Robust Phase Retrieval by Alternating Minimization

TL;DR

This paper introduces a robust alternating minimization algorithm for phase retrieval that effectively handles sparse noise, with proven convergence guarantees and superior performance over existing methods.

Contribution

The paper proposes a novel robust alternating minimization method for phase retrieval with theoretical convergence analysis and practical efficiency improvements.

Findings

Guaranteed linear convergence under Gaussian measurements

Achieves order-optimal sample complexity

Outperforms existing robust phase retrieval algorithms in experiments

Abstract

We consider a least absolute deviation (LAD) approach to the robust phase retrieval problem that aims to recover a signal from its absolute measurements corrupted with sparse noise. To solve the resulting non-convex optimization problem, we propose a robust alternating minimization (Robust-AM) derived as an unconstrained Gauss-Newton method. To solve the inner optimization arising in each step of Robust-AM, we adopt two computationally efficient methods for linear programs. We provide a non-asymptotic convergence analysis of these practical algorithms for Robust-AM under the standard Gaussian measurement assumption. These algorithms, when suitably initialized, are guaranteed to converge linearly to the ground truth at an order-optimal sample complexity with high probability while the support of sparse noise is arbitrarily fixed and the sparsity level is no larger than $1/4$. Additionally, through comprehensive numerical experiments on synthetic and image datasets, we show that Robust-AM outperforms existing methods for robust phase retrieval offering comparable theoretical performance