In this work we propose a nonconvex two-stage \underline{s}tochastic \underline{a}lternating \underline{m}inimizing (SAM) method for sparse phase retrieval. The proposed algorithm is guaranteed to have an exact recovery from $O(s\log n)$ samples if provided the initial guess is in a local neighbour of the ground truth. Thus, the proposed algorithm is two-stage, first we estimate a desired initial guess (e.g. via a spectral method), and then we introduce a randomized alternating minimization strategy for local refinement. Also, the hard-thresholding pursuit algorithm is employed to solve the sparse constraint least square subproblems. We give the theoretical justifications that SAM find the underlying signal exactly in a finite number of iterations (no more than $O(\log m)$ steps) with high probability. Further, numerical experiments illustrates that SAM requires less measurements than state-of-the-art algorithms for sparse phase retrieval problem.

翻译：在这个工作中,我们建议了一种非混凝土的两阶段\ 下线\ 下线\ 线{ a} 交替的\ 下线{ m} 最小化方法( SAM), 用于稀薄的阶段检索。 如果最初的猜测是在地面真相的当地附近, 则提议的算法保证能够从$O( s\log n) 样本中准确回收。 因此, 提议的算法是两阶段的, 首先我们估计了想要的初始猜法( 例如通过光谱法), 然后我们为本地的精细化采用了随机化的交替最小化策略 。 另外, 硬藏追踪算法被用来解决稀薄的制约最小的子问题。 我们给出理论理由, 使 SAM 找到的基本信号完全在一定的迭代数中( 不超过$O( log m) 步) 。 此外, 数字实验表明 SAM 需要比 最先进的算法少的测量方法来解决稀薄的阶段检索问题。