The recovery of complex sparse signals from few phaseless measurements

TL;DR

This paper demonstrates that complex sparse signals can be stably recovered from a minimal number of phaseless measurements using  minimization, establishing theoretical bounds for measurement requirements.

Contribution

It provides the first theoretical estimation of the measurement number needed for stable recovery of complex sparse signals from Gaussian quadratic measurements.

Findings

Stable recovery with  minimization from O(k  n/k) measurements.

Gaussian measurements satisfy restricted isometry property over rank-2 and sparse matrices.

First theoretical bounds for measurement numbers in complex sparse signal recovery.

Abstract

We study the stable recovery of complex $k$-sparse signals from as few phaseless measurements as possible. The main result is to show that one can employ $\ell_1$ minimization to stably recover complex $k$-sparse signals from $m\geq O(k\log (n/k))$ complex Gaussian random quadratic measurements with high probability. To do that, we establish that Gaussian random measurements satisfy the restricted isometry property over rank-$2$ and sparse matrices with high probability. This paper presents the first theoretical estimation of the measurement number for stably recovering complex sparse signals from complex Gaussian quadratic measurements.