Near-optimal phase retrieval of sparse vectors

TL;DR

This paper introduces a new algorithm for sparse phase retrieval that guarantees near-optimal recovery with a number of measurements proportional to the sparsity, outperforming traditional convex methods.

Contribution

It presents a polarization-based algorithm for sparse phase retrieval with measurement complexity scaling linearly with sparsity, unlike existing convex approaches.

Findings

Algorithm guarantees recovery with fewer measurements

Measurement complexity scales linearly with sparsity

Outperforms convex methods in sparse recovery

Abstract

In many areas of imaging science, it is difficult to measure the phase of linear measurements. As such, one often wishes to reconstruct a signal from intensity measurements, that is, perform phase retrieval. In several applications the signal in question is believed to be sparse. In this paper, we use ideas from the recently developed polarization method for phase retrieval and provide an algorithm that is guaranteed to recover a sparse signal from a number of phaseless linear measurements that scales linearly with the sparsity of the signal (up to logarithmic factors). This is particularly remarkable since it is known that a certain popular class of convex methods is not able to perform recovery unless the number of measurements scales with the square of the sparsity of the signal. This is a shorter version of a more complete publication that will appear elsewhere.