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 $\mathcal{O}\big(\log (\Vert\boldsymbol{x}^{\natural} \Vert /x_{\min}^{\natural})\big)$ iterations, provided a sample complexity of $\mathcal{O}(s^2\log n)$, where $\boldsymbol{x}^{\natural} \in \mathbb{R}^n$ represents an $s$-sparse ground truth signal. Numerical experiments demonstrate that our algorithm outperforms state-of-the-art methods in terms of achieving a significantly faster convergence rate.
A Fast and Provable Algorithm for Sparse Phase Retrieval
Jian-Feng CAI Yu Long Ruixue WEN Jiaxi Ying