PhaseMax: Convex Phase Retrieval via Basis Pursuit

TL;DR

PhaseMax introduces a convex, non-lifting approach to phase retrieval that leverages basis pursuit algorithms, providing theoretical guarantees and practical effectiveness for recovering signals from magnitude-only measurements.

Contribution

It presents a novel convex formulation called PhaseMax that operates without lifting and connects phase retrieval to basis pursuit, enabling new algorithmic and theoretical insights.

Findings

PhaseMax achieves high success probability with random measurements.

The dual problem to PhaseMax is basis pursuit, linking phase retrieval to sparse recovery methods.

Numerical results confirm the theoretical recovery guarantees and practical effectiveness.

Abstract

We consider the recovery of a (real- or complex-valued) signal from magnitude-only measurements, known as phase retrieval. We formulate phase retrieval as a convex optimization problem, which we call PhaseMax. Unlike other convex methods that use semidefinite relaxation and lift the phase retrieval problem to a higher dimension, PhaseMax is a "non-lifting" relaxation that operates in the original signal dimension. We show that the dual problem to PhaseMax is Basis Pursuit, which implies that phase retrieval can be performed using algorithms initially designed for sparse signal recovery. We develop sharp lower bounds on the success probability of PhaseMax for a broad range of random measurement ensembles, and we analyze the impact of measurement noise on the solution accuracy. We use numerical results to demonstrate the accuracy of our recovery guarantees, and we showcase the efficacy and limits of PhaseMax in practice.