Stable Signal Recovery from Phaseless Measurements

Abstract

The aim of this paper is to study the stability of the ℓ<inf>1</inf> 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 m= O(klog (N/ k)) measurements are enough to guarantee the ℓ<inf>1</inf> minimization to recover k-sparse signals stably provided the measurement matrix A satisfies the strong RIP property. We second investigate the phaseless instance-optimality presenting a null space property of the measurement matrix A under which there exists a decoder Δ so that the phaseless instance-optimality holds. We use the result to study the phaseless instance-optimality for the ℓ<inf>1</inf> norm. This builds a parallel for compressive phase retrieval with the classical compressive sensing.