Sample-Efficient Sparse Phase Retrieval via Stochastic Alternating Minimization

Abstract

In this work we propose a nonconvex two-stage stochastic alternating minimizing (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.