Exploration of Balanced Metrics on Symmetric Positive Definite Matrices

TL;DR

This paper introduces the principle of balanced metrics on SPD matrices, proposing new metric families that relate existing metrics and exploring their connection to dual connections in information geometry.

Contribution

It proposes the principle of balanced metrics on SPD matrices and introduces two new families of such metrics, linking them to dual connections in information geometry.

Findings

Introduces the principle of balanced metrics for SPD matrices.

Defines two new families: mixed-power-Euclidean and mixed-power-affine.

Discusses the relation to dual connections in information geometry.

Abstract

Symmetric Positive Definite (SPD) matrices have been used in many fields of medical data analysis. Many Riemannian metrics have been defined on this manifold but the choice of the Riemannian structure lacks a set of principles that could lead one to choose properly the metric. This drives us to introduce the principle of balanced metrics that relate the affine-invariant metric with the Euclidean and inverse-Euclidean metric, or the Bogoliubov-Kubo-Mori metric with the Euclidean and log-Euclidean metrics. We introduce two new families of balanced metrics, the mixed-power-Euclidean and the mixed-power-affine metrics and we discuss the relation between this new principle of balanced metrics and the concept of dual connections in information geometry.