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In this paper, we consider the sparse phase retrieval problem, recovering an$s$-sparse signal $\bm{x}^{\natural}\in\mathbb{R}^n$ from $m$ phaseless samples$y_i=|\langle\bm{x}^{\natural},\bm{a}_i\rangle|$ for $i=1,\ldots,m$. Existingsparse phase retrieval algorithms are usually first-order and hence converge atmost linearly. Inspired by the hard thresholding pursuit (HTP) algorithm incompressed sensing, we propose an efficient second-order algorithm for sparsephase retrieval. Our proposed algorithm is theoretically guaranteed to give anexact sparse signal recovery in finite (in particular, at most $O(\log m +\log(\|\bm{x}^{\natural}\|_2/|x_{\min}^{\natural}|))$) steps, when$\{\bm{a}_i\}_{i=1}^{m}$ are i.i.d. standard Gaussian random vector with $m\simO(s\log(n/s))$ and the initialization is in a neighborhood of the underlyingsparse signal. Together with a spectral initialization, our algorithm isguaranteed to have an exact recovery from $O(s^2\log n)$ samples. Since thecomputational cost per iteration of our proposed algorithm is the same order aspopular first-order algorithms, our algorithm is extremely efficient.Experimental results show that our algorithm can be several times faster thanexisting sparse phase retrieval algorithms.

Sparse Signal Recovery from Phaseless Measurements via Hard Thresholding Pursuit