Sparse Phase Retrieval: Convex Algorithms and Limitations

TL;DR

This paper introduces a new algorithm for sparse phase retrieval that surpasses previous methods in recovering signals with higher sparsity levels from power spectral density measurements, addressing limitations of convex algorithms.

Contribution

Develops a reweighted l1-minimization algorithm that improves sparse signal recovery in phase retrieval, overcoming the square-root bottleneck of existing convex methods.

Findings

Signals up to $o(\sqrt{n})$ sparsity can be recovered by SDP methods.

The new algorithm recovers higher sparsity signals efficiently.

Measurement complexity is reduced to $O(k^2 \, log n)$ with random measurements and $O(k \, log n)$ with designed measurements.

Abstract

We consider the problem of recovering signals from their power spectral density. This is a classical problem referred to in literature as the phase retrieval problem, and is of paramount importance in many fields of applied sciences. In general, additional prior information about the signal is required to guarantee unique recovery as the mapping from signals to power spectral density is not one-to-one. In this paper, we assume that the underlying signals are sparse. Recently, semidefinite programming (SDP) based approaches were explored by various researchers. Simulations of these algorithms strongly suggest that signals upto $o(\sqrt{n})$ sparsity can be recovered by this technique. In this work, we develop a tractable algorithm based on reweighted $l_1$-minimization that recovers a sparse signal from its power spectral density for significantly higher sparsities, which is unprecedented. We discuss the square-root bottleneck of the existing convex algorithms and show that a $k$-sparse signal can be efficiently recovered using $O(k^2logn)$ phaseless Fourier measurements. We also show that a $k$-sparse signal can be recovered using only $O(k log n)$ phaseless measurements if we are allowed to design the measurement matrices.