Provable Phase Retrieval with Mirror Descent

TL;DR

This paper introduces a mirror descent algorithm for phase retrieval that removes the need for Lipschitz continuity, demonstrating high-probability recovery and local linear convergence for Gaussian and structured measurements.

Contribution

It proposes a novel mirror descent approach for phase retrieval that relaxes classical assumptions and achieves provable convergence for different measurement models.

Findings

High-probability recovery with Gaussian measurements

Dimension-independent local linear convergence

Effective image reconstruction in optics

Abstract

In this paper, we consider the problem of phase retrieval, which consists of recovering an $n$-dimensional real vector from the magnitude of its $m$ linear measurements. We propose a mirror descent (or Bregman gradient descent) algorithm based on a wisely chosen Bregman divergence, hence allowing to remove the classical global Lipschitz continuity requirement on the gradient of the non-convex phase retrieval objective to be minimized. We apply the mirror descent for two random measurements: the \iid standard Gaussian and those obtained by multiple structured illuminations through Coded Diffraction Patterns (CDP). For the Gaussian case, we show that when the number of measurements $m$ is large enough, then with high probability, for almost all initializers, the algorithm recovers the original vector up to a global sign change. For both measurements, the mirror descent exhibits a local linear convergence behaviour with a dimension-independent convergence rate. Our theoretical results are finally illustrated with various numerical experiments, including an application to the reconstruction of images in precision optics.