Bures-Wasserstein minimizing geodesics between covariance matrices of different ranks

TL;DR

This paper characterizes the geodesics in the space of covariance matrices under the Bures-Wasserstein metric, especially between matrices of different ranks, providing explicit formulas and conditions for uniqueness.

Contribution

It completes the understanding of geodesics across different rank strata, offering explicit formulas and parametrizations for minimizing geodesics between covariance matrices.

Findings

Explicit formulas for exponential map and logarithms on each stratum.

Characterization of all minimizing geodesics between covariance matrices of any ranks.

Uniqueness of geodesics depends on the ranks of the matrices.

Abstract

The set of covariance matrices equipped with the Bures-Wasserstein distance is the orbit space of the smooth, proper and isometric action of the orthogonal group on the Euclidean space of square matrices. This construction induces a natural orbit stratification on covariance matrices, which is exactly the stratification by the rank. Thus, the strata are the manifolds of symmetric positive semi-definite (PSD) matrices of fixed rank endowed with the Bures-Wasserstein Riemannian metric. In this work, we study the geodesics of the Bures-Wasserstein distance. Firstly, we complete the literature on geodesics in each stratum by clarifying the set of preimages of the exponential map and by specifying the injection domain. We also give explicit formulae of the horizontal lift, the exponential map and the Riemannian logarithms that were kept implicit in previous works. Secondly, we give the expression of all the minimizing geodesic segments joining two covariance matrices of any rank. More precisely, we show that the set of all minimizing geodesics between two covariance matrices $\Sigma$ and $\Lambda$ is parametrized by the closed unit ball of $\mathbb{R}^{(k-r)\times(l-r)}$ for the spectral norm, where $k, l, r$ are the respective ranks of $\Sigma$, $\Lambda$, $\Sigma$$\Lambda$. In particular, the minimizing geodesic is unique if and only if $r = \min(k, l)$. Otherwise, there are infinitely many.