Stable Signal Recovery from Phaseless Measurements

TL;DR

This paper investigates the stability of $ ext{l}_1$ minimization in compressive phase retrieval, establishing measurement bounds and null space properties to ensure accurate sparse signal recovery from phaseless measurements.

Contribution

It extends instance-optimality concepts to real phase retrieval and provides conditions under which stable recovery is guaranteed using $ ext{l}_1$ minimization.

Findings

O($k \, ext{log}(N/k)$) measurements suffice for stable recovery.

Null space property ensures phaseless instance-optimality.

Parallel results to classical compressive sensing are established.

Abstract

The aim of this paper is to study the stability of the $\ell_1$ minimization for the compressive phase retrieval and to extend the instance-optimality in compressed sensing to the real phase retrieval setting. We first show that the $m={\mathcal O}(k\log(N/k))$ measurements is enough to guarantee the $\ell_1$ minimization to recover $k$-sparse signals stably provided the measurement matrix $A$ satisfies the strong RIP property. We second investigate the phaseless instance-optimality with presenting a null space property of the measurement matrix $A$ under which there exists a decoder $\Delta$ so that the phaseless instance-optimality holds. We use the result to study the phaseless instance-optimality for the $\ell_1$ norm. The results build a parallel for compressive phase retrieval with the classical compressive sensing.