Noisy phase retrieval from subgaussian measurements

TL;DR

This paper develops robust, regularization-free algorithms for noisy phase retrieval from subgaussian measurements, achieving linear convergence without truncation or penalty terms, and provides the first theoretical guarantees for non-peaky signals with Bernoulli measurements.

Contribution

Introduces novel spectral initialization and analyzes Wirtinger flow for phase retrieval under subgaussian noise, with improved step sizes and no regularization, covering non-peaky signals and Bernoulli measurements.

Findings

Achieves linear convergence with sample complexity $m \,\ge\, O(n \log^3 m)$

Algorithms are regularization-free and allow large step sizes $O(1)$

Provides first theoretical guarantees for non-peaky signals from Bernoulli measurements.

Abstract

This paper aims to address the phase retrieval problem from subgaussian measurements with arbitrary noise, with a focus on devising robust and efficient algorithms for solving non-convex problems. To ensure uniqueness of solutions in the subgaussian setting, we explore two commonly used assumptions: either the subgaussian measurements satisfy a fourth-moment condition or the target signals exhibit non-peakiness. For each scenario, we introduce a novel spectral initialization method that yields robust initial estimates. Building on this, we employ leave-one-out arguments to show that the classical Wirtinger flow algorithm achieves a linear rate of convergence for both real-valued and complex-valued cases, provided the sampling complexity $m\ge O(n \log^3 m)$, where $n$ is the dimension of the underlying signals. In contrast to existing work, our algorithms are regularization-free, requiring no truncation, trimming, or additional penalty terms, and they permit the algorithm step sizes as large as $O(1)$, compared to the $O(1/n)$ in previous literature. Furthermore, our results accommodate arbitrary noise vectors that meet certain statistical conditions, covering a wide range of noise scenarios, with sub-exponential noise as a notable special case. The effectiveness of our algorithms is validated through various numerical experiments. We emphasize that our findings provide the first theoretical guarantees for recovering non-peaky signals using non-convex methods from Bernoulli measurements, which is of independent interest.