Non-Convex Structured Phase Retrieval

TL;DR

This paper reviews non-convex methods for phase retrieval that leverage signal structure, such as sparsity or low rank, to reduce measurement requirements with theoretical guarantees.

Contribution

It provides an overview of non-convex algorithms for structured phase retrieval and discusses their sample complexity guarantees under simple assumptions.

Findings

Non-convex approaches can achieve sample-efficient phase retrieval.

Structural assumptions like sparsity and low rank improve measurement efficiency.

The paper summarizes recent advances and guarantees in the field.

Abstract

Phase retrieval (PR), also sometimes referred to as quadratic sensing, is a problem that occurs in numerous signal and image acquisition domains ranging from optics, X-ray crystallography, Fourier ptychography, sub-diffraction imaging, and astronomy. In each of these domains, the physics of the acquisition system dictates that only the magnitude (intensity) of certain linear projections of the signal or image can be measured. Without any assumptions on the unknown signal, accurate recovery necessarily requires an over-complete set of measurements. The only way to reduce the measurements/sample complexity is to place extra assumptions on the unknown signal/image. A simple and practically valid set of assumptions is obtained by exploiting the structure inherently present in many natural signals or sequences of signals. Two commonly used structural assumptions are (i) sparsity of a given signal/image or (ii) a low rank model on the matrix formed by a set, e.g., a time sequence, of signals/images. Both have been explored for solving the PR problem in a sample-efficient fashion. This article describes this work, with a focus on non-convex approaches that come with sample complexity guarantees under simple assumptions. We also briefly describe other different types of structural assumptions that have been used in recent literature.