Generalized BuresWasserstein Geometry for Positive Definite Matrices
Abstract
This paper proposes a generalized BuresWasserstein (BW) Riemannian geometry for the manifold of symmetric positive definite matrices. We explore the generalization of the BW geometry in three different ways: 1) by generalizing the Lyapunov operator in the metric, 2) by generalizing the orthogonal Procrustes distance, and 3) by generalizing the Wasserstein distance between the Gaussians. We show that they all lead to the same 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 derive expressions for the distance, geodesic, exponential/logarithm maps, LeviCivita connection, and sectional curvature under the generalized BW geometry. We also present applications and experiments that illustrate the efficacy of the proposed geometry.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.10464
 Bibcode:
 2021arXiv211010464H
 Keywords:

 Mathematics  Functional Analysis;
 Mathematics  Differential Geometry;
 Mathematics  Optimization and Control;
 Mathematics  Statistics Theory;
 Statistics  Machine Learning