A Fast and Provable Algorithm for Sparse Phase Retrieval

We study the sparse phase retrieval problem, which seeks to recover a sparse signal from a limited set of magnitude-only measurements. In contrast to prevalent sparse phase retrieval algorithms that primarily use first-order methods, we propose an innovative second-order algorithm that employs a New...

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Bibliographic Details
Published inarXiv.org
Main Authors Jian-Feng, Cai, Long, Yu, Wen, Ruixue, Ying, Jiaxi
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 19.03.2024
Subjects
Algorithms
Complexity
Convergence
Iterative methods
Phase retrieval
Signal reconstruction
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Summary:We study the sparse phase retrieval problem, which seeks to recover a sparse signal from a limited set of magnitude-only measurements. In contrast to prevalent sparse phase retrieval algorithms that primarily use first-order methods, we propose an innovative second-order algorithm that employs a Newton-type method with hard thresholding. This algorithm overcomes the linear convergence limitations of first-order methods while preserving their hallmark per-iteration computational efficiency. We provide theoretical guarantees that our algorithm converges to the \(s\)-sparse ground truth signal \(\mathbf{x}^{\natural} \in \mathbb{R}^n\) (up to a global sign) at a quadratic convergence rate after at most \(O(\log (\Vert\mathbf{x}^{\natural} \Vert /x_{\min}^{\natural}))\) iterations, using \(\Omega(s^2\log n)\) Gaussian random samples. Numerical experiments show that our algorithm achieves a significantly faster convergence rate than state-of-the-art methods.
ISSN:2331-8422