Riemannian metrics on positive definite matrices related to means

TL;DR

This paper explores a class of Riemannian metrics on positive definite matrices defined via kernel functions, analyzing their geometric properties, special cases, and how they relate to well-known metrics like Fisher-Rao and quantum Fisher information.

Contribution

It introduces a unified framework for kernel-based Riemannian metrics on positive definite matrices, including new insights into their geometric structure and special cases such as the geometric and logarithmic means.

Findings

Characterization of geodesic curves and distances for these metrics

Identification of special cases like the geometric and logarithmic means

Demonstration that certain metrics are preserved under specific mappings

Abstract

The Riemannian metric on the manifold of positive definite matrices is defined by a kernel function $\phi$ in the form $K_D^\phi(H,K)=\sum_{i,j}\phi(\lambda_i,\lambda_j)^{-1} Tr P_iHP_jK$ when $\sum_i\lambda_iP_i$ is the spectral decomposition of the foot point $D$ and the Hermitian matrices $H,K$ are tangent vectors. For such kernel metrics the tangent space has an orthogonal decomposition. The pull-back of a kernel metric under a mapping $D\mapsto G(D)$ is a kernel metric as well. Several Riemannian geometries of the literature are particular cases, for example, the Fisher-Rao metric for multivariate Gaussian distributions and the quantum Fisher information. In the paper the case $\phi(x,y)=M(x,y)^\theta$ is mostly studied when $M(x,y)$ is a mean of the positive numbers $x$ and $y$. There are results about the geodesic curves and geodesic distances. The geometric mean, the logarithmic mean and the root mean are important cases.