The geometry of mixed-Euclidean metrics on symmetric positive definite matrices

TL;DR

This paper introduces a unified framework for defining and analyzing mixed-Euclidean metrics on the manifold of SPD matrices, revealing their geometric properties and relationships with divergence measures.

Contribution

It formalizes the principle of deformed metrics, introduces the Mixed-Euclidean metrics, and explores their geometric properties and connections to divergence measures in information geometry.

Findings

Mixed-Euclidean metrics generalize MPE metrics.

Sectional curvature of these metrics can be negative or positive.

Explicit formulas for the Riemann curvature tensor are provided.

Abstract

Several Riemannian metrics and families of Riemannian metrics were defined on the manifold of Symmetric Positive Definite (SPD) matrices. Firstly, we formalize a common general process to define families of metrics: the principle of deformed metrics. We relate the recently introduced family of alpha-Procrustes metrics to the general class of mean kernel metrics by providing a sufficient condition under which elements of the former belongs to the latter. Secondly, we focus on the principle of balanced bilinear forms that we recently introduced. We give a new sufficient condition under which the balanced bilinear form is a metric. It allows us to introduce the Mixed-Euclidean (ME) metrics which generalize the Mixed-Power-Euclidean (MPE) metrics. We unveal their link with the (u, v)-divergences and the ($\alpha$, $\beta$)-divergences of information geometry and we provide an explicit formula of the Riemann curvature tensor. We show that the sectional curvature of all ME metrics can take negative values and we show experimentally that the sectional curvature of all MPE metrics but the log-Euclidean, power-Euclidean and power-affine metrics can take positive values.