On the Bures–Wasserstein distance between positive definite matrices

By Rajendra Bhatia, Tanvi Jain, Yongdo Lim. Expositiones mathematicae, 2017

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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.

Subfields: Applied Mathematics, Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/24c289930f198ae00483c8d9c52456b9b20723d8

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Citations in the corpus

Statistical inference for Bures–Wasserstein barycenters (2019, background): “According to [Bhatia et al., 2018, Theorem 7] a function h(X) = trX1/2 is strictly concave on H+(d), hence the function Q 7→ d2BW (Q,S) = trS + trQ− 2 tr ( S1/2QS1/2 )1/2 is convex onH+(d) for any positive semi-definite S.”
Computational Optimal Transport (2018, background): “30): Kinderlehrer and Walkington [1999], Blanchet et al. [2008], Agueh and Bowles [2013], Matthes and Osberger [2014], Blanchet and Carlier [2015].”
Wasserstein Riemannian geometry of Gaussian densities (2018, background): “also [8], which in turn adapts to the Gaussian case the original work [29, §4].”