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Needham Reiterates Instructure Holdings (INST) Buy Recommendation

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Dale E. Bowen; CFO; Instructure Holdings, Inc. David Banks; VP of IR; Instructure Holdings, Inc. Peter Walker Stephen M. Daly; CEO & Director; Instructure Holdings ...

Q3 2023 Instructure Holdings Inc Earnings Call

Instructure Holdings, Inc. beats earnings expectations. Reported EPS is $0.25, expectations were $0.2. Operator: Ladies and gentlemen, thank you for standing by and welcome to Instructure’s Third Quarter 2023 Earnings Conference Call. At this time ...

Instructure Holdings, Inc. (NYSE:INST) Q3 2023 Earnings Call Transcript

Massive open online course provider Coursera has reported a third-quarter 2023 net loss of $32.1 million. Stay up to date with all our events, awards and publications. Information you provide us with will be kept private at all times, and will be used for ...

US: Instructure buys academic credentialing platform

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Instructure Holdings (NYSE:INST) on Monday said it has signed an agreement to acquire academic credential management platform and network Parchment. The transaction is valued at about $835 million. Including a tax benefit resulting from the transaction,

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Instructure Holdings

instructure

Table of Links Abstract and 1. Introduction Preliminaries


Proposed Approach
3.1 Notation
3.2 Nueral Networks on SPD Manifolds
3.3 MLR in Structure Spaces
3.4 Neural Networks on Grassmann Manifolds


Experiments


Conclusion and References A. Notations B. MLR in Structure Spaces C. Formulation of MLR from the Perspective of Distances to Hyperplanes D. Human Action Recognition E. Node Classification F. Limitations of our work G. Some Related Definitions H. Computation of Canonical Representation I. Proof of Proposition 3.2 J. Proof of Proposition 3.4 K. Proof of Proposition 3.5 L. Proof of Proposition 3.6 M. Proof of Proposition 3.11 N. Proof of Proposition 3.12 A NOTATIONS Tab. 3 presents the main notations used in our paper. B MLR IN STRUCTURE SPACES Algorithm 1 summarizes all steps for the computation of pseudo-gyrodistances in Theorem 3.11. Details of some steps are given below: Authors:
(1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (xuan-son.nguyen@ensea.fr);
(2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (son.nguyen@ensea.fr);
(3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (aymeric.histace@ensea.fr). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction Preliminaries Proposed Approach
3.1 Notation
3.2 Nueral Networks on SPD Manifolds
3.3 MLR in Structure Spaces
3.4 Neural Networks on Grassmann Manifolds Experiments Conclusion and References Preliminaries Preliminaries Preliminaries Proposed Approach 3.1 Notation 3.2 Nueral Networks on SPD Manifolds 3.3 MLR in Structure Spaces 3.4 Neural Networks on Grassmann Manifolds Proposed Approach 3.1 Notation 3.1 Notation 3.2 Nueral Networks on SPD Manifolds 3.2 Nueral Networks on SPD Manifolds 3.3 MLR in Structure Spaces 3.3 MLR in Structure Spaces 3.4 Neural Networks on Grassmann Manifolds 3.4 Neural Networks on Grassmann Manifolds Experiments Experiments Experiments Conclusion and References Conclusion and References Conclusion and References A. Notations A. Notations B. MLR in Structure Spaces B. MLR in Structure Spaces C. Formulation of MLR from the Perspective of Distances to Hyperplanes C. Formulation of MLR from the Perspective of Distances to Hyperplanes D. Human Action Recognition D. Human Action Recognition E. Node Classification E. Node Classification F. Limitations of our work F. Limitations of our work G. Some Related Definitions G. Some Related Definitions H. Computation of Canonical Representation H. Computation of Canonical Representation I. Proof of Proposition 3.2 I. Proof of Proposition 3.2 J. Proof of Proposition 3.4 J. Proof of Proposition 3.4 K. Proof of Proposition 3.5 K. Proof of Proposition 3.5 L. Proof of Proposition 3.6 L. Proof of Proposition 3.6 M. Proof of Proposition 3.11 M. Proof of Proposition 3.11 N. Proof of Proposition 3.12 N. Proof of Proposition 3.12 A NOTATIONS Tab. 3 presents the main notations used in our paper. B MLR IN STRUCTURE SPACES Algorithm 1 summarizes all steps for the computation of pseudo-gyrodistances in Theorem 3.11. Details of some steps are given below: Authors: (1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (xuan-son.nguyen@ensea.fr); (2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (son.nguyen@ensea.fr); (3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (aymeric.histace@ensea.fr). Authors: Authors: (1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (xuan-son.nguyen@ensea.fr); (2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (son.nguyen@ensea.fr); (3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (aymeric.histace@ensea.fr). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. available on arxiv available on arxiv

Key Notations and Algorithm for Computing Pseudo-Gyrodistances in Structure Spaces

Table of Links Abstract and 1. Introduction Preliminaries


Proposed Approach
3.1 Notation
3.2 Nueral Networks on SPD Manifolds
3.3 MLR in Structure Spaces
3.4 Neural Networks on Grassmann Manifolds


Experiments


Conclusion and References A. Notations B. MLR in Structure Spaces C. Formulation of MLR from the Perspective of Distances to Hyperplanes D. Human Action Recognition E. Node Classification F. Limitations of our work G. Some Related Definitions H. Computation of Canonical Representation I. Proof of Proposition 3.2 J. Proof of Proposition 3.4 K. Proof of Proposition 3.5 L. Proof of Proposition 3.6 M. Proof of Proposition 3.11 N. Proof of Proposition 3.12 I PROOF OF PROPOSITION 3.2 Proof. We first recall the definition of the binary operation ⊕g in Nguyen (2022b). J PROOF OF PROPOSITION 3.4 Proof. The first part of Proposition 3.4 can be easily verified using the definition of the SPD inner product (see Definition G.4) and that of Affine-Invariant metrics (Pennec et al., 2020) (see Chapter 3). To prove the second part of Proposition 3.4, we will use the notion of SPD pseudogyrodistance (Nguyen & Yang, 2023) in our interpretation of FC layers on SPD manifolds, i.e., the signed distance is replaced with the signed SPD pseudo-gyrodistance in the interpretation given in Section 3.2.1. First, we need the following result from Nguyen & Yang (2023). We consider two cases: K PROOF OF PROPOSITION 3.5 Proof. This proposition is a direct consequence of Proposition 3.4 for β = 0. L PROOF OF PROPOSITION 3. Proof. The first part of Proposition 3.6 can be easily verified using the definition of the SPD inner product (see Definition G.4) and that of Log-Cholesky metrics (Lin, 2019). To prove the second part of Proposition 3.6, we first recall the following result from Nguyen & Yang (2023 According to our interpretation of FC layers, We consider two cases M PROOF OF THEOREM 3.1 N PROOF OF PROPOSITION 3.12 Proof. We need the following result from Nguyen & Yang (2023). Authors:
(1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (xuan-son.nguyen@ensea.fr);
(2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (son.nguyen@ensea.fr);
(3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (aymeric.histace@ensea.fr). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction Preliminaries Proposed Approach
3.1 Notation
3.2 Nueral Networks on SPD Manifolds
3.3 MLR in Structure Spaces
3.4 Neural Networks on Grassmann Manifolds Experiments Conclusion and References Preliminaries Preliminaries Preliminaries Proposed Approach 3.1 Notation 3.2 Nueral Networks on SPD Manifolds 3.3 MLR in Structure Spaces 3.4 Neural Networks on Grassmann Manifolds Proposed Approach 3.1 Notation 3.1 Notation 3.2 Nueral Networks on SPD Manifolds 3.2 Nueral Networks on SPD Manifolds 3.3 MLR in Structure Spaces 3.3 MLR in Structure Spaces 3.4 Neural Networks on Grassmann Manifolds 3.4 Neural Networks on Grassmann Manifolds Experiments Experiments Experiments Conclusion and References Conclusion and References Conclusion and References A. Notations A. Notations B. MLR in Structure Spaces B. MLR in Structure Spaces C. Formulation of MLR from the Perspective of Distances to Hyperplanes C. Formulation of MLR from the Perspective of Distances to Hyperplanes D. Human Action Recognition D. Human Action Recognition E. Node Classification E. Node Classification F. Limitations of our work F. Limitations of our work G. Some Related Definitions G. Some Related Definitions H. Computation of Canonical Representation H. Computation of Canonical Representation I. Proof of Proposition 3.2 I. Proof of Proposition 3.2 J. Proof of Proposition 3.4 J. Proof of Proposition 3.4 K. Proof of Proposition 3.5 K. Proof of Proposition 3.5 L. Proof of Proposition 3.6 L. Proof of Proposition 3.6 M. Proof of Proposition 3.11 M. Proof of Proposition 3.11 N. Proof of Proposition 3.12 N. Proof of Proposition 3.12 I PROOF OF PROPOSITION 3.2 Proof . We first recall the definition of the binary operation ⊕g in Nguyen (2022b). Proof J PROOF OF PROPOSITION 3.4 Proof . The first part of Proposition 3.4 can be easily verified using the definition of the SPD inner product (see Definition G.4) and that of Affine-Invariant metrics (Pennec et al., 2020) (see Chapter 3). Proof To prove the second part of Proposition 3.4, we will use the notion of SPD pseudogyrodistance (Nguyen & Yang, 2023) in our interpretation of FC layers on SPD manifolds, i.e., the signed distance is replaced with the signed SPD pseudo-gyrodistance in the interpretation given in Section 3.2.1. First, we need the following result from Nguyen & Yang (2023). We consider two cases: K PROOF OF PROPOSITION 3.5 Proof . This proposition is a direct consequence of Proposition 3.4 for β = 0. Proof L PROOF OF PROPOSITION 3. Proof . The first part of Proposition 3.6 can be easily verified using the definition of the SPD inner product (see Definition G.4) and that of Log-Cholesky metrics (Lin, 2019). Proof To prove the second part of Proposition 3.6, we first recall the following result from Nguyen & Yang (2023 According to our interpretation of FC layers, We consider two cases M PROOF OF THEOREM 3.1 N PROOF OF PROPOSITION 3.12 Proof . We need the following result from Nguyen & Yang (2023). Proof Authors: (1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (xuan-son.nguyen@ensea.fr); (2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (son.nguyen@ensea.fr); (3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (aymeric.histace@ensea.fr). Authors: Authors: (1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (xuan-son.nguyen@ensea.fr); (2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (son.nguyen@ensea.fr); (3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (aymeric.histace@ensea.fr). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. available on arxiv available on arxiv

Mathematical Proofs for SPD Inner Products and Pseudo-Gyrodistances in Manifold Layers

Table of Links Abstract and 1. Introduction Preliminaries


Proposed Approach
3.1 Notation
3.2 Nueral Networks on SPD Manifolds
3.3 MLR in Structure Spaces
3.4 Neural Networks on Grassmann Manifolds


Experiments


Conclusion and References A. Notations B. MLR in Structure Spaces C. Formulation of MLR from the Perspective of Distances to Hyperplanes D. Human Action Recognition E. Node Classification F. Limitations of our work G. Some Related Definitions H. Computation of Canonical Representation I. Proof of Proposition 3.2 J. Proof of Proposition 3.4 K. Proof of Proposition 3.5 L. Proof of Proposition 3.6 M. Proof of Proposition 3.11 N. Proof of Proposition 3.12 H COMPUTATION OF CANONICAL REPRESENTATIONS Authors:
(1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (xuan-son.nguyen@ensea.fr);
(2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (son.nguyen@ensea.fr);
(3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (aymeric.histace@ensea.fr). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction Preliminaries Proposed Approach
3.1 Notation
3.2 Nueral Networks on SPD Manifolds
3.3 MLR in Structure Spaces
3.4 Neural Networks on Grassmann Manifolds Experiments Conclusion and References Preliminaries Preliminaries Preliminaries Proposed Approach 3.1 Notation 3.2 Nueral Networks on SPD Manifolds 3.3 MLR in Structure Spaces 3.4 Neural Networks on Grassmann Manifolds Proposed Approach 3.1 Notation 3.1 Notation 3.2 Nueral Networks on SPD Manifolds 3.2 Nueral Networks on SPD Manifolds 3.3 MLR in Structure Spaces 3.3 MLR in Structure Spaces 3.4 Neural Networks on Grassmann Manifolds 3.4 Neural Networks on Grassmann Manifolds Experiments Experiments Experiments Conclusion and References Conclusion and References Conclusion and References A. Notations A. Notations B. MLR in Structure Spaces B. MLR in Structure Spaces C. Formulation of MLR from the Perspective of Distances to Hyperplanes C. Formulation of MLR from the Perspective of Distances to Hyperplanes D. Human Action Recognition D. Human Action Recognition E. Node Classification E. Node Classification F. Limitations of our work F. Limitations of our work G. Some Related Definitions G. Some Related Definitions H. Computation of Canonical Representation H. Computation of Canonical Representation I. Proof of Proposition 3.2 I. Proof of Proposition 3.2 J. Proof of Proposition 3.4 J. Proof of Proposition 3.4 K. Proof of Proposition 3.5 K. Proof of Proposition 3.5 L. Proof of Proposition 3.6 L. Proof of Proposition 3.6 M. Proof of Proposition 3.11 M. Proof of Proposition 3.11 N. Proof of Proposition 3.12 N. Proof of Proposition 3.12 H COMPUTATION OF CANONICAL REPRESENTATIONS Authors: (1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (xuan-son.nguyen@ensea.fr); (2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (son.nguyen@ensea.fr); (3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (aymeric.histace@ensea.fr). Authors: Authors: (1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (xuan-son.nguyen@ensea.fr); (2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (son.nguyen@ensea.fr); (3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (aymeric.histace@ensea.fr). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. available on arxiv available on arxiv

Canonical Representations and Computation in SPD Matrices

Table of Links Abstract and 1. Introduction Preliminaries


Proposed Approach
3.1 Notation
3.2 Nueral Networks on SPD Manifolds
3.3 MLR in Structure Spaces
3.4 Neural Networks on Grassmann Manifolds


Experiments


Conclusion and References A. Notations B. MLR in Structure Spaces C. Formulation of MLR from the Perspective of Distances to Hyperplanes D. Human Action Recognition E. Node Classification F. Limitations of our work G. Some Related Definitions H. Computation of Canonical Representation I. Proof of Proposition 3.2 J. Proof of Proposition 3.4 K. Proof of Proposition 3.5 L. Proof of Proposition 3.6 M. Proof of Proposition 3.11 N. Proof of Proposition 3.12 G SOME RELATED DEFINITIONS G.1 GYROGROUPS AND GYROVECTOR SPACES Gyrovector spaces form the setting for hyperbolic geometry in the same way that vector spaces form the setting for Euclidean geometry (Ungar, 2002; 2005; 2014). We recap the definitions of gyrogroups and gyrocommutative gyrogroups proposed in Ungar (2002; 2005; 2014). For greater mathematical detail and in-depth discussion, we refer the interested reader to these papers. Definition G.1 (Gyrogroups (Ungar, 2014)). A pair (G, ⊕*) is a groupoid in the sense that it is a nonempty set, G, with a binary operation, ⊕. A groupoid (G,* ⊕*) is a gyrogroup if its binary operation satisfies the following axioms for a, b, c ∈ G:* (G1) There is at least one element e ∈ G called a left identity such that e ⊕ a = a. (G2) There is an element ⊖a ∈ G called a left inverse of a such that ⊖a ⊕ a = e. (G3) There is an automorphism gyr[a, b] : G → G for each a, b ∈ G such that a ⊕ (b ⊕ c) = (a ⊕ b) ⊕ gyr[a, b]c (Left Gyroassociative Law). The automorphism gyr[a, b] is called the gyroautomorphism, or the gyration of G generated by a, b. (G4) gyr[a, b] = gyr[a ⊕ b, b] (Left Reduction Property). Definition G.2 (Gyrocommutative Gyrogroups (Ungar, 2014)). A gyrogroup (G, ⊕*) is gyrocommutative if it satisfies* a ⊕ b = gyr[a, b](b ⊕ a) (Gyrocommutative Law). The following definition of gyrovector spaces is slightly different from Definition 3.2 in Ungar (2014). Definition G.3 (Gyrovector Spaces). A gyrocommutative gyrogroup (G, ⊕) equipped with a scalar multiplication Definition G.3 (Gyrovector Spaces). A gyrocommutative gyrogroup (G, ⊕*) equipped with a scalar multiplication* (t, x) → t ⊙ x : R × G → G is called a gyrovector space if it satisfies the following axioms for s, t ∈ R and a, b, c ∈ G*:* (V1) 1 ⊙ a = a, 0 ⊙ a = t ⊙ e = e, and (−1) ⊙ a = ⊖a. (V2) (s + t) ⊙ a = s ⊙ a ⊕ t ⊙ a. (V3) (st) ⊙ a = s ⊙ (t ⊙ a). (V4) gyr[a, b](t ⊙ c) = t ⊙ gyr[a, b]c. (V5) gyr[s ⊙ a, t ⊙ a] = Id, where Id is the identity map. G.2 AI GYROVECTOR SPACES G.3 LE GYROVECTOR SPACES G.4 LC GYROVECTOR SPACES G.5 GRASSMANN MANIFOLDS IN THE PROJECTOR PERSPECTIVE G.6 GRASSMANN MANIFOLDS IN THE ONB PERSPECTIVE G.7 THE SPD AND GRASSMANN INNER PRODUCTS G.9 THE GYRODISTANCE FUNCTION IN STRUCTURE SPACES G.10 THE PSEUDO-GYRODISTANCE FUNCTION IN STRUCTURE SPACES Authors:
(1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (xuan-son.nguyen@ensea.fr);
(2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (son.nguyen@ensea.fr);
(3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (aymeric.histace@ensea.fr). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction Preliminaries Proposed Approach
3.1 Notation
3.2 Nueral Networks on SPD Manifolds
3.3 MLR in Structure Spaces
3.4 Neural Networks on Grassmann Manifolds Experiments Conclusion and References Preliminaries Preliminaries Preliminaries Proposed Approach 3.1 Notation 3.2 Nueral Networks on SPD Manifolds 3.3 MLR in Structure Spaces 3.4 Neural Networks on Grassmann Manifolds Proposed Approach 3.1 Notation 3.1 Notation 3.2 Nueral Networks on SPD Manifolds 3.2 Nueral Networks on SPD Manifolds 3.3 MLR in Structure Spaces 3.3 MLR in Structure Spaces 3.4 Neural Networks on Grassmann Manifolds 3.4 Neural Networks on Grassmann Manifolds Experiments Experiments Experiments Conclusion and References Conclusion and References Conclusion and References A. Notations A. Notations B. MLR in Structure Spaces B. MLR in Structure Spaces C. Formulation of MLR from the Perspective of Distances to Hyperplanes C. Formulation of MLR from the Perspective of Distances to Hyperplanes D. Human Action Recognition D. Human Action Recognition E. Node Classification E. Node Classification F. Limitations of our work F. Limitations of our work G. Some Related Definitions G. Some Related Definitions H. Computation of Canonical Representation H. Computation of Canonical Representation I. Proof of Proposition 3.2 I. Proof of Proposition 3.2 J. Proof of Proposition 3.4 J. Proof of Proposition 3.4 K. Proof of Proposition 3.5 K. Proof of Proposition 3.5 L. Proof of Proposition 3.6 L. Proof of Proposition 3.6 M. Proof of Proposition 3.11 M. Proof of Proposition 3.11 N. Proof of Proposition 3.12 N. Proof of Proposition 3.12 G SOME RELATED DEFINITIONS G.1 GYROGROUPS AND GYROVECTOR SPACES Gyrovector spaces form the setting for hyperbolic geometry in the same way that vector spaces form the setting for Euclidean geometry (Ungar, 2002; 2005; 2014). We recap the definitions of gyrogroups and gyrocommutative gyrogroups proposed in Ungar (2002; 2005; 2014). For greater mathematical detail and in-depth discussion, we refer the interested reader to these papers. Definition G.1 (Gyrogroups (Ungar, 2014)) . A pair (G, ⊕*) is a groupoid in the sense that it is a nonempty set, G, with a binary operation, ⊕. A groupoid (G,* ⊕*) is a gyrogroup if its binary operation satisfies the following axioms for a, b, c ∈ G:* Definition G.1 (Gyrogroups (Ungar, 2014)) A pair (G, (G1) There is at least one element e ∈ G called a left identity such that e ⊕ a = a. (G1) There is at least one element e ∈ G called a left identity such that e ⊕ a = a. (G2) There is an element ⊖ a ∈ G called a left inverse of a such that ⊖ a ⊕ a = e. (G2) There is an element a ∈ G called a left inverse of a such that a a = e. (G3) There is an automorphism gyr[a, b] : G → G for each a, b ∈ G such that (G3) There is an automorphism gyr[a, b] : G → G for each a, b ∈ G such that a ⊕ (b ⊕ c) = (a ⊕ b) ⊕ gyr[a, b]c ( Left Gyroassociative Law ). Left Gyroassociative Law The automorphism gyr[a, b] is called the gyroautomorphism, or the gyration of G generated by a, b. (G4) gyr[a, b] = gyr[a ⊕ b, b] (Left Reduction Property). The automorphism gyr[a, b] is called the gyroautomorphism, or the gyration of G generated by a, b. (G4) gyr[a, b] = gyr[a ⊕ b, b] (Left Reduction Property). Definition G.2 (Gyrocommutative Gyrogroups (Ungar, 2014)). A gyrogroup (G, ⊕*) is gyrocommutative if it satisfies* Definition G.2 (Gyrocommutative Gyrogroups (Ungar, 2014)). A gyrogroup (G, a ⊕ b = gyr[a, b](b ⊕ a) ( Gyrocommutative Law ). Gyrocommutative Law The following definition of gyrovector spaces is slightly different from Definition 3.2 in Ungar (2014). Definition G.3 (Gyrovector Spaces). A gyrocommutative gyrogroup (G, ⊕) equipped with a scalar multiplication Definition G.3 (Gyrovector Spaces). A gyrocommutative gyrogroup (G, ⊕*) equipped with a scalar multiplication* Definition G.3 (Gyrovector Spaces). A gyrocommutative gyrogroup (G, ( t, x ) → t ⊙ x : R × G → G t, x t x : R × G → G is called a gyrovector space if it satisfies the following axioms for s, t ∈ R and a, b, c ∈ G*:* is called a gyrovector space if it satisfies the following axioms for s, t and a, b, c (V1) 1 ⊙ a = a, 0 ⊙ a = t ⊙ e = e, and (−1) ⊙ a = ⊖a. (V1) and (V2) (s + t) ⊙ a = s ⊙ a ⊕ t ⊙ a. (V2) (V3) (st) ⊙ a = s ⊙ (t ⊙ a). (V3) (V4) gyr[a, b](t ⊙ c) = t ⊙ gyr[a, b]c. (V4) (V5) gyr[s ⊙ a, t ⊙ a] = Id, where Id is the identity map. (V5) where is the identity map. G.2 AI GYROVECTOR SPACES G.3 LE GYROVECTOR SPACES G.4 LC GYROVECTOR SPACES G.5 GRASSMANN MANIFOLDS IN THE PROJECTOR PERSPECTIVE G.6 GRASSMANN MANIFOLDS IN THE ONB PERSPECTIVE G.7 THE SPD AND GRASSMANN INNER PRODUCTS G.9 THE GYRODISTANCE FUNCTION IN STRUCTURE SPACES G.10 THE PSEUDO-GYRODISTANCE FUNCTION IN STRUCTURE SPACES Authors: (1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (xuan-son.nguyen@ensea.fr); (2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (son.nguyen@ensea.fr); (3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (aymeric.histace@ensea.fr). Authors: Authors: (1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (xuan-son.nguyen@ensea.fr); (2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (son.nguyen@ensea.fr); (3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (aymeric.histace@ensea.fr). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. available on arxiv available on arxiv

What Are Gyrocommutative Gyrogroups?

Table of Links Abstract and 1. Introduction Preliminaries


Proposed Approach
3.1 Notation
3.2 Nueral Networks on SPD Manifolds
3.3 MLR in Structure Spaces
3.4 Neural Networks on Grassmann Manifolds


Experiments


Conclusion and References A. Notations B. MLR in Structure Spaces C. Formulation of MLR from the Perspective of Distances to Hyperplanes D. Human Action Recognition E. Node Classification F. Limitations of our work G. Some Related Definitions H. Computation of Canonical Representation I. Proof of Proposition 3.2 J. Proof of Proposition 3.4 K. Proof of Proposition 3.5 L. Proof of Proposition 3.6 M. Proof of Proposition 3.11 N. Proof of Proposition 3.12 F LIMITATIONS OF OUR WORK Our SPD network GyroSpd++ relies on different Riemannian metrics across the layers, i.e., the convolutional layer is based on Affine-Invariant metrics while the MLR layer is based on LogEuclidean metrics. Although we have provided the experimental results demonstrating that GyroSpd++ achieves good performance on all the datasets compared to state-of-the-art methods, it is not clear if our design is optimal for the human action recognition task. When it comes to building a deep SPD architecture, it is useful to provide insights into Riemannian metrics one should use for each network block in order to obtain good performance on a target task. In our Grassmann network Gr-GCN++, the feature transformation and bias and nonlinearity operations are performed on Grassmann manifolds, while the aggregation operation is performed in tangent spaces. Previous works (Dai et al., 2021; Chen et al., 2022) on HNNs have shown that this hybrid method limits the modeling ability of networks. Therefore, it is desirable to develop GCNs where all the operations are formalized on Grassmann manifolds. Authors:
(1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (xuan-son.nguyen@ensea.fr);
(2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (son.nguyen@ensea.fr);
(3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (aymeric.histace@ensea.fr). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction Preliminaries Proposed Approach
3.1 Notation
3.2 Nueral Networks on SPD Manifolds
3.3 MLR in Structure Spaces
3.4 Neural Networks on Grassmann Manifolds Experiments Conclusion and References Preliminaries Preliminaries Preliminaries Proposed Approach 3.1 Notation 3.2 Nueral Networks on SPD Manifolds 3.3 MLR in Structure Spaces 3.4 Neural Networks on Grassmann Manifolds Proposed Approach 3.1 Notation 3.1 Notation 3.2 Nueral Networks on SPD Manifolds 3.2 Nueral Networks on SPD Manifolds 3.3 MLR in Structure Spaces 3.3 MLR in Structure Spaces 3.4 Neural Networks on Grassmann Manifolds 3.4 Neural Networks on Grassmann Manifolds Experiments Experiments Experiments Conclusion and References Conclusion and References Conclusion and References A. Notations A. Notations B. MLR in Structure Spaces B. MLR in Structure Spaces C. Formulation of MLR from the Perspective of Distances to Hyperplanes C. Formulation of MLR from the Perspective of Distances to Hyperplanes D. Human Action Recognition D. Human Action Recognition E. Node Classification E. Node Classification F. Limitations of our work F. Limitations of our work G. Some Related Definitions G. Some Related Definitions H. Computation of Canonical Representation H. Computation of Canonical Representation I. Proof of Proposition 3.2 I. Proof of Proposition 3.2 J. Proof of Proposition 3.4 J. Proof of Proposition 3.4 K. Proof of Proposition 3.5 K. Proof of Proposition 3.5 L. Proof of Proposition 3.6 L. Proof of Proposition 3.6 M. Proof of Proposition 3.11 M. Proof of Proposition 3.11 N. Proof of Proposition 3.12 N. Proof of Proposition 3.12 F LIMITATIONS OF OUR WORK Our SPD network GyroSpd++ relies on different Riemannian metrics across the layers, i.e., the convolutional layer is based on Affine-Invariant metrics while the MLR layer is based on LogEuclidean metrics. Although we have provided the experimental results demonstrating that GyroSpd++ achieves good performance on all the datasets compared to state-of-the-art methods, it is not clear if our design is optimal for the human action recognition task. When it comes to building a deep SPD architecture, it is useful to provide insights into Riemannian metrics one should use for each network block in order to obtain good performance on a target task. In our Grassmann network Gr-GCN++, the feature transformation and bias and nonlinearity operations are performed on Grassmann manifolds, while the aggregation operation is performed in tangent spaces. Previous works (Dai et al., 2021; Chen et al., 2022) on HNNs have shown that this hybrid method limits the modeling ability of networks. Therefore, it is desirable to develop GCNs where all the operations are formalized on Grassmann manifolds. Authors: (1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (xuan-son.nguyen@ensea.fr); (2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (son.nguyen@ensea.fr); (3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (aymeric.histace@ensea.fr). Authors: Authors: (1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (xuan-son.nguyen@ensea.fr); (2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (son.nguyen@ensea.fr); (3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (aymeric.histace@ensea.fr). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. available on arxiv available on arxiv

The Limitations of GyroSpd++ and Gr-GCN++ in Human Action Recognition and Graph Embedding Tasks

Table of Links Abstract and 1. Introduction Preliminaries


Proposed Approach
3.1 Notation
3.2 Nueral Networks on SPD Manifolds
3.3 MLR in Structure Spaces
3.4 Neural Networks on Grassmann Manifolds


Experiments


Conclusion and References A. Notations B. MLR in Structure Spaces C. Formulation of MLR from the Perspective of Distances to Hyperplanes D. Human Action Recognition E. Node Classification F. Limitations of our work G. Some Related Definitions H. Computation of Canonical Representation I. Proof of Proposition 3.2 J. Proof of Proposition 3.4 K. Proof of Proposition 3.5 L. Proof of Proposition 3.6 M. Proof of Proposition 3.11 N. Proof of Proposition 3.12 E NODE CLASSIFICATION E.1 DATASETS Airport (Chami et al., 2019) It is a flight network dataset from OpenFlights.org where nodes represent airports, edges represent the airline Routes, and node labels are the populations of the country where the airport belongs. Pubmed (Namata et al., 2012b) It is a standard benchmark describing citation networks where nodes represent scientific papers in the area of medicine, edges are citations between them, and node labels are academic (sub)areas. Cora (Sen et al., 2008) It is a citation network where nodes represent scientific papers in the area of machine learning, edges are citations between them, and node labels are academic (sub)areas. The statistics of the three datasets are summarized in Tab. 11. E.2 IMPLEMENTATION DETAILS E.2.1 SETUP E.2.2 GRASSMANN LOGARITHMIC MAP IN THE ONB PERSPECTIVE The Grassmann logarithmic map in the ONB perspective is given (Edelman et al., 1998) by E.2.3 GR-GCN++ E.2.4 GR-GCN-ONB E.2.5 OPTIMIZATION E.3 MORE EXPERIMENTAL RESULTS E.3.1 ABLATION STUDY Projector vs. ONB perspective More results of Gr-GCN++ and Gr-GCN-ONB are presented in Tabs. 12 and 13. As can be observed, Gr-GCN++ outperforms Gr-GCN-ONB in all cases. In particular, the former outperforms the latter by large margins on Airport and Cora datasets. Results show that while both the networks learn node embeddings on Grassmann manifolds, the choice of perspective for representing these embeddings and the associated parameters can have a significant impact on the network performance. E.3.2 COMPARISON OF GR-GCN++ AGAINST STATE-OF-THE-ART METHODS Tab. 14 shows results of Gr-GCN++ and some state-of-the-art methods on the three datasets. The hyperbolic networks outperform their SPD and Grassmann counterparts on Airport dataset with high hyperbolicity (Chami et al., 2019). This agrees with previous works (Chami et al., 2019; Zhang et al., 2022) that report good performances of hyperbolic embeddings on tree-like datasets. However, our network and its SPD counterpart SPD-GCN outperform their competitors on Pubmed and Cora datasets with low hyperbolicities. Compared to SPD-GCN, Gr-GCN++ always gives more consistent results. Authors:
(1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (xuan-son.nguyen@ensea.fr);
(2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (son.nguyen@ensea.fr);
(3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (aymeric.histace@ensea.fr). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction Preliminaries Proposed Approach
3.1 Notation
3.2 Nueral Networks on SPD Manifolds
3.3 MLR in Structure Spaces
3.4 Neural Networks on Grassmann Manifolds Experiments Conclusion and References Preliminaries Preliminaries Preliminaries Proposed Approach 3.1 Notation 3.2 Nueral Networks on SPD Manifolds 3.3 MLR in Structure Spaces 3.4 Neural Networks on Grassmann Manifolds Proposed Approach 3.1 Notation 3.1 Notation 3.2 Nueral Networks on SPD Manifolds 3.2 Nueral Networks on SPD Manifolds 3.3 MLR in Structure Spaces 3.3 MLR in Structure Spaces 3.4 Neural Networks on Grassmann Manifolds 3.4 Neural Networks on Grassmann Manifolds Experiments Experiments Experiments Conclusion and References Conclusion and References Conclusion and References A. Notations A. Notations B. MLR in Structure Spaces B. MLR in Structure Spaces C. Formulation of MLR from the Perspective of Distances to Hyperplanes C. Formulation of MLR from the Perspective of Distances to Hyperplanes D. Human Action Recognition D. Human Action Recognition E. Node Classification E. Node Classification F. Limitations of our work F. Limitations of our work G. Some Related Definitions G. Some Related Definitions H. Computation of Canonical Representation H. Computation of Canonical Representation I. Proof of Proposition 3.2 I. Proof of Proposition 3.2 J. Proof of Proposition 3.4 J. Proof of Proposition 3.4 K. Proof of Proposition 3.5 K. Proof of Proposition 3.5 L. Proof of Proposition 3.6 L. Proof of Proposition 3.6 M. Proof of Proposition 3.11 M. Proof of Proposition 3.11 N. Proof of Proposition 3.12 N. Proof of Proposition 3.12 E NODE CLASSIFICATION E.1 DATASETS Airport (Chami et al., 2019) It is a flight network dataset from OpenFlights.org where nodes represent airports, edges represent the airline Routes, and node labels are the populations of the country where the airport belongs. Airport (Chami et al., 2019) Pubmed (Namata et al., 2012b) It is a standard benchmark describing citation networks where nodes represent scientific papers in the area of medicine, edges are citations between them, and node labels are academic (sub)areas. Pubmed (Namata et al., 2012b) Cora (Sen et al., 2008) It is a citation network where nodes represent scientific papers in the area of machine learning, edges are citations between them, and node labels are academic (sub)areas. Cora (Sen et al., 2008) The statistics of the three datasets are summarized in Tab. 11. E.2 IMPLEMENTATION DETAILS E.2.1 SETUP E.2.1 SETUP E.2.2 GRASSMANN LOGARITHMIC MAP IN THE ONB PERSPECTIVE E.2.2 GRASSMANN LOGARITHMIC MAP IN THE ONB PERSPECTIVE The Grassmann logarithmic map in the ONB perspective is given (Edelman et al., 1998) by E.2.3 GR-GCN++ E.2.3 GR-GCN++ E.2.4 GR-GCN-ONB E.2.4 GR-GCN-ONB E.2.5 OPTIMIZATION E.2.5 OPTIMIZATION E.3 MORE EXPERIMENTAL RESULTS E.3.1 ABLATION STUDY E.3.1 ABLATION STUDY Projector vs. ONB perspective More results of Gr-GCN++ and Gr-GCN-ONB are presented in Tabs. 12 and 13. As can be observed, Gr-GCN++ outperforms Gr-GCN-ONB in all cases. In particular, the former outperforms the latter by large margins on Airport and Cora datasets. Results show that while both the networks learn node embeddings on Grassmann manifolds, the choice of perspective for representing these embeddings and the associated parameters can have a significant impact on the network performance. Projector vs. ONB perspective E.3.2 COMPARISON OF GR-GCN++ AGAINST STATE-OF-THE-ART METHODS E.3.2 COMPARISON OF GR-GCN++ AGAINST STATE-OF-THE-ART METHODS Tab. 14 shows results of Gr-GCN++ and some state-of-the-art methods on the three datasets. The hyperbolic networks outperform their SPD and Grassmann counterparts on Airport dataset with high hyperbolicity (Chami et al., 2019). This agrees with previous works (Chami et al., 2019; Zhang et al., 2022) that report good performances of hyperbolic embeddings on tree-like datasets. However, our network and its SPD counterpart SPD-GCN outperform their competitors on Pubmed and Cora datasets with low hyperbolicities. Compared to SPD-GCN, Gr-GCN++ always gives more consistent results. Authors: (1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (xuan-son.nguyen@ensea.fr); (2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (son.nguyen@ensea.fr); (3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (aymeric.histace@ensea.fr). Authors: Authors: (1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (xuan-son.nguyen@ensea.fr); (2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (son.nguyen@ensea.fr); (3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (aymeric.histace@ensea.fr). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. available on arxiv available on arxiv

Evaluating Gr-GCN++ for Node Classification Across Various Datasets: Results and Comparisons

Table of Links Abstract and 1. Introduction Preliminaries


Proposed Approach
3.1 Notation
3.2 Nueral Networks on SPD Manifolds
3.3 MLR in Structure Spaces
3.4 Neural Networks on Grassmann Manifolds


Experiments


Conclusion and References A. Notations B. MLR in Structure Spaces C. Formulation of MLR from the Perspective of Distances to Hyperplanes D. Human Action Recognition E. Node Classification F. Limitations of our work G. Some Related Definitions H. Computation of Canonical Representation I. Proof of Proposition 3.2 J. Proof of Proposition 3.4 K. Proof of Proposition 3.5 L. Proof of Proposition 3.6 M. Proof of Proposition 3.11 N. Proof of Proposition 3.12 C FORMULATION OF MLR FROM THE PERSPECTIVE OF DISTANCES TO HYPERPLANES D HUMAN ACTION RECOGNITION D.1 DATASETS HDM05 (Muller et al., 2007) It has 2337 sequences of 3D skeleton data classified into 130 classes. Each frame contains the 3D coordinates of 31 body joints. We use all the action classes and follow the experimental protocol in Harandi et al. (2018) in which 2 subjects are used for training and the remaining 3 subjects are used for testing. FPHA (Garcia-Hernando et al., 2018) It has 1175 sequences of 3D skeleton data classified into 45 classes. Each frame contains the 3D coordinates of 21 hand joints. We follow the experimental protocol in Garcia-Hernando et al. (2018) in which 600 sequences are used for training and 575 sequences are used for testing. NTU60 (Shahroudy et al., 2016) It has 56880 sequences of 3D skeleton data classified into 60 classes. Each frame contains the 3D coordinates of 25 or 50 body joints. We use the mutual actions and follow the cross-subject experimental protocol in Shahroudy et al. (2016) in which data from 20 subjects are used for training, and those from the other 20 subjects are used for testing. D.2 IMPLEMENTATION DETAILS D.2.1 SETUP D.2.2 INPUT DATA For SPDNet and SPDNetBN, each sequence is represented by a covariance matrix (Huang & Gool, 2017; Brooks et al., 2019). The sizes of the covariance matrices are 93×93, 60×60, and 150×150 for HDM05, FPHA, and NTU60 datasets, respectively. For SPDNet, the same architecture as the one in Huang & Gool (2017) is used with three Bimap layers. For SPDNetBN, the same architecture as the one in Brooks et al. (2019) is used with three Bimap layers. The sizes of the transformation matrices for the experiments on HDM05, FPHA, and NTU60 datasets are set to 93 × 93, 60 × 60, and 150 × 150, respectively D.2.3 CONVOLUTIONAL LAYERS D.2.4 OPTIMIZATION For parameters that are SPD matrices, we model them on the space of symmetric matrices, and then apply the exponential map at the identity. Thus, we can optimize all parameters on Euclidean spaces without having to resort to techniques developed on Riemannian manifolds. D.3 TIME COMPLEXITY ANALYSIS D.4 MORE EXPERIMENTAL RESULTS D.4.1 ABLATION STUDY Tab. 4 reports the mean accuracies and standard deviations of GyroSpd++ with respect to different settings of β on the three datasets. GyroSpd++ with the setting β = 0 generally works well on all the datasets. Setting k = 3 improves the accuracy of GyroSpd++ on NTU60 dataset. We also observe that setting k to a high value, e.g., k = 10 lowers the accuracies of GyroSpd++ on the datasets. Output dimension of convolutional layers Tab. 5 presents results and computation times of GyroSpd++ with respect to different settings of the output dimension of the convolutional layer on FPHA dataset. Results show that the setting m = 21 clearly outperforms the setting m = 10 in terms of mean accuracy and standard deviation. However, compared to the setting m = 21, the setting m = 30 only increases the training and testing times without improving the mean accuracy of GyroSpd++. Design of Riemannian metrics for network blocks The use of different Riemannian metrics for the convolutional and MLR layers of GyroSpd++ results in different variants of the same architecture. Results of some of these variants on FPHA dataset are shown in Tab. 6. It is noted that our architecture gives the best performance in terms of mean accuracy, while the architecture with Log-Cholesky geometry for the MLR layer performs the worst in terms of mean accuracy. D.4.2 COMPARISON OF GYROSPD++ AGAINST STATE-OF-THE-ART METHODS Finally, we present a comparison of computation times of SPD neural networks in Tab. 10. Authors:
(1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (xuan-son.nguyen@ensea.fr);
(2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (son.nguyen@ensea.fr);
(3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (aymeric.histace@ensea.fr). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. [3] https://github.com/dalab/hyperbolic_nn. [4] https://github.com/kenziyuliu/MS-G3D. [5] https://github.com/zhysora/FR-Head. [6] https://github.com/Chiaraplizz/ST-TR. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction Preliminaries Proposed Approach
3.1 Notation
3.2 Nueral Networks on SPD Manifolds
3.3 MLR in Structure Spaces
3.4 Neural Networks on Grassmann Manifolds Experiments Conclusion and References Preliminaries Preliminaries Preliminaries Proposed Approach 3.1 Notation 3.2 Nueral Networks on SPD Manifolds 3.3 MLR in Structure Spaces 3.4 Neural Networks on Grassmann Manifolds Proposed Approach 3.1 Notation 3.1 Notation 3.2 Nueral Networks on SPD Manifolds 3.2 Nueral Networks on SPD Manifolds 3.3 MLR in Structure Spaces 3.3 MLR in Structure Spaces 3.4 Neural Networks on Grassmann Manifolds 3.4 Neural Networks on Grassmann Manifolds Experiments Experiments Experiments Conclusion and References Conclusion and References Conclusion and References A. Notations A. Notations B. MLR in Structure Spaces B. MLR in Structure Spaces C. Formulation of MLR from the Perspective of Distances to Hyperplanes C. Formulation of MLR from the Perspective of Distances to Hyperplanes D. Human Action Recognition D. Human Action Recognition E. Node Classification E. Node Classification F. Limitations of our work F. Limitations of our work G. Some Related Definitions G. Some Related Definitions H. Computation of Canonical Representation H. Computation of Canonical Representation I. Proof of Proposition 3.2 I. Proof of Proposition 3.2 J. Proof of Proposition 3.4 J. Proof of Proposition 3.4 K. Proof of Proposition 3.5 K. Proof of Proposition 3.5 L. Proof of Proposition 3.6 L. Proof of Proposition 3.6 M. Proof of Proposition 3.11 M. Proof of Proposition 3.11 N. Proof of Proposition 3.12 N. Proof of Proposition 3.12 C FORMULATION OF MLR FROM THE PERSPECTIVE OF DISTANCES TO HYPERPLANES D HUMAN ACTION RECOGNITION D.1 DATASETS HDM05 (Muller et al., 2007) It has 2337 sequences of 3D skeleton data classified into 130 classes. Each frame contains the 3D coordinates of 31 body joints. We use all the action classes and follow the experimental protocol in Harandi et al. (2018) in which 2 subjects are used for training and the remaining 3 subjects are used for testing. HDM05 (Muller et al., 2007) FPHA (Garcia-Hernando et al., 2018) It has 1175 sequences of 3D skeleton data classified into 45 classes. Each frame contains the 3D coordinates of 21 hand joints. We follow the experimental protocol in Garcia-Hernando et al. (2018) in which 600 sequences are used for training and 575 sequences are used for testing. FPHA (Garcia-Hernando et al., 2018) NTU60 (Shahroudy et al., 2016) It has 56880 sequences of 3D skeleton data classified into 60 classes. Each frame contains the 3D coordinates of 25 or 50 body joints. We use the mutual actions and follow the cross-subject experimental protocol in Shahroudy et al. (2016) in which data from 20 subjects are used for training, and those from the other 20 subjects are used for testing. NTU60 (Shahroudy et al., 2016) D.2 IMPLEMENTATION DETAILS D.2.1 SETUP D.2.1 SETUP D.2.2 INPUT DATA D.2.2 INPUT DATA For SPDNet and SPDNetBN, each sequence is represented by a covariance matrix (Huang & Gool, 2017; Brooks et al., 2019). The sizes of the covariance matrices are 93×93, 60×60, and 150×150 for HDM05, FPHA, and NTU60 datasets, respectively. For SPDNet, the same architecture as the one in Huang & Gool (2017) is used with three Bimap layers. For SPDNetBN, the same architecture as the one in Brooks et al. (2019) is used with three Bimap layers. The sizes of the transformation matrices for the experiments on HDM05, FPHA, and NTU60 datasets are set to 93 × 93, 60 × 60, and 150 × 150, respectively D.2.3 CONVOLUTIONAL LAYERS D.2.3 CONVOLUTIONAL LAYERS D.2.4 OPTIMIZATION D.2.4 OPTIMIZATION For parameters that are SPD matrices, we model them on the space of symmetric matrices, and then apply the exponential map at the identity. Thus, we can optimize all parameters on Euclidean spaces without having to resort to techniques developed on Riemannian manifolds. D.3 TIME COMPLEXITY ANALYSIS D.4 MORE EXPERIMENTAL RESULTS D.4.1 ABLATION STUDY D.4.1 ABLATION STUDY Tab. 4 reports the mean accuracies and standard deviations of GyroSpd++ with respect to different settings of β on the three datasets. GyroSpd++ with the setting β = 0 generally works well on all the datasets. Setting k = 3 improves the accuracy of GyroSpd++ on NTU60 dataset. We also observe that setting k to a high value, e.g., k = 10 lowers the accuracies of GyroSpd++ on the datasets. Output dimension of convolutional layers Tab. 5 presents results and computation times of GyroSpd++ with respect to different settings of the output dimension of the convolutional layer on FPHA dataset. Results show that the setting m = 21 clearly outperforms the setting m = 10 in terms of mean accuracy and standard deviation. However, compared to the setting m = 21, the setting m = 30 only increases the training and testing times without improving the mean accuracy of GyroSpd++. Output dimension of convolutional layers Design of Riemannian metrics for network blocks The use of different Riemannian metrics for the convolutional and MLR layers of GyroSpd++ results in different variants of the same architecture. Results of some of these variants on FPHA dataset are shown in Tab. 6. It is noted that our architecture gives the best performance in terms of mean accuracy, while the architecture with Log-Cholesky geometry for the MLR layer performs the worst in terms of mean accuracy. Design of Riemannian metrics for network blocks D.4.2 COMPARISON OF GYROSPD++ AGAINST STATE-OF-THE-ART METHODS D.4.2 COMPARISON OF GYROSPD++ AGAINST STATE-OF-THE-ART METHODS Finally, we present a comparison of computation times of SPD neural networks in Tab. 10. Authors: (1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (xuan-son.nguyen@ensea.fr); (2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (son.nguyen@ensea.fr); (3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (aymeric.histace@ensea.fr). Authors: Authors: (1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (xuan-son.nguyen@ensea.fr); (2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (son.nguyen@ensea.fr); (3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (aymeric.histace@ensea.fr). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. available on arxiv available on arxiv [3] https://github.com/dalab/hyperbolic_nn. [4] https://github.com/kenziyuliu/MS-G3D. [5] https://github.com/zhysora/FR-Head. [6] https://github.com/Chiaraplizz/ST-TR.

Exploring the Impact of Riemannian Metrics in Human Action Recognition Tasks Using GyroSpd++

Bridging Geometry and Deep Learning: Key Developments in SPD and Grassmann Networks

Table of Links Abstract and 1. Introduction Preliminaries


Proposed Approach
3.1 Notation
3.2 Nueral Networks on SPD Manifolds
3.3 MLR in Structure Spaces
3.4 Neural Networks on Grassmann Manifolds


Experiments


Conclusion and References A. Notations B. MLR in Structure Spaces C. Formulation of MLR from the Perspective of Distances to Hyperplanes D. Human Action Recognition E. Node Classification F. Limitations of our work G. Some Related Definitions H. Computation of Canonical Representation I. Proof of Proposition 3.2 J. Proof of Proposition 3.4 K. Proof of Proposition 3.5 L. Proof of Proposition 3.6 M. Proof of Proposition 3.11 N. Proof of Proposition 3.12 4 EXPERIMENTS 4.1 HUMAN ACTION RECOGNITION We use three datasets, i.e., HDM05 (Muller et al., 2007), FPHA (Garcia-Hernando et al., 2018), and ¨ NTU RBG+D 60 (NTU60) (Shahroudy et al., 2016). We compare our networks against the following state-of-the-art models: SPDNet (Huang & Gool, 2017)[1], SPDNetBN (Brooks et al., 2019)[2], SPSDAI (Nguyen, 2022a), GyroAI-HAUNet (Nguyen, 2022b), and MLR-AI (Nguyen & Yang, 2023). 4.1.1 ABLATION STUDY Convolutional layers in SPD neural networks Our network GyroSpd++ has a MLR layer stacked on top of a convolutional layer (see Fig. 1). The motivation for using a convolutional layer is that it can extract global features from local ones (covariance matrices computed from joint coordinates within sub-sequences of an action sequence). We use Affine-Invariant metrics for the convolutional layer and Log-Euclidean metrics for the MLR layer. Results in Tab. 1 show that GyroSpd++ consistently outperforms the SPD baselines in terms of mean accuracy. Results of GyroSpd++ with different designs of Riemannian metrics for its layers are given in Appendix D.4.1. MLR in structure spaces We build GyroSpsd++ by replacing the MLR layer of GyroSpd++ with a MLR layer proposed in Section 3.3. Results of GyroSpsd++ are given in Tab. 1. Except SPSDAI, GyroSpsd++ outperforms the other baselines on HDM05 dataset in terms of mean accuracy. Furthermore, GyroSpsd++ outperforms GyroSpd++ and all the baselines on FPHA and NTU60 datasets in terms of mean accuracy. These results show that MLR is effective when being designed in structure spaces from a gyrovector space perspective. 4.2 NODE CLASSIFICATION We use three datasets, i.e., Airport (Zhang & Chen, 2018), Pubmed (Namata et al., 2012a), and Cora (Sen et al., 2008), each of them contains a single graph with thousands of labeled nodes. We compare our network Gr-GCN++ (see Fig. 1) against its variant Gr-GCN-ONB (see Appendix E.2.4) based on the ONB perspective. Results are shown in Tab. 2. Both networks give the best performance for n = 14 and p = 7. It can be seen that Gr-GCN++ outperforms Gr-GCN-ONB in all cases. The performance gaps are significant on Pubmed and Cora datasets. Authors:
(1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (xuan-son.nguyen@ensea.fr);
(2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (son.nguyen@ensea.fr);
(3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (aymeric.histace@ensea.fr). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. [1] https://github.com/zhiwu-huang/SPDNet. [2] https://papers.nips.cc/paper/2019/hash/6e69ebbfad976d4637bb4b39de261bf7-Abstract. html. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction Preliminaries Proposed Approach
3.1 Notation
3.2 Nueral Networks on SPD Manifolds
3.3 MLR in Structure Spaces
3.4 Neural Networks on Grassmann Manifolds Experiments Conclusion and References Preliminaries Preliminaries Preliminaries Proposed Approach 3.1 Notation 3.2 Nueral Networks on SPD Manifolds 3.3 MLR in Structure Spaces 3.4 Neural Networks on Grassmann Manifolds Proposed Approach 3.1 Notation 3.1 Notation 3.2 Nueral Networks on SPD Manifolds 3.2 Nueral Networks on SPD Manifolds 3.3 MLR in Structure Spaces 3.3 MLR in Structure Spaces 3.4 Neural Networks on Grassmann Manifolds 3.4 Neural Networks on Grassmann Manifolds Experiments Experiments Experiments Conclusion and References Conclusion and References Conclusion and References A. Notations A. Notations B. MLR in Structure Spaces B. MLR in Structure Spaces C. Formulation of MLR from the Perspective of Distances to Hyperplanes C. Formulation of MLR from the Perspective of Distances to Hyperplanes D. Human Action Recognition D. Human Action Recognition E. Node Classification E. Node Classification F. Limitations of our work F. Limitations of our work G. Some Related Definitions G. Some Related Definitions H. Computation of Canonical Representation H. Computation of Canonical Representation I. Proof of Proposition 3.2 I. Proof of Proposition 3.2 J. Proof of Proposition 3.4 J. Proof of Proposition 3.4 K. Proof of Proposition 3.5 K. Proof of Proposition 3.5 L. Proof of Proposition 3.6 L. Proof of Proposition 3.6 M. Proof of Proposition 3.11 M. Proof of Proposition 3.11 N. Proof of Proposition 3.12 N. Proof of Proposition 3.12 4 EXPERIMENTS 4.1 HUMAN ACTION RECOGNITION We use three datasets, i.e., HDM05 (Muller et al., 2007), FPHA (Garcia-Hernando et al., 2018), and ¨ NTU RBG+D 60 (NTU60) (Shahroudy et al., 2016). We compare our networks against the following state-of-the-art models: SPDNet (Huang & Gool, 2017)[1], SPDNetBN (Brooks et al., 2019)[2], SPSDAI (Nguyen, 2022a), GyroAI-HAUNet (Nguyen, 2022b), and MLR-AI (Nguyen & Yang, 2023). 4.1.1 ABLATION STUDY 4.1.1 ABLATION STUDY Convolutional layers in SPD neural networks Our network GyroSpd++ has a MLR layer stacked on top of a convolutional layer (see Fig. 1). The motivation for using a convolutional layer Convolutional layers in SPD neural networks is that it can extract global features from local ones (covariance matrices computed from joint coordinates within sub-sequences of an action sequence). We use Affine-Invariant metrics for the convolutional layer and Log-Euclidean metrics for the MLR layer. Results in Tab. 1 show that GyroSpd++ consistently outperforms the SPD baselines in terms of mean accuracy. Results of GyroSpd++ with different designs of Riemannian metrics for its layers are given in Appendix D.4.1. MLR in structure spaces We build GyroSpsd++ by replacing the MLR layer of GyroSpd++ with a MLR layer proposed in Section 3.3. Results of GyroSpsd++ are given in Tab. 1. Except SPSDAI, GyroSpsd++ outperforms the other baselines on HDM05 dataset in terms of mean accuracy. Furthermore, GyroSpsd++ outperforms GyroSpd++ and all the baselines on FPHA and NTU60 datasets in terms of mean accuracy. These results show that MLR is effective when being designed in structure spaces from a gyrovector space perspective. MLR in structure spaces 4.2 NODE CLASSIFICATION We use three datasets, i.e., Airport (Zhang & Chen, 2018), Pubmed (Namata et al., 2012a), and Cora (Sen et al., 2008), each of them contains a single graph with thousands of labeled nodes. We compare our network Gr-GCN++ (see Fig. 1) against its variant Gr-GCN-ONB (see Appendix E.2.4) based on the ONB perspective. Results are shown in Tab. 2. Both networks give the best performance for n = 14 and p = 7. It can be seen that Gr-GCN++ outperforms Gr-GCN-ONB in all cases. The performance gaps are significant on Pubmed and Cora datasets. Authors: (1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (xuan-son.nguyen@ensea.fr); (2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (son.nguyen@ensea.fr); (3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (aymeric.histace@ensea.fr). Authors: Authors: (1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (xuan-son.nguyen@ensea.fr); (2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (son.nguyen@ensea.fr); (3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (aymeric.histace@ensea.fr). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. available on arxiv available on arxiv [1] https://github.com/zhiwu-huang/SPDNet. [2] https://papers.nips.cc/paper/2019/hash/6e69ebbfad976d4637bb4b39de261bf7-Abstract. html.

New Riemannian Networks Outperform Traditional Models in Action Recognition and Node Classification

Новите риманови мрежи превъзхождат традиционните модели при разпознаване на действия и класификация на възли

Nye Riemannske netværk overgår traditionelle modeller i handlingsgenkendelse og nodeklassificering

Las nuevas redes de Riemann superan a los modelos tradicionales en el reconocimiento de acciones y la clasificación de nodos

Uudet Riemannin verkot ylittävät perinteiset mallit toiminnantunnistuksessa ja solmuluokituksessa

רשתות רימן חדשות עולות על המודלים המסורתיים בזיהוי פעולה ובסיווג צמתים

Nove Riemannove mreže nadmašuju tradicionalne modele u prepoznavanju radnji i klasifikaciji čvorova

Le nuove reti riemanniane superano i modelli tradizionali nel riconoscimento delle azioni e nella classificazione dei nodi

新しいリーマンネットワークは、アクション認識とノード分類において従来のモデルよりも優れている

Жаңа Римандық желілер әрекетті тану және түйіндерді жіктеу бойынша дәстүрлі үлгілерден асып түседі

Nauji Riemanno tinklai pranoksta tradicinius modelius veiksmų atpažinimo ir mazgų klasifikavimo srityje

नयाँ Riemannian नेटवर्कहरूले कार्य पहिचान र नोड वर्गीकरणमा परम्परागत मोडेलहरूलाई उत्कृष्ट प्रदर्शन गर्दछ

新型黎曼网络在动作识别和节点分类方面优于传统模型

Table of Links Abstract and 1. Introduction Preliminaries


Proposed Approach
3.1 Notation
3.2 Nueral Networks on SPD Manifolds
3.3 MLR in Structure Spaces
3.4 Neural Networks on Grassmann Manifolds


Experiments


Conclusion and References A. Notations B. MLR in Structure Spaces C. Formulation of MLR from the Perspective of Distances to Hyperplanes D. Human Action Recognition E. Node Classification F. Limitations of our work G. Some Related Definitions H. Computation of Canonical Representation I. Proof of Proposition 3.2 J. Proof of Proposition 3.4 K. Proof of Proposition 3.5 L. Proof of Proposition 3.6 M. Proof of Proposition 3.11 N. Proof of Proposition 3.12 3.4 NEURAL NETWORKS ON GRASSMANN MANIFOLDS In this section, we present a method for computing the Grassmann logarithmic map in the projector perspective. We then propose GCNs on Grassmann manifolds. 3.4.1 GRASSMANN LOGARITHMIC MAP IN THE PROJECTOR PERSPECTIVE The Grassmann logarithmic map is given (Batzies et al., 2015; Bendokat et al., 2020) by Proof See Appendix N. 3.4.2 GRAPH CONVOLUTIONAL NETWORKS ON GRASSMANN MANIFOLDS The Grassmann logarithmic maps in the aggregation operation are obtained using Proposition 3.12. Another approach for embedding graphs on Grassmann manifolds has also been proposed in Zhou et al. (2022). However, unlike our method, this method creates a Grassmann representation for a graph via a SVD of the matrix formed from node embeddings previously learned by a Euclidean neural network. Therefore, it is not designed to learn node embeddings on Grassmann manifolds. Authors:
(1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (xuan-son.nguyen@ensea.fr);
(2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (son.nguyen@ensea.fr);
(3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (aymeric.histace@ensea.fr). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction Preliminaries Proposed Approach
3.1 Notation
3.2 Nueral Networks on SPD Manifolds
3.3 MLR in Structure Spaces
3.4 Neural Networks on Grassmann Manifolds Experiments Conclusion and References Preliminaries Preliminaries Preliminaries Proposed Approach 3.1 Notation 3.2 Nueral Networks on SPD Manifolds 3.3 MLR in Structure Spaces 3.4 Neural Networks on Grassmann Manifolds Proposed Approach 3.1 Notation 3.1 Notation 3.2 Nueral Networks on SPD Manifolds 3.2 Nueral Networks on SPD Manifolds 3.3 MLR in Structure Spaces 3.3 MLR in Structure Spaces 3.4 Neural Networks on Grassmann Manifolds 3.4 Neural Networks on Grassmann Manifolds Experiments Experiments Experiments Conclusion and References Conclusion and References Conclusion and References A. Notations A. Notations B. MLR in Structure Spaces B. MLR in Structure Spaces C. Formulation of MLR from the Perspective of Distances to Hyperplanes C. Formulation of MLR from the Perspective of Distances to Hyperplanes D. Human Action Recognition D. Human Action Recognition E. Node Classification E. Node Classification F. Limitations of our work F. Limitations of our work G. Some Related Definitions G. Some Related Definitions H. Computation of Canonical Representation H. Computation of Canonical Representation I. Proof of Proposition 3.2 I. Proof of Proposition 3.2 J. Proof of Proposition 3.4 J. Proof of Proposition 3.4 K. Proof of Proposition 3.5 K. Proof of Proposition 3.5 L. Proof of Proposition 3.6 L. Proof of Proposition 3.6 M. Proof of Proposition 3.11 M. Proof of Proposition 3.11 N. Proof of Proposition 3.12 N. Proof of Proposition 3.12 3.4 NEURAL NETWORKS ON GRASSMANN MANIFOLDS In this section, we present a method for computing the Grassmann logarithmic map in the projector perspective. We then propose GCNs on Grassmann manifolds. 3.4.1 GRASSMANN LOGARITHMIC MAP IN THE PROJECTOR PERSPECTIVE 3.4.1 GRASSMANN LOGARITHMIC MAP IN THE PROJECTOR PERSPECTIVE The Grassmann logarithmic map is given (Batzies et al., 2015; Bendokat et al., 2020) by Proof See Appendix N. Proof 3.4.2 GRAPH CONVOLUTIONAL NETWORKS ON GRASSMANN MANIFOLDS 3.4.2 GRAPH CONVOLUTIONAL NETWORKS ON GRASSMANN MANIFOLDS The Grassmann logarithmic maps in the aggregation operation are obtained using Proposition 3.12. Another approach for embedding graphs on Grassmann manifolds has also been proposed in Zhou et al. (2022). However, unlike our method, this method creates a Grassmann representation for a graph via a SVD of the matrix formed from node embeddings previously learned by a Euclidean neural network. Therefore, it is not designed to learn node embeddings on Grassmann manifolds. Authors: (1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (xuan-son.nguyen@ensea.fr); (2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (son.nguyen@ensea.fr); (3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (aymeric.histace@ensea.fr). Authors: Authors: (1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (xuan-son.nguyen@ensea.fr); (2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (son.nguyen@ensea.fr); (3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (aymeric.histace@ensea.fr). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. available on arxiv available on arxiv

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