Bures-Wasserstein Barycenters and Low-Rank Matrix Recovery

TL;DR

This paper introduces a novel approach to low-rank positive semidefinite matrix recovery using optimal transport, framing it as a Wasserstein barycenter problem, leading to new geometric algorithms with proven convergence.

Contribution

It establishes a new variational formulation linking matrix recovery to Wasserstein barycenters, enabling the development of efficient geometric first-order methods with convergence guarantees.

Findings

New methodology outperforms existing methods in simulations

Proves strong convergence guarantees for the proposed algorithms

Reveals a geometric perspective on low-rank matrix recovery

Abstract

We revisit the problem of recovering a low-rank positive semidefinite matrix from rank-one projections using tools from optimal transport. More specifically, we show that a variational formulation of this problem is equivalent to computing a Wasserstein barycenter. In turn, this new perspective enables the development of new geometric first-order methods with strong convergence guarantees in Bures-Wasserstein distance. Experiments on simulated data demonstrate the advantages of our new methodology over existing methods.