Compressive Phase Retrieval From Squared Output Measurements Via Semidefinite Programming

TL;DR

This paper introduces a semidefinite programming approach for recovering sparse signals from squared output measurements without phase information, extending compressive sensing to phase retrieval scenarios with theoretical guarantees and practical validation.

Contribution

It proposes a novel semidefinite programming method for sparse phase retrieval from squared measurements, expanding compressive sensing applications to phase-omitted data.

Findings

Exact sparse signal recovery with high sampling rates

Theoretical guarantees for phase retrieval accuracy

Validated through simulations and practical experiments

Abstract

Given a linear system in a real or complex domain, linear regression aims to recover the model parameters from a set of observations. Recent studies in compressive sensing have successfully shown that under certain conditions, a linear program, namely, l1-minimization, guarantees recovery of sparse parameter signals even when the system is underdetermined. In this paper, we consider a more challenging problem: when the phase of the output measurements from a linear system is omitted. Using a lifting technique, we show that even though the phase information is missing, the sparse signal can be recovered exactly by solving a simple semidefinite program when the sampling rate is sufficiently high, albeit the exact solutions to both sparse signal recovery and phase retrieval are combinatorial. The results extend the type of applications that compressive sensing can be applied to those where only output magnitudes can be observed. We demonstrate the accuracy of the algorithms through theoretical analysis, extensive simulations and a practical experiment.