) equipped with scalar multiplication @ is called a nonreductive gyrovector space if it satisfies axioms (V1), (V2), (V3), (V4), and (V5).
Definition 3.18 gives the expression of gyroautomorphism in Grassmann manifolds.
Definition 3.18 (Grassmann Gyrocommutative and Gyrononreductive Gyrogroups). Define a binary operation Byr and a scalar multiplication &gr by Eqs. (8) and (9), respectively. For P.Q GIa., assuming that Lap and P are not in each other's cut locus, Ln,p and Q are not in each other's 's cut locus, In.p and @gr Q are not in each other's 's cut locus, then a gyroautomorphism generated by P and Q can be defined
gyr gr
P. Q For (P,
(10)
where For (P, Q) is given by
F (P, Q) exp( yr Q, In
exp((Q, In,pl).
(11)
5

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The following identities are required for the verification of some axioms of nonreductive gyrovector spaces for Grassmann manifolds subject to certain conditions (Theorem 3.20). These identities are new to the best of our knowledge.
Lemma 3.19. For P, P.F F identities hold:
Grn.p' O
Tp Grn.p N, and 8 R, the following E
Oz A
1. Logo (OPO*) O Logs (P)O";
2. Exp oPP (040*) O Expp (A)O";
3. P| = (I, ( ( (I, - -2P);
4. exp(s A. P|) (I,, 2P) exp( -|A (I, 2P).
Proof See the supplemental material.
We are now ready to state the main results of this section.
Theorem 3.20. Groupoids (Grn.p @grr form gyrocommtative and gyrononreductive gyrogroups.
Furthermore, gyrocommutative and gyrononeductivv gyrogroups (Gra.p: Sor) with the scalar mul- tiplication @gr satisfy axioms (VI) and (V4), and axioms (V2), (V3), and (V5) under the following conditions:
(V2) (s + Bgr P 8gr P Bor Egr for P E Gra. t E R, and R such that:
min 2 Im(A:)) |s|
π
where is an eigenvalue of [P, p. and lmn(ds) is the imaginary part of X;-
(V3) (st) &gr P= s gg (t @r P) P Grn.p- s SER, R, and E R. such that:
min x
Im(A)))
(V5) gyT Bgr P. P Id Gra.p and s, t such that:
max[[s], 14, t < min As 2 Imn())]
π
Proof See the supplemental material.
The following corollaries are useful practice the verification axioms (V2), (V3), and (V5). Corollary 3.21. Gyrocommutative and gyrononreductive gyrogroups (Grn.ps @ gr) with the scalar multiplication @gr satisfy axioms (V2), (V3), and (V5) if the right-hane sides of the inequalities stated in Theorem 3.20 are replaced with
π
P. I Inplll
where ||-I| denotes the Hilbert-Schmidt norm.
Proof See the supplemental material.
Corollary 3.22. Gyrocomnutative and gyrononreductive gyrogroups (GIn,ps θgr) with the scalar multiplication &gr satisfy axioms (V2), (V3), and (V5) under the following conditions: (V2) |s $ (V3) |t] < 1; (V5) max[[s], [t], + t|} < 1.
Proof
See the supplemental material.
It follows from Corollary 3.22 that if max([s], [4] + +() < 1, then Grassmann manifolds, when equipped with the basic operations defined in Eqs. (8) and (9), verify all the axioms of nonreductive gyrovector spaces.
4 Applications
To showcase our approach, we propose new methods for human activity understanding and question answering. We refer the interested reader the supplemental material for more applications.

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4.1 Human Activity Understanding
In this section, we develop class of RNNNs on SPD manifolds for human activity understanding. It is worth mentioning that the operations defined in Section 3.1 as well as those constructed in Section 4.1.2 can be used build any type of neural networks on SPD manifolds, e.g., convolutional neural networks. However, since RNNs are based on update equations that involve the basic operations on matrices, they are well suited for validating our approach.
4.1.1 Problem Formulation
Human activities can be recognized from low/mid level features [35, 36] or high-level poses [23, 32, 33, 34]. We use 3D skeleton data as they have shown to outperform low/mid level features for the considered task [23]. The goal is to build a model that for each sequence of 3D positions of body (hand) joints identifies the action performed by the person (or group of persons) in the sequence.
4.1.2 Proposed Method
We will make use of the basic operations on AI and LE gyrovector spaces and the concept of gy- roderivative in these spaces (see the supplemental material) that is similar to hyperbolic derivative [4] in Möbius gyrovector spaces. We also need to generalize some operations of Euclidean RNNs to the SPD manifold setting. Here focus operations, i.e., matrix scaling and pointwise nonlinearity.
Matrix Scaling
I Sym We R" W 0, then the matrix scaling W @% is given by
W
x spd P = U diag( W * V)UT,
where U diag( (T)UT is the eigenvalue decomposition P, and W V is the element-wise multi- plication.
Pointwise nonlinearity If y is a pointwise nonlinear activation function, then the pointwise non- linearity o (P) given by
@ (P) diag(max(cL,, pb (V)))
where 0 is rectification threshold, and U diag( (V) is the eigenvalue decomposition of P.
By adapting a class of models that are invariant time rescaling [42] the SPD manifold setting (see the supplemental material), we obtain the following update equations for our models:
P 40 0. (Wn, Bapd H W2 8xpd X.),
(12)
H @ a o ((eH.-)) P.).
(13)
where X, E Sym, is the input at frame f, H H, E Sym are the hidden states at frames and f, respectively, Wh, W. E IR, and a E R is learnable parameter.
4.1.3 Implementation Details
In order to retain the correlation of neighboring joints [5, 49] and to increase feature interactions encoded by covariance matrices, we first identify closest left (right) neighbor of every joint based on their distance to the hip (wrist) joint', and then combine the 3D coordinates of each joint and those of its left (right) neighbor create feature vector for the joint. For a given frame t, a mean vector μ, and a covariance matrix Σ, are computed from the set of feature vectors of the frame and then combined [30] to create SPD matrix
= Σg + (1e) (n:)²*
14
The lower part of matrix log( Y, is flattened obtain vector ve. All vectors v, within time window [t, + c - 1] where c a constant are used to compute a covariance matrix as Z, = Li=t (v; vv, ",, where v, >Ete-l v. Matrix Z, is then the input data at frame of the networks. Our network GyroAI-HAÜNE
illustrated in Fig. la. For classification, the network output is projected to the tangent space at the identity matrix using the logarithmic map. The lower part of the resulting matrix is flattened and then fed fully-connecter layer.
For joints having more than 2 neighbors, of them can be chosen.
7

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Y
Y Y+-o-1
d+1
dl. flay!!
(d+1)(d+2)
Gray
2
Z
Z
Ereandding
Queation Embeeding
Twen orbedsin
Theer inteedine
Erroddingg
++1
Pestue trower
Mogatte
(a) Architecture of GyroAI-HAUNet (d 6)
(b) Architecture GyroGR QANet
Figure 1: Our proposed architectures for human action understanding and question answering.
Table 1: Accuracy comparison (%) of our networks against ffate-o-the-aae art SPD neural networks.
SPDNet
SPDNeiBN
SPD.SRU
HAUNE
Gy
HAUNet
Datnset
M 1
M 3
M
M =
M
M
M
M 3
M
M -
HDMOS
58.44
72.75
62.54
76.25
42,26
54.79
61.50
78.14
57.01
74.53
#HIDMOS
0.77
0.13
0.82
0.05
0.32
0.05
0.31
0.05
MB
FPHA
87.65
88.17
88.52
91.83
78.57
85,16
89.73
96.00
83.03
89.94
АЕРHA
0,04 MB
0.28 MB
0.05 MB
0.31 MB
0.019 MB
0.114 MB
0,018 MB
0.110 MB
0.018 MB
0.110 MB
NTU60
73.26
77.38
75.84
79.52
66.25
75.34
83,12
94.72
77.25
89.44
#NTU60
0.03 MB
0.20 MB
0.04 MB
0.28 MB
0.004 MB
0.027 MB
0.004 MB
0.026 MB
0.004 MB
0.026 MB
4.1.4
Results
We use three datasets, i.e., HDM05 [16], FPHA [13], and NTU RGB+D 60 (NTU60) [38]. These datasets include three different types of activities: body actions (HDM05), hand actions (FPHA), and interaction actions (NTU60). In the same spirit of previous works [6. 20, 40], we are interested in comparing manifold networks with focus on SPD manifolds, and do not necessarily seek state-of- the-art performance for the target task. We use temporal pyramid representation for each sequence. At temporal pyramid M, given sequence is partitioned into M subsequences of equal size. Each subsequence is then fed to a model with its own parameter set. The outputs from all the models are concatenated to create final representation of the sequence. We run each model three times and report the best accuracy from these three runs [12]. The experimental settings can be found in the supplemental material. This document also reports the mean accuracies and standard deviations of some representative methods, an ablation study, and a comparison of our networks against Euclidean RNNs, transformers, HNNs, and graph neural networks (GNNs).
Comparison against SPD Neural Networks Our networks, referred as GyroAI-HAUNet and GyroLE-HAUNG are compared against SPDNet [20], SPDNetBN [6], and SPD-SRU [8]. Results of these networks are obtained using their official 2..e with default parameter settings. Results for M = 1 and M = 3 are given in Tab. 1. On HDM05 dataset, GyroAA--HAUN outperforms SPDNet and SPD-SRU, and performs worse than SPDNetBN when M 1. However, when M 3, AAUNee- gives the best result among the competing networks. On FPHA and NTU60 datasets, GyroAI-HAUNet gives the best results for both M = and M = 3. In terms of model size, GyroAI-HAUNet and GyroLE-HAUNe require far fewer parameters than SPDNet and SPDNetBN in all cases. For example, on FPHA dataset, when M 3, GyroAI-HAUNet outperforms SPDNetBN by 4.17% with 2.8 times fewer parameters, Also, on NTU60 dataset, when M 3, GynoAI-HAUNet outperforms SPDNetBN by 15.20% with 10.7 times fewer parameters.
'nttps///papezz http://hiuuhhuag///aa SPDNet html
'nttpa://github wttptttuu..cmmheeeeeetee cow/zhivu
2019
'httpa://giii //githbb/bbbbb...../m cattub..oo///eeeei com/zhemxinggiaa. /t/////DSSS//S//S S.DD-SRR//ree/maattte For all the networks, setting M > did not yield better results the three datasets.
8

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4.2 Question Answering
In this section, we consider learning word embeddings in GTa.p. We also investigate the use of product manifolds GIar, Sym, for training word embeddings. This idea is mainly motivated by the work of [14] that shows the efficacy of mixed-curvature representations for graph embeddings.
4.2.1 Problem Formulation
Let Q list questions, AA be list of answers to the questions in Q. Each question g Q has a list of candidate answers in A. The candidate set comes with their relevancy judgements, where answers that are correct (positive) have labels equal I and otherwise. The goal is to build a model that for each query q and its list of candidate answers generates an optimal ranking such that correct answers appear at top the list [29, 43].
4.2.2 Proposed Method
The core idea is to learn a scoring function [3] given as
Oya a) -wyd(Q A) + wb:
where Q and A are embeddings of question and answer a, respectively, wy, wh E IK are parameters of the model, and d(-) is distance function in the target manifolds. We define matrix scaling operation as follows.
Matrix Scaling
We parameterize point P GIn. by matrix B E Mp.n-p such that
0
B
P.
I.Ill
- -BT
Then point can be computed (see Eq. (8) and the supplemental material) by
P exp([P, Inpp)) exp(- In,p)) exp ([-
B
0.
B
0
In.p exp
-B"
The matrix scaling S is defined as
A P exp (I-(A= (BT
A
I In.p exp (- (-[CCA-
A+B
gr
0
where A E Mp.n-p. p- The embedding of question is computed from those of its tokens as
Q 0gr (IS 8 T1) Byr (S 8 T2) DDr (S 8 T()).
where T3 GIa.p. i 1, I,,.., are embeddings of the tokens in question 9. S Mp.n-p and B E Grn,p are parameters of the model. The embedding of answer a is the summation those of its tokens using operation @grr For P. Q GI,,pp the distance function d(...) is defined as
d(P, Q) = dgr (P, Q = PP QlI
We also train word embeddings in product manifold Grnj.p Sym using the distance function
d((P gr: P Pepa); (Qur: Qepa)) = dgr (P gr: Qgr) + Tdl spa
(P Pepds Qepa):
where P gr: Qgr E GTa P apdd Qapd Sym (ppa(..)) is the Finsler distance function proposed in [29], and is constant. This results in three models based on the scaling, rotation, and reflection transformations of [29]. Our Grassmann model is shown in Fig. lb.
4.2.3 Results
We use two datasets, i.e., TrecQA [47] (clean version) and WikiQA [50]. The mean average precision (MAP) and mean reciprocal rank (MRR) are used as evaluation metries [43]. We compare our m against the SPD models of [29]. Since the codes of these models are not publicly available,
CCode available ht httpp::/// // wttuu..coo/eeeeeeeeeeeeeeeeeeeee wwt.coo/////e.e com/spratmtt/qq.

---

Table 2: Comparison of our models against the SPD models of [29]. The SPD models learn
embeddings in f4 Our Grassmann model learns embeddings in 1T Our models based on the product manifold learn embeddings in Gr14,7 Symg. Results are computed over three runs.
TreQQA
WikiQA
Time (TrecQA.
DOF
Model
MAP
MRR
MAP
MRR
Train/epoch
Test
105
SPD
47.77 0.18
$7.54 0.32
59.42
60.57 0.05
15.35
1.41
105
SPD
48.35 + 1.24
57.64 + 2.23
60.68 + 0.42
62.01 = 0.34
15.56
1.43
105
SPD
48.07 + 0.89
54.49 0.87
59.37 1.25
60.74 1.50
15.98
1.41
105
SPD Ros
49.02 = 1.62
58.72 + 1.85
60.60 +: 0.67
62.28 = 0.74
16.09
1.43
105
Bet
48.79 + 0.91
57.12 + 1.23
59.07 0.53
60.58 0.52
15.98
1.41
105
SPD Ber
48.33 + 0.48
56. + 0.97
59.93 +: 0.18
61.80 = 0.39
16.09
1.43
GyroGR- Qeteet
1.29
1.45
85
GyroRR-SPDD Se. QANet
50.10 - 0.30
57,70 + 0.93
60.62 +: 0.25
62.42 = 0.16
11.62
0.90
GynoGR-SPD Mos QANet
50.27 + 0.56
58.62 - 1.35
59.78 1 0.15
61.66 0.23
11.72
0.90
85
GyroGR-SPD Ref QANet
48.83 + 1.89
58.11 + 0.87
60.41 + 0.39
61.86 = 0.35
11.72
0.90
we implemented them by following closely the instructions [29]. The implementation details and experimental settings can be found in the supplemental material.
Tab. 2 reports the means and standard deviations of MAP and MRR from three runs (DOF stands for degrees of freedom). Our Grassmann model BeroGR-QANCE performs favorably against most of the SPD models on TrecQA dataset. Also, results of our models based on the product manifold show the efficacy of mixed-curvature representations [14] in question answering. Note that the numbers of DOF of our models are smaller than those of the SPD models. We also note that techniques in [19, 28] could potentially improve the performance of our models.
Limitations of Our Work
To develop our SPD models, we only construct the basic operations and two other operations, i.e., matrix scaling and pointwise nonlinearity (see Section 4.1.1). Other operations [39] should also be designed in order to improve the representation power of our networks. Also, the question how to generalize a broader class of DNNs to the SPD manifold setting should be addressed in future work.
Our experiments for question answering have shown the usefulness of product manifolds Grm.p x Sym for learning word, entity, and relation embeddings. More investigation is needed to see product of multiple dmallsr-dimension SPD and Grassmann manifolds will improve performance [14]. It would also be interesting to find out patterns that show the relationship between the performance of the embeddings and the theoretical curvature of the manifolds in our problems [9].
6 Conclusion
We have shown that the Al and LE geometries of SPD manifolds have strong connection to hyperbolic geometry, and Grassmann manifolds share very similar properties with gyrovector spaces. We have presented new methods for generalizing Fuclidean neural networks to the SPD and Grassmann manifold settings. Our experimental results on human activity understanding and question answering have demonstrated the effectiveness of the proposed approach.
Acknowledgments
We are grateful for the constructive comments and feedback from the anonymous reviewers. We thank Amir Shahroudy, Jun Liu, Tian-Tsoon Ng, Gang Wang, Guillermo Yuan, Seungryul Baek, and Tae-Kyun Kim for giving us access to the NTU RGB+D and FPHA datasets.
Shanxin
10

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Checklist
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(a) Do the main claims made in the abstract and introduction accurately reflect the paper's
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(b) Did you describe the limitations of your work? [Yes] See the supplemental material.
(c) Did you discuss any potential negative societal impacts of your work? [No]
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(a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material or as a URL)? [Yes] We included the codes, data, and instructions for part of our experiments as a URL and in the supplemental material.
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13


Reviews:
[{"review_id": "Q2FMgrXWft", "reviewer": "Reviewer_L1ia", "summary": "This paper investigates the space of SPD matrices and Grassmann manifolds as gyro-vector spaces. The authors showed the basic definitions of these algebraic structures and proved that the SPD manifold satisfies the axioms in two different ways, i.e., the so-called AI and LE geometries.  The authors also showed that Grassmann manifolds satisfy similar properties.\n\nAs applications, the authors rebuilt SPD networks using the proposed operations of SPD matrices and showed performance improvements over SPDNets on actions recognization. The authors also use Grassmann manifolds for word embedding, showing marginal improvements over SPD models.", "questions": "I am convinced that the authors imported new geometric tools into deep learning, which can be potentially useful. However, it is not clear why they are needed, and how they are used in the applications.\n\nSection 3 introduced the gyro-vector space structures of SPD and Grassmann manifolds. The author gives a list of definitions and properties. It should be better emphasized which of these definitions are new. If these are existing definitions, proper citations are needed in the body of the definitions. Similar to the formal statements: does section 3 merely review existing properties of such spaces or there are new discoveries?\n\nThe author vaguely mentioned that \"We note that the expressions of the binary operation, scalar multiplication ... have already been found. However, none of them pointed out the relations in Eqs. (1) and (2)\". It is not clear what \"relations\" this sentence refers to.\n\nMore importantly, why the gyrovector spaces are important to redefine the structure of such manifolds? What are the motivations from the application perspective? What limitations of existing techniques call for these gyrovector spaces?\n\nSection 4 Applications. There should be details in implementing the modified SPD networks, which are missing. I suggest the authors explain in detail and formally the background problem (classification), the original SPD net, and how the gyro-vector space is plugged into this framework.", "limitations": "This work is mostly thoretical. In the applications, the authors proposed variations of existing deep learning techniques for the applications of word embedding and action recognition. The authors' experiments do not advance the state-of-the-art performance scores from the application perspective. The negative societal impact is therefore limited by these existing techniques.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 1, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Pro:\n- new geometric tools for deep learning\n\nCon:\n- the writing lacks motivation and clarity. (see the questions section below)\n- some implementation details are missing in section 4", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "N4RbF1Qqigl", "reviewer": "Reviewer_DGz9", "summary": "The paper studies the gyro-structure of several matrix manifolds. In the process, the authors show that the affine-invariant (AI) and log-Euclidean (LE) geometries of SPD manifolds are closely related to hyperbolic geometries, and Grassmann manifolds share similar properties with gyrovector spaces. Several interesting applications based on these geometries are presented and compared against other models.", "questions": "I have developed some uncertainty about the magnitude of significance of such works. The gyrovector operations seem to be basically standard geometric operations that have been understood for quite a while. The packaging of such standard operations in the gyrovector formalism seems reasonably straightforward, albeit necessary in certain contexts. There does not appear to be much particularly profound new analysis or geometry which makes this translation into the gyro-structure language possible. This may be a misunderstanding on my part which may only be clarified by a thorough reading of the supplementary materials. Unfortunately, given the volume of such supplementary material and the reviewing demands of the conference, it is often not possible to perform such a detailed reading in the available time.\n\nIf the authors could provide a response to dispel my impression noted above, I would be grateful. They may want to emphasise it in the main body of the paper as well.", "limitations": "These need to be discussed clearly in the main paper and not just as supplementary material.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The paper is well-structured and is written in a highly readable and professional style. It builds on several recent works on the topic effectively.  The applications and associated experimental results are interesting and appear convincing.\n\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "-TnJpqaTIyD", "reviewer": "Reviewer_65HJ", "summary": "This paper studied potential gyrovector space structures for SPD manifolds with AI and LE geometries and the Grassman manifolds. The authors defined gyroautomorphisms for these geometries and then verified that SPD manifolds can form gyrovector spaces with AI and LE geometries. For Grassman manifold, the gyrogroup is not gyroreductive, but it is gyrocommunitative. From these defined gyrovector spaces, the authors derived operations that can be used for neural networks. They empirically validated that such manifold networks can be applied to applications such as multi-time-series data analysis and word embeddings. ", "questions": "- Contributions (2) and (3), Why AI geometry is left out?\n- Definition 3.1, How is r useful?", "limitations": "N/A", "rating": 7, "confidence": 2, "soundness": 4, "presentation": 2, "contribution": 4, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\n1. The paper contains some original ideas. \n2. The theoretical analyses seem to be well supported by the empirical results provided.\n\nWeaknesses:\n\n- This paper is difficult to read, partly because of the prerequisite concepts, and partly because of the authors' writing style. The authors can help the reader by making sure that new concepts are properly defined and motivated. The information in this paper can be better organised so that the key messages are clear and consistent.\n\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept