We consider an $ell_2$-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 $m,n ightarrow infty$, $m ightarrow delta$ 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 $m=left ( rac{64}{pi^2}-4ight)npprox 2.5n$ 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 $ell_2$ 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.
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机译：我们考虑一个 ell_2 $正则化的非凸优化问题，用于从嘈杂的无相位观测中恢复信号。我们设计和研究旨在解决此优化问题的消息传递算法的性能。我们考虑渐近设置$ m，n rightarrow infty $，$ m / n rightarrow delta $并获得尖锐的性能边界，其中$ m $是测量数，$ n $是信号维数。我们表明，对于复杂信号，该算法仅用$ m = left（ frac {64} { pi ^ 2} -4 right）n 约2.5n $次测量即可执行准确的恢复。此外，我们对算法对噪声的敏感性进行了深入分析。我们重点介绍有关消息传递算法的以下事实：（i）即使在无噪声的环境中，将$ ell_2 $正则化添加到非凸损失函数中也可能是有益的; （ii）频谱初始化对算法的性能影响很小。
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