# Bures-Wasserstein Barycenters and Low-Rank Matrix Recovery

arXiv: https://arxiv.org/abs/2210.14671 · HTML: https://mlsanity.com/paper/2210.14671/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Oct 2022.

## Built on by (most cited first, 5 of 5)

- [Forward-backward Gaussian variational inference via JKO in the Bures-Wasserstein Space](https://mlsanity.com/paper/2304.05398/) (2023)
- [Acceleration and Implicit Regularization in Gaussian Phase Retrieval](https://mlsanity.com/paper/2311.12888/) (2023)
- [A Smoothing Newton Method for Rank-one Matrix Recovery](https://mlsanity.com/paper/2507.23017/) (2025)
- [Fr'echet Regression on the Bures-Wasserstein Manifold](https://mlsanity.com/paper/2604.03566/) (2026)
- [Projected Riemannian Gradient Descent for the Bures-Wasserstein Barycenter: Dimension-Independent Linear Convergence at Unit Step Size](https://mlsanity.com/paper/2609.03762/) (2026)

## Abstract (arXiv)

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.