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Sparse signal recovery from phaseless measurements via hard thresholding pursuit
Applied and Computational Harmonic Analysis  (IF3.055),  Pub Date : 2021-10-12, DOI: 10.1016/j.acha.2021.10.002
Jian-Feng Cai, Jingzhi Li, Xiliang Lu, Juntao You

In this paper, we consider the sparse phase retrieval problem, recovering an s-sparse signal ${\mathbit{x}}^{♮}\in {\mathbb{R}}^{n}$ from m phaseless samples ${y}_{i}=|〈{\mathbit{x}}^{♮},{\mathbit{a}}_{i}〉|$ for $i=1,\dots ,m$. Existing sparse phase retrieval algorithms are usually first-order and hence converge at most linearly. Inspired by the hard thresholding pursuit (HTP) algorithm in compressed sensing, we propose an efficient second-order algorithm for sparse phase retrieval. Our proposed algorithm is theoretically guaranteed to give an exact sparse signal recovery in finite (in particular, at most $O\left(\mathrm{log}m+\mathrm{log}\left({‖{\mathbit{x}}^{♮}‖}_{2}/|{x}_{\mathrm{min}}^{♮}|\right)\right)$ steps, when ${\left\{{\mathbit{a}}_{i}\right\}}_{i=1}^{m}$ are i.i.d. standard Gaussian random vector with $m\sim O\left(s\mathrm{log}\left(n/s\right)\right)$ and the initialization is in a neighborhood of the underlying sparse signal. Together with a spectral initialization, our algorithm is guaranteed to have an exact recovery from $O\left({s}^{2}\mathrm{log}n\right)$ samples. Since the computational cost per iteration of our proposed algorithm is the same order as popular first-order algorithms, our algorithm is extremely efficient. Experimental results show that our algorithm can be several times faster than existing sparse phase retrieval algorithms.