An associated bundle approach to the Bures--Wasserstein geometry of fixed rank covariance matrices

TL;DR

This paper explores the geometry of fixed-rank covariance matrices under the Bures--Wasserstein metric, using an associated bundle model to derive geodesic equations and clarify geometric structures.

Contribution

It introduces a new bundle-based framework for analyzing the Bures--Wasserstein geometry of fixed-rank covariance matrices, linking Grassmannian and matrix logarithms.

Findings

Derived differential equations for Bures--Wasserstein geodesics.

Proved fibers are totally geodesic.

Established correspondence between Grassmannian and Bures--Wasserstein logarithms.

Abstract

The Bures--Wasserstein geometry of covariance matrices provides a canonical distance on the statistical manifold of centred Gaussian measures and lies at the intersection of information geometry, quantum information, and optimal transport. The space of covariance matrices admits a natural stratified structure whose strata consist of fixed-rank covariance matrices. In this paper we focus on the rank-$k$ stratum $\Sym^+(n,k)$ and revisit its geometry through the diffeomorphic associated-bundle model $\Sym^+(n,k)\cong\St(n,k)\times_{O(k)}\Sym^{+}(k)$. Working in this bundle picture, we (i) derive a system of differential equations for Bures--Wasserstein geodesics, (ii) prove that the fibers are totally geodesic and (iii) establish a one-to-one correspondence between Grassmannian logarithms and Bures--Wasserstein logarithms on $\Sym^+(n,k)$, and hence between minimizing geodesics in the two spaces. This alternative viewpoint clarifies the role of the underlying base $\Gr(k,n)$ in the Bures--Wasserstein geometry of low-rank covariance matrices and sets the stage for further investigations into structured covariance models.