Tractable Gaussian Phase Retrieval with Heavy Tails and Adversarial Corruption with Near-Linear Sample Complexity
Santanu Das, Jatin Batra
TL;DR
This work addresses robust phase retrieval under Gaussian measurements with heavy-tailed noise and a constant corruption fraction, introducing the first polynomial-time, near-linear-sample algorithms. The approach hinges on a robust spectral initialization derived from truncated covariance of ya and robust PCA (Kong 2020), followed by robust gradient descent, and extends to non-zero-mean noise via a symmetrized blind-deconvolution reduction. It achieves an estimation error dist$(x,x^*) = O\left(\frac{\sigma}{\|x^*\|}\sqrt{\epsilon}\right)$ with sample complexity $m = \tilde{O}(n)$ and runtimes $\tilde{O}(m^2 n)$, while generalizing the initialization to non-zero-mean noise. The results reveal a deep connection between robust phase retrieval and stability-based robust PCA, offering a pathway to near-linear-time, robust non-convex estimation and suggesting several open questions on stability conditions and fundamental limits in robust high-dimensional inference.
Abstract
Phase retrieval is the classical problem of recovering a signal $x^* \in \mathbb{R}^n$ from its noisy phaseless measurements $y_i = \langle a_i, x^* \rangle^2 + ζ_i$ (where $ζ_i$ denotes noise, and $a_i$ is the sensing vector) for $i \in [m]$. The problem of phase retrieval has a rich history, with a variety of applications such as optics, crystallography, heteroscedastic regression, astrophysics, etc. A major consideration in algorithms for phase retrieval is robustness against measurement errors. In recent breakthroughs in algorithmic robust statistics, efficient algorithms have been developed for several parameter estimation tasks such as mean estimation, covariance estimation, robust principal component analysis (PCA), etc. in the presence of heavy-tailed noise and adversarial corruptions. In this paper, we study efficient algorithms for robust phase retrieval with heavy-tailed noise when a constant fraction of both the measurements $y_i$ and the sensing vectors $a_i$ may be arbitrarily adversarially corrupted. For this problem, Buna and Rebeschini (AISTATS 2025) very recently gave an exponential time algorithm with sample complexity $O(n \log n)$. Their algorithm needs a robust spectral initialization, specifically, a robust estimate of the top eigenvector of a covariance matrix, which they deemed to be beyond known efficient algorithmic techniques (similar spectral initializations are a key ingredient of a large family of phase retrieval algorithms). In this work, we make a connection between robust spectral initialization and recent algorithmic advances in robust PCA, yielding the first polynomial-time algorithms for robust phase retrieval with both heavy-tailed noise and adversarial corruptions, in fact with near-linear (in $n$) sample complexity.