Phase Recovery, MaxCut and Complex Semidefinite Programming

TL;DR

This paper introduces a new semidefinite programming relaxation called PhaseCut for phase retrieval, solving it efficiently with a block coordinate descent algorithm, and demonstrates its effectiveness through numerical experiments.

Contribution

It formulates phase retrieval as a complex semidefinite program and proposes a convergent algorithm, providing a novel approach to solving phase retrieval problems.

Findings

PhaseCut outperforms greedy algorithms in phase retrieval tasks.

The proposed algorithm converges reliably in numerical tests.

The method is effective across different phase retrieval problem types.

Abstract

Phase retrieval seeks to recover a signal x from the amplitude |Ax| of linear measurements. We cast the phase retrieval problem as a non-convex quadratic program over a complex phase vector and formulate a tractable relaxation (called PhaseCut) similar to the classical MaxCut semidefinite program. We solve this problem using a provably convergent block coordinate descent algorithm whose structure is similar to that of the original greedy algorithm in Gerchberg-Saxton, where each iteration is a matrix vector product. Numerical results show the performance of this approach over three different phase retrieval problems, in comparison with greedy phase retrieval algorithms and matrix completion formulations.