Building Neural Networks on Matrix Manifolds: A Gyrovector Space Approach

TL;DR

This paper extends gyrovector space theory to matrix manifolds like SPD and Grassmann, enabling the construction of neural networks on these manifolds with demonstrated effectiveness in action recognition and knowledge graph tasks.

Contribution

It generalizes gyrovector space concepts to SPD and Grassmann manifolds and introduces new neural network models for these spaces.

Findings

Effective neural network models for SPD and Grassmann manifolds.

Improved performance in human action recognition.

Successful application to knowledge graph completion.

Abstract

Matrix manifolds, such as manifolds of Symmetric Positive Definite (SPD) matrices and Grassmann manifolds, appear in many applications. Recently, by applying the theory of gyrogroups and gyrovector spaces that is a powerful framework for studying hyperbolic geometry, some works have attempted to build principled generalizations of Euclidean neural networks on matrix manifolds. However, due to the lack of many concepts in gyrovector spaces for the considered manifolds, e.g., the inner product and gyroangles, techniques and mathematical tools provided by these works are still limited compared to those developed for studying hyperbolic geometry. In this paper, we generalize some notions in gyrovector spaces for SPD and Grassmann manifolds, and propose new models and layers for building neural networks on these manifolds. We show the effectiveness of our approach in two applications, i.e., human action recognition and knowledge graph completion.