The nonsmooth landscape of phase retrieval

TL;DR

This paper analyzes the nonsmooth phase retrieval problem, showing that a subgradient method converges linearly near solutions and that stationary points become concentrated as measurements increase, with supporting experiments.

Contribution

It provides theoretical convergence guarantees for a subgradient method and characterizes the distribution of stationary points in phase retrieval.

Findings

Subgradient method converges linearly near solutions.

Stationary points concentrate on a codimension two set with more measurements.

Experiments validate the theoretical results.

Abstract

We consider a popular nonsmooth formulation of the real phase retrieval problem. We show that under standard statistical assumptions, a simple subgradient method converges linearly when initialized within a constant relative distance of an optimal solution. Seeking to understand the distribution of the stationary points of the problem, we complete the paper by proving that as the number of Gaussian measurements increases, the stationary points converge to a codimension two set, at a controlled rate. Experiments on image recovery problems illustrate the developed algorithm and theory.