PhaseMax: Convex Phase Retrieval via Basis Pursuit

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

PhasePack: A Phase Retrieval Library

Phase retrieval deals with the estimation of complex-valued signals solely from the magnitudes of linear measurements. While there has been a recent explosion in the development of phase retrieval algorithms, the lack of a common interface has made it difficult to compare new methods against the state-of-the-art. The purpose of PhasePack is to create a common software interface for a wide range of phase retrieval algorithms and to provide a common testbed using both synthetic data and empirical imaging datasets. PhasePack is able to benchmark a large number of recent phase retrieval methods against one another to generate comparisons using a range of different performance metrics. The software package handles single method testing as well as multiple method comparisons. The algorithm implementations in PhasePack differ slightly from their original descriptions in the literature in order to achieve faster speed and improved robustness. In particular, PhasePack uses adaptive stepsizes, line-search methods, and fast eigensolvers to speed up and automate convergence