This paper considers the problem of phase retrieval, where the goal is to recover a signal $\mathit{z}\in {\mathbb{C}}^n$ from the observations $y_i=|{\mathit{a}}_i^{⁎}\mathit{z}|$, $i=1,2,\cdots ,m$. While many algorithms have been proposed, the alternating minimization algorithm is still one of the most commonly used and the simplest methods. Existing works have proved that when the observation vectors ${\{{\mathit{a}}_i\}}_{i=1}^m$ are sampled from a complex normal distribution $CN(0,\mathbf{I})$, the alternating minimization algorithm recovers the underlying signal with a good initialization when $m=O(n)$, or with random initialization when $m=O(n^2)$, and it is conjectured that random initialization succeeds with $m=O(n)$ [26]. This work proposes a modified alternating minimization method in a batch setting and proves that when $m=O(n{\log }^{5}n)$, the proposed algorithm with random initialization recovers the underlying signal with high probability. The proof is based on the observation that after each iteration of alternating minimization, with high probability, the correlation between the direction of the estimated signal and the direction of the underlying signal increases.

本文考虑了相位恢复的问题，其目标是恢复信号 $\mathit{ž}\in {\mathbb{C}}^{ñ}$ 从观察 ${ÿ}_{\mathrm{一世}}=|{\mathit{一种}}_{\mathrm{一世}}^{⁎}\mathit{ž}|$， $\mathrm{一世}=\mathrm{1个}，2，\cdots ，米$。尽管已经提出了许多算法，但是交替最小化算法仍然是最常用和最简单的方法之一。现有工作证明，当观测向量${\{{\mathit{一种}}_{\mathrm{一世}}\}}_{\mathrm{一世}=\mathrm{1个}}^{米}$ 从复杂的正态分布中采样 $Cñ（0，\mathbf{一世}）$，交替最小化算法可在以下情况下通过良好的初始化恢复基础信号： $米=Ø（ñ）$，或在以下情况下使用随机初始化 $米=Ø（{ñ}^2）$，并且推测随机初始化成功 $米=Ø（ñ）$[26]。这项工作提出了一种改进的在批次设置中交替最小化的方法，并证明了当$米=Ø（ñ{\mathrm{日志}}^5ñ）$，所提出的随机初始化算法以较高的概率恢复基础信号。该证明基于以下观察：在交替最小化的每次迭代之后，概率很高，估计信号的方向与基础信号的方向之间的相关性增加。