In this paper, we consider the sparse phase retrieval problem, recovering an s-sparse signal ${\mathit{x}}^{♮}\in {\mathbb{R}}^n$ from m phaseless samples $y_i=|〈{\mathit{x}}^{♮},{\mathit{a}}_i〉|$ 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 $O(\log m+\log ({\Vert {\mathit{x}}^{♮}\Vert }_2/|x_{\min }^{♮}|))$ steps, when ${\{{\mathit{a}}_i\}}_{i=1}^m$ are i.i.d. standard Gaussian random vector with $m\sim O(s\log (n/s))$ 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 $O(s^2\log n)$ 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.

在本文中，我们考虑稀疏相位检索问题，恢复一个s -稀疏信号${\mathit{X}}^{♮}\in {\mathbb{电阻}}^n$来自m 个无相样品${是}_{\mathrm{一世}}=|〈{\mathit{X}}^{♮},{\mathit{一种}}_{\mathrm{一世}}〉|$ 为了 $\mathrm{一世}=1,\dots ,米$. 现有的稀疏相位检索算法通常是一阶的，因此最多线性收敛。受压缩感知中的硬阈值追踪（HTP）算法的启发，我们提出了一种用于稀疏相位检索的高效二阶算法。我们提出的算法理论上保证在有限（特别是，最多$哦(\mathrm{日志}米+\mathrm{日志}({\Vert {\mathit{X}}^{♮}\Vert }_2/|X_{\mathrm{分钟}}^{♮}|))$ 步骤，当 ${\{{\mathit{一种}}_{\mathrm{一世}}\}}_{\mathrm{一世}=1}^{米}$ 是 iid 标准高斯随机向量 $米～哦(秒\mathrm{日志}(n/秒))$并且初始化在底层稀疏信号的邻域中。与频谱初始化一起，我们的算法保证从$哦({秒}^2\mathrm{日志}n)$样品。由于我们提出的算法每次迭代的计算成本与流行的一阶算法相同，因此我们的算法非常有效。实验结果表明，我们的算法可以比现有的稀疏相位检索算法快几倍。