Optimization-Based AMP for Phase Retrieval: The Impact of Initialization and $\ell_{2}$ Regularization

We consider an <inline-formula> <tex-math notation="LaTeX">$\ell _{2}$ </tex-math></inline-formula>-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 <inline-formula> <tex-math notation="LaTeX">$m,n \rightarrow \infty $ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$m/n \rightarrow \delta $ </tex-math></inline-formula> and obtain sharp performance bounds, where <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> is the number of measurements and <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> is the signal dimension. We show that for complex signals, the algorithm can perform accurate recovery with only <inline-formula> <tex-math notation="LaTeX">$m = (({64}/{\pi ^{2}})-4)n \approx 2.5n$ </tex-math></inline-formula> measurements. Also, we provide a sharp analysis on the sensitivity of the algorithm to noise. We highlight the following facts about our message passing algorithm: 1) adding <inline-formula> <tex-math notation="LaTeX">$\ell _{2}$ </tex-math></inline-formula> regularization to the non-convex loss function can be beneficial and 2) spectral initialization has a marginal impact on the performance of the algorithm. The sharp analyses, in this paper, not only enable us to compare the performance of our method with other phase recovery schemes but also shed light on designing better iterative algorithms for other non-convex optimization problems.