Sample-Efficient Sparse Phase Retrieval via Stochastic Alternating Minimization

• Published in: IEEE transactions on signal processing Vol. 70; pp. 1 - 15
• Main Authors: Cai, Jian-Feng, Jiao, Yuling, Lu, Xiliang, You, Juntao
• Format: Journal Article
• Language: English
• Published: New York IEEE 2022; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Approximation algorithms; Complexity theory; finite-step convergence; Indexes; Iterative methods; Minimization; nonconvex algorithm; Phase measurement; Phase retrieval; Signal processing algorithms; Sparse matrices; Sparse phase retrieval; Spectral methods; stochastic method
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2022.3214091

In this work, we propose a nonconvex stochastic alternating minimizing (SAM) method for sparse phase retrieval, where an <inline-formula><tex-math notation="LaTeX">s</tex-math></inline-formula>-sparse vector of length <inline-formula><tex-math notation="LaTeX">n</tex-math></inline-formula> is recovered from <inline-formula><tex-math notation="LaTeX">m</tex-math></inline-formula> phaseless linear measurements. In each iteration of SAM, a batch of measurements is chosen randomly to form a sparse constrained least square subproblem, and then we employ a hard-thresholding pursuit algorithm to solve the resulting subproblem. We prove that the proposed SAM algorithm finds the target vector in at most <inline-formula><tex-math notation="LaTeX">O(\log m)</tex-math></inline-formula> steps from <inline-formula><tex-math notation="LaTeX">\Omega (s\log n)</tex-math></inline-formula> samples if provided the initial guess is in a neighbour of the ground truth. Thus, together with a desired initial guess (e.g. via a spectral method), our proposed SAM algorithm is guaranteed to have a successful sparse phase retrieval with finitely many iterations. Further, numerical experiments illustrates that SAM requires less measurements than state-of-the-art algorithms for sparse phase retrieval problem.