Learning with symmetric positive definite matrices via generalized Bures-Wasserstein geometry

TL;DR

This paper introduces a generalized Bures-Wasserstein geometry for symmetric positive definite matrices, expanding the geometric framework to enhance machine learning applications and demonstrating its advantages through experiments.

Contribution

The paper proposes a novel generalized Bures-Wasserstein geometry parameterized by a matrix, extending the existing BW geometry and enabling new differential geometric tools for machine learning.

Findings

The generalized geometry encompasses the BW geometry as a special case.

Experiments show improved performance over traditional BW geometry.

The framework facilitates new applications in machine learning.

Abstract

Learning with symmetric positive definite (SPD) matrices has many applications in machine learning. Consequently, understanding the Riemannian geometry of SPD matrices has attracted much attention lately. A particular Riemannian geometry of interest is the recently proposed Bures-Wasserstein (BW) geometry which builds on the Wasserstein distance between the Gaussian densities. In this paper, we propose a novel generalization of the BW geometry, which we call the GBW geometry. The proposed generalization is parameterized by a symmetric positive definite matrix $\mathbf{M}$ such that when $\mathbf{M} = \mathbf{I}$, we recover the BW geometry. We provide a rigorous treatment to study various differential geometric notions on the proposed novel generalized geometry which makes it amenable to various machine learning applications. We also present experiments that illustrate the efficacy of the proposed GBW geometry over the BW geometry.