On the Bures–Wasserstein distance between positive definite matrices

Authors

Rajendra Bhatia, Ashoka University
T. Jain, Indian Statistical Institute (Delhi Centre)
Yongdo Lim, Sungkyunkwan University

Article Type

Research Article

Publication Title

Expositiones Mathematicae

Abstract

The metric d(A,B)=trA+trB−2tr(A1∕2BA1∕2)1∕21∕2 on the manifold of n×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.

First Page

165

Last Page

191

DOI

10.1016/j.exmath.2018.01.002

Publication Date

6-1-2019

Comments

Open Access, Green

Recommended Citation

Bhatia, Rajendra; Jain, T.; and Lim, Yongdo, "On the Bures–Wasserstein distance between positive definite matrices" (2019). Journal Articles. 833.
https://digitalcommons.isical.ac.in/journal-articles/833