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

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.

References in corpus (8)

• Manopt, a Matlab toolbox for optimization on manifolds
• Conic geometric optimisation on the manifold of positive definite matrices
• Riemannian Geometry of Symmetric Positive Definite Matrices via Cholesky Decomposition
• Geometric Mean Metric Learning
• Subspace Robust Wasserstein Distances
• Is affine invariance well defined on SPD matrices? A principled continuum of metrics
• On Riemannian Optimization over Positive Definite Matrices with the Bures-Wasserstein Geometry
• Projection Robust Wasserstein Barycenters

Cited by in corpus (1)

• Riemannian-geometric generalizations of quantum fidelities and Bures-Wasserstein distance