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

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