Sample-Efficient Sparse Phase Retrieval via Stochastic Alternating Minimization

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 in local neighbour of ground truth. Thus, two-stage, first estimate desired (e.g. via spectral method), and then introduce randomized alternating minimization strategy refinement. Also, hard-thresholding pursuit employed solve constraint least square subproblems. We give theoretical justifications that SAM find underlying signal exactly finite number iterations (no more than $O(\log m)$ steps) with high probability. Further, numerical experiments illustrates requires less measurements state-of-the-art algorithms retrieval problem.