Theoretical Statistics and Mathematics Unit, ISI Delhi

On the Bures-Wasserstein distance between positive definite matrices

by Rajendra Bhatia, Tanvi Jain and Yongdo Lim

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.

isid/ms/2018/01 [fulltext]

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