Jian-Feng Cai, Jingzhi Li, Xiliang Lu, Juntao You

In this paper, we consider the sparse phase retrieval problem, recovering an *s*-sparse signal ${\mathit{x}}^{\u266e}\in {\mathbb{R}}^{n}$ from *m* phaseless samples ${y}_{i}=|\u3008{\mathit{x}}^{\u266e},{\mathit{a}}_{i}\u3009|$ 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 $O(\mathrm{log}m+\mathrm{log}({\Vert {\mathit{x}}^{\u266e}\Vert}_{2}/|{x}_{\mathrm{min}}^{\u266e}|))$ steps, when ${\{{\mathit{a}}_{i}\}}_{i=1}^{m}$ are i.i.d. standard Gaussian random vector with $m\sim O(s\mathrm{log}(n/s))$ 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 $O({s}^{2}\mathrm{log}n)$ 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.