Abstract

We consider an -regularized non-convex optimization problem for recovering signals from their noisy phaseless observations. We design and study the performance of a message passing algorithm that aims to solve this optimization problem. We consider the asymptotic setting m, n ∞ , m/n  δ and obtain sharp performance bounds, where m is the number of measurements and n is the signal dimension. We show that for complex signals the algorithm can perform accurate recovery with only m = measurements. Also, we provide sharp analysis on the sensitivity of the algorithm to noise. We highlight the following facts about our message passing algorithm: (i) Adding regularization to the non-convex loss function can be beneficial even in the noiseless setting; (ii) spectral initialization has marginal impact on the performance of the algorithm.