O(n)-invariant Riemannian metrics on SPD matrices

TL;DR

This paper explores and extends classes of O(n)-invariant Riemannian metrics on SPD matrices, introducing new properties like cometric-stability and providing formulas for curvature and parallel transport.

Contribution

It characterizes super-classes of kernel metrics, investigates key properties, and introduces the concept of cometric-stability for geodesic computation.

Findings

Characterization of O(n)-invariant metrics on SPD matrices

Introduction of cometric-stability property

Formulas for sectional curvature and parallel transport

Abstract

Symmetric Positive Definite (SPD) matrices are ubiquitous in data analysis under the form of covariance matrices or correlation matrices. Several O(n)-invariant Riemannian metrics were defined on the SPD cone, in particular the kernel metrics introduced by Hiai and Petz. The class of kernel metrics interpolates between many classical O(n)-invariant metrics and it satisfies key results of stability and completeness. However, it does not contain all the classical O(n)-invariant metrics. Therefore in this work, we investigate super-classes of kernel metrics and we study which key results remain true. We also introduce an additional key result called cometric-stability, a crucial property to implement geodesics with a Hamiltonian formulation. Our method to build intermediate embedded classes between O(n)-invariant metrics and kernel metrics is to give a characterization of the whole class of O(n)-invariant metrics on SPD matrices and to specify requirements on metrics one by one until we reach kernel metrics. As a secondary contribution, we synthesize the literature on the main O(n)-invariant metrics, we provide the complete formula of the sectional curvature of the affine-invariant metric and the formula of the geodesic parallel transport between commuting matrices for the Bures-Wasserstein metric.