Sample-Efficient Sparse Phase Retrieval via Stochastic Alternating Minimization

TL;DR

This paper introduces a stochastic alternating minimization method for sparse phase retrieval that guarantees exact recovery with fewer samples and fewer measurements than existing algorithms, combining spectral initialization and hard-thresholding.

Contribution

The paper presents a novel two-stage stochastic alternating minimization algorithm with theoretical guarantees for sparse phase retrieval, improving sample efficiency and recovery accuracy.

Findings

Exact recovery with $O(s\, ext{log}\,n)$ samples.

Finite iteration convergence with high probability.

Requires fewer measurements than state-of-the-art methods.

Abstract

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.