We study the sparse phase retrieval problem, which aims to recover a sparse signal from a limited number of phaseless measurements. Existing algorithms for sparse phase retrieval primarily rely on first-order methods with linear convergence rate. In this paper, we propose an efficient second-order algorithm based on Newton projection, which maintains the same per-iteration computational complexity as popular first-order methods. The proposed algorithm is theoretically guaranteed to converge to the ground truth (up to a global sign) at a quadratic convergence rate after at most $O\big(\log (\Vert \mathbf{x}^{\natural} \, \Vert /x_{\min}^{\natural})\big)$ iterations, provided a sample complexity of $O(s^2\log n)$, where $\mathbf{x}^{\natural} \in \mathbb{R}^n$ represents an $s$-sparse ground truth signal. Numerical experiments demonstrate that our algorithm not only outperforms state-of-the-art methods in terms of achieving a significantly faster convergence rate, but also excels in attaining a higher success rate for exact signal recovery from noise-free measurements and providing enhanced signal reconstruction in noisy scenarios.

翻译：我们研究稀疏相位恢复问题，其目标是从有限数量的无相位测量中恢复稀疏信号。现有的稀疏相位恢复算法主要依赖具有线性收敛速度的一阶方法。本文提出了一种基于牛顿投影的高效二阶算法，该算法在每次迭代中保持与流行一阶方法相同的计算复杂度。所提算法理论上保证在最多 $O\big(\log (\Vert \mathbf{x}^{\natural} \, \Vert /x_{\min}^{\natural})\big)$ 次迭代后以二次收敛速度收敛到真实信号（最多差一个全局符号），前提是样本复杂度为 $O(s^2\log n)$，其中 $\mathbf{x}^{\natural} \in \mathbb{R}^n$ 表示一个 $s$-稀疏的真实信号。数值实验表明，我们的算法不仅在实现显著更快的收敛速度方面优于当前最优方法，而且在无噪声测量中实现精确信号恢复的成功率更高，以及在噪声场景中提供更强的信号重建性能。