What Are Gyrocommutative Gyrogroups?

Table of Links

Abstract and 1. Introduction

2. Preliminaries

3. 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

4. Experiments

5. 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