Bures-Wasserstein Barycenters and Low-Rank Matrix Recovery

Understand

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.