A Smoothing Newton Method for Rank-one Matrix Recovery

--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. Phase retrieval--the problem of recovering a real or complex signal from magnitude-only measurements--is a fundamental problem in signal processing. Its applications range from X-ray crystallography to astronomical imaging, where measurement systems capture a form of intensity [Harrison, 1993, Fienup, 1982, Fienup and Dainty, 1987, Miao et al., 1998]. The seemingly simple constraint of measuring magnitudes transforms what would be a linear problem into a challenging nonlinear and nonconvex optimization problem. More critically, the direct optimization formulation using least squares yields a nonconvex objective function, making it difficult to solve effectively. The phase retrieval problem is a specific instance of a broader class of low-rank matrix sensing problems that arise throughout signal processing [Recht et al., 2010].