Inexact Alternating Optimization for Phase Retrieval In the Presence of Outliers

TL;DR

This paper introduces an inexact alternating optimization approach for phase retrieval that effectively handles outliers by using an $ ext{l}_p$-norm estimator, outperforming existing methods under heavy-tailed noise conditions.

Contribution

It proposes a novel $ ext{l}_p$-norm based estimator and two algorithms within an inexact alternating optimization framework for robust phase retrieval with outliers.

Findings

Algorithms approach the Cramér-Rao bound.

Outperform state-of-the-art methods in heavy-tailed noise.

Convergence properties are established.

Abstract

Phase retrieval has been mainly considered in the presence of Gaussian noise. However, the performance of the algorithms proposed under the Gaussian noise model severely degrades when grossly corrupted data, i.e., outliers, exist. This paper investigates techniques for phase retrieval in the presence of heavy-tailed noise -- which is considered a better model for situations where outliers exist. An $\ell_p$-norm ($0<p<2$) based estimator is proposed for fending against such noise, and two-block inexact alternating optimization is proposed as the algorithmic framework to tackle the resulting optimization problem. Two specific algorithms are devised by exploring different local approximations within this framework. Interestingly, the core conditional minimization steps can be interpreted as iteratively reweighted least squares and gradient descent. Convergence properties of the algorithms are discussed, and the Cram\'er-Rao bound (CRB) is derived. Simulations demonstrate that the proposed algorithms approach the CRB and outperform state-of-the-art algorithms in heavy-tailed noise.