On the Bures-Wasserstein distance between positive definite matrices

TL;DR

This paper explores the Bures-Wasserstein distance on positive definite matrices, providing simplified proofs, developing a theory of matrix means and barycenters, and discussing algorithms for their computation within matrix analysis.

Contribution

It introduces a unified analysis of the Bures-Wasserstein metric, develops a theory of matrix means and barycenters, and discusses fixed point algorithms for computing Wasserstein barycenters.

Findings

Simplified proofs of properties of the Bures-Wasserstein metric

Development of a theory of matrix means and barycenters

Discussion of fixed point iteration algorithms for barycenter computation

Abstract

The metric $d(A,B)=\left[ \tr\, A+\tr\, B-2\tr(A^{1/2}BA^{1/2})^{1/2}\right]^{1/2}$ on the manifold of $n\times n$ positive definite matrices arises in various optimisation problems, in quantum information and in the theory of optimal transport. It is also related to Riemannian geometry. In the first part of this paper we study this metric from the perspective of matrix analysis, simplifying and unifying various proofs. Then we develop a theory of a mean of two, and a barycentre of several, positive definite matrices with respect to this metric. We explain some recent work on a fixed point iteration for computing this Wasserstein barycentre. Our emphasis is on ideas natural to matrix analysis.