Robust Gradient Descent for Phase Retrieval
Alex Buna, Patrick Rebeschini
TL;DR
The paper studies robust phase retrieval where measurements $y=(\boldsymbol{a}^\top \boldsymbol{x^*})^2+z$ are contaminated by heavy-tailed noise and adversarial outliers. It introduces a robust gradient-descent framework built on Wirtinger Flow, using robust mean estimators in place of empirical gradients and two spectral initialization strategies (MeanEstStab and CovEst) to handle zero-mean and unknown-mean noise. Theoretical guarantees show good initialization within a strongly convex region and linear convergence of the iterative updates, with sample complexities $m_0=O(n^2\log n)$ (MeanEstStab) or $m_0=O(n)$ (CovEst) for initialization and per-iteration samples $\tilde{m}=O(n\log n)$, yielding near-optimal total complexity. For unknown mean noise, a preprocessing step reduces to the zero-mean setting while preserving guarantees. Overall, the work advances robust non-convex phase retrieval by providing practical algorithms with explicit finite-sample performance under heavy-tailed noise and arbitrary contamination.
Abstract
Recent progress in robust statistical learning has mainly tackled convex problems, like mean estimation or linear regression, with non-convex challenges receiving less attention. Phase retrieval exemplifies such a non-convex problem, requiring the recovery of a signal from only the magnitudes of its linear measurements, without phase (sign) information. While several non-convex methods, especially those involving the Wirtinger Flow algorithm, have been proposed for noiseless or mild noise settings, developing solutions for heavy-tailed noise and adversarial corruption remains an open challenge. In this paper, we investigate an approach that leverages robust gradient descent techniques to improve the Wirtinger Flow algorithm's ability to simultaneously cope with fourth moment bounded noise and adversarial contamination in both the inputs (covariates) and outputs (responses). We address two scenarios: known zero-mean noise and completely unknown noise. For the latter, we propose a preprocessing step that alters the problem into a new format that does not fit traditional phase retrieval approaches but can still be resolved with a tailored version of the algorithm for the zero-mean noise context.