Fisher-Rao distance on the covariance cone

TL;DR

This paper provides a rigorous proof of the Fisher-Rao geodesic distance on the manifold of zero-mean multivariate Gaussians using fundamental Riemannian geometry principles.

Contribution

It offers the first formal proof of the Fisher-Rao distance on the covariance cone, clarifying its geometric structure.

Findings

Fisher-Rao distance characterized on Gaussian manifold

Proof utilizes basic Riemannian geometry

Clarifies geometric properties of covariance space

Abstract

The Fisher-Rao geodesic distance on the statistical manifold consisting of zero-mean p-dimensional multivariate Gaussians appears without proof in several places (such as Steven Smith's "Covariance, Subspace, and Intrinsic Cramer-Rao Bounds"). In this paper, we give a proof using basic Riemannian geometry.