Outlier-Robust Phase Retrieval in Nearly-Linear Time

Haichen Dong, Alessio Mazzetto, Yu Cheng, Rong Ge

Keywords: high-dimensional robust statistics, learning theory, phase retrieval

TL;DR: We provide a robust algorithm for the phase retrieval problem when a small fraction of the input is adversarially corrupted.

Abstract: Phase retrieval is a fundamental problem in signal processing, where the goal is to recover a (complex-valued) signal from phaseless intensity measurements. In this paper, we propose and study the (real-valued) outlier-robust phase retrieval problem. Specifically, we seek to recover a vector $x \in \mathbb{R}^d$ from $n$ intensity measurements $y_i = (a_i^\top x)^2$, where a small fraction of the $(a_i,y_i)$ pairs are adversarially corrupted. Our main result is a near-sample-optimal and nearly-linear-time algorithm that provably recovers the ground-truth vector. Our algorithm first solves a lightweight convex program to find an initial point close to the ground truth, and then runs a robust version of gradient descent to achieve exact recovery. Our approach is conceptually simple and provides a framework for developing robust algorithms for other non-convex optimization problems.

Primary Area: Theory (e.g., control theory, learning theory, algorithmic game theory)

Submission Number: 20452