Approximate Message Passing for Amplitude Based Optimization

We consider an ${\mathrm{\ell }}_2$-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 $m,n \rightarrow \infty$,$m/n \rightarrow \delta$andobtainsharpperformancebounds,where$m$isthenumberofmeasurementsand$n$ is the signal dimension. We show that for complex signals the algorithm can perform accurate recovery with only $m=(\frac{64}{{\pi }^2}-4)n\approx 2.5n$ 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 ${\mathrm{\ell }}_2$ 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.