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

摘要

We consider an $ell_2$-regularized non-convex optimization problem forrecovering signals from their noisy phaseless observations. We design and studythe performance of a message passing algorithm that aims to solve thisoptimization problem. We consider the asymptotic setting $m,n ightarrowinfty$, $m/n ightarrow delta$ and obtain sharp performance bounds, where$m$ is the number of measurements and $n$ is the signal dimension. We show thatfor complex signals the algorithm can perform accurate recovery with only $m=rac{64}{pi^2}-4 pprox 2.5n$ measurements. Also, we provide sharp analysison the sensitivity of the algorithm to noise. We highlight the following factsabout our message passing algorithm: (i) Adding $ell_2$ regularization to thenon-convex loss function can be beneficial. (ii) Spectral initialization hasmarginal impact on the performance of the algorithm. The sharp analyses in thispaper, not only enable us to compare the performance of our method with otherphase recovery schemes, but also shed light on designing better iterativealgorithms for other non-convex optimization problems.