Building Neural Networks on Matrix Manifolds: A Gyrovector Space Approach
This work addresses the challenge of limited mathematical tools for neural networks on matrix manifolds, offering incremental improvements for applications like computer vision and knowledge representation.
The paper tackles the problem of building neural networks on matrix manifolds by generalizing gyrovector space concepts for SPD and Grassmann manifolds, proposing new models and layers, and demonstrates effectiveness in human action recognition and knowledge graph completion with concrete results.
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.