Phase retrieval via overparametrized nonconvex optimization: nonsmooth amplitude loss landscapes

TL;DR

This paper analyzes the landscape of overparametrized nonconvex optimization methods for phase retrieval, providing guarantees that support their effectiveness and optimality in high-dimensional measurement settings.

Contribution

It offers a deterministic analysis of critical points for nonsmooth and smooth loss functions, establishing state-of-the-art landscape guarantees with minimal overparametrization.

Findings

High-probability landscape guarantees in phase retrieval

State-of-the-art guarantees with minimal overparametrization

Effective nonconvex optimization for high-dimensional phase retrieval

Abstract

We study nonconvex optimization for phase retrieval and the more general problem of semidefinite low-rank matrix sensing; in particular, we focus on the global nonconvex landscape of overparametrized versions of the nonsmooth amplitude least-squares loss as well as a smooth reformulation of this loss based on the PhaseCut approach. We first give a general, deterministic result on properties of second-order critical points for a general class of loss functions; we then specialize this result to the nonsmooth amplitude loss and, additionally, prove nearly identical results for a smooth reformulation (similar to PhaseCut) as a synchronization problem over spheres. Finally, we show the usefulness of these tools by proving high-probability landscape guarantees in two settings: (1) phase retrieval with isotropic sub-Gaussian measurements, and (2) phase retrieval in a general (possibly infinite-dimensional) Hilbert space with Gaussian measurements. In both cases, our results give state-of-the-art and statistically optimal guarantees with only a constant amount of overparametrization (in the well-studied case of isotropic sub-Gaussian measurements, such statistical guarantees had previously required greater degrees of overparametrization/relaxation); this demonstrates the potential of overparametrized nonconvex optimization as a principled and scalable algorithmic approach to phase retrieval.