Stable Recovery of Weighted Sparse Signals from Phaseless Measurements via Weighted l1 Minimization

TL;DR

This paper extends the theoretical framework of phaseless compressed sensing by incorporating prior support information, demonstrating conditions under which weighted l1 minimization guarantees stable recovery of sparse signals from magnitude-only measurements.

Contribution

It introduces the weighted null space property and strong weighted restricted isometry property as new conditions for stable recovery in phaseless compressed sensing with prior support knowledge.

Findings

Weighted null space property is necessary and sufficient for weighted l1 recovery.

Strong weighted RIP ensures stable signal recovery from phaseless measurements.

Theoretical guarantees extend to signals with prior support information.

Abstract

The goal of phaseless compressed sensing is to recover an unknown sparse or approximately sparse signal from the magnitude of its measurements. However, it does not take advantage of any support information of the original signal. Therefore, our main contribution in this paper is to extend the theoretical framework for phaseless compressed sensing to incorporate with prior knowledge of the support structure of the signal. Specifically, we investigate two conditions that guarantee stable recovery of a weighted $k$-sparse signal via weighted l1 minimization without any phase information. We first prove that the weighted null space property (WNSP) is a sufficient and necessary condition for the success of weighted l1 minimization for weighted k-sparse phase retrievable. Moreover, we show that if a measurement matrix satisfies the strong weighted restricted isometry property (SWRIP), then the original signal can be stably recovered from the phaseless measurements.