A Smoothing Newton Method for Rank-one Matrix Recovery

TL;DR

This paper introduces a smoothing Newton method for rank-one matrix recovery that offers stable, superlinear convergence, improving upon existing algorithms like BWGD by addressing their instability and limited convergence guarantees.

Contribution

The authors develop a smoothing framework that regularizes the nonsmooth, nonconvex objective in phase retrieval, enabling a stable Newton-based method with proven superlinear convergence.

Findings

The proposed method achieves superior stability in synthetic experiments.

It maintains fast convergence comparable to or better than existing methods.

The framework extends to general rank-one matrix recovery problems.

Abstract

We consider the phase retrieval problem, which involves recovering a rank-one positive semidefinite matrix from rank-one measurements. A recently proposed algorithm based on Bures-Wasserstein gradient descent (BWGD) exhibits superlinear convergence, but it is unstable, and existing theory can only prove local linear convergence for higher rank matrix recovery. We resolve this gap by revealing that BWGD implements Newton's method with a nonsmooth and nonconvex objective. We develop a smoothing framework that regularizes the objective, enabling a stable method with rigorous superlinear convergence guarantees. Experiments on synthetic data demonstrate this superior stability while maintaining fast convergence.