The Gyro-Structure of Some Matrix Manifolds

Xuan Son Nguyen

Keywords: manifold learning, representation learning, deep learning, gyrovector spaces

TL;DR: This paper studies the gyrovector space structure (gyro-structure) of some matrix manifolds

Abstract: In this paper, we study the gyrovector space structure (gyro-structure) of matrix manifolds. Our work is motivated by the success of hyperbolic neural networks (HNNs) that have demonstrated impressive performance in a variety of applications. At the heart of HNNs is the theory of gyrovector spaces that provides a powerful tool for studying hyperbolic geometry. Here we focus on two matrix manifolds, i.e., Symmetric Positive Definite (SPD) and Grassmann manifolds, and consider connecting the Riemannian geometry of these manifolds with the basic operations, i.e., the binary operation and scalar multiplication on gyrovector spaces. Our work reveals some interesting facts about SPD and Grassmann manifolds. First, SPD matrices with an Affine-Invariant (AI) or a Log-Euclidean (LE) geometry have rich structure with strong connection to hyperbolic geometry. Second, linear subspaces, when equipped with our proposed basic operations, form what we call gyrocommutative and gyrononreductive gyrogroups. Furthermore, they share remarkable analogies with gyrovector spaces. We demonstrate the applicability of our approach for human activity understanding and question answering.