Phase Retrieval via Linear Programming: Fundamental Limits and Algorithmic Improvements

TL;DR

This paper provides an exact asymptotic analysis of the PhaseMax convex approach for phase retrieval with Gaussian measurements, reveals a phase transition, and introduces an improved nonconvex algorithm called PhaseLamp.

Contribution

It offers the first exact performance analysis of PhaseMax, uncovers a phase transition phenomenon, and proposes a new algorithm, PhaseLamp, with better recovery performance.

Findings

Exact asymptotic performance bounds for PhaseMax

Identification of a sharp phase transition in recovery success

Introduction of the PhaseLamp algorithm with improved performance

Abstract

A recently proposed convex formulation of the phase retrieval problem estimates the unknown signal by solving a simple linear program. This new scheme, known as PhaseMax, is computationally efficient compared to standard convex relaxation methods based on lifting techniques. In this paper, we present an exact performance analysis of PhaseMax under Gaussian measurements in the large system limit. In contrast to previously known performance bounds in the literature, our results are asymptotically exact and they also reveal a sharp phase transition phenomenon. Furthermore, the geometrical insights gained from our analysis led us to a novel nonconvex formulation of the phase retrieval problem and an accompanying iterative algorithm based on successive linearization and maximization over a polytope. This new algorithm, which we call PhaseLamp, has provably superior recovery performance over the original PhaseMax method.