We study the sparse phase retrieval problem, recovering an $s$-sparse length-$n$ signal from $m$ magnitude-only measurements. Two-stage non-convex approaches have drawn much attention in recent studies for this problem. Despite non-convexity, many two-stage algorithms provably converge to the underlying solution linearly when appropriately initialized. However, in terms of sample complexity, the bottleneck of those algorithms often comes from the initialization stage. Although the refinement stage usually needs only $m=\Omega(s\log n)$ measurements, the widely used spectral initialization in the initialization stage requires $m=\Omega(s^2\log n)$ measurements to produce a desired initial guess, which causes the total sample complexity order-wisely more than necessary. To reduce the number of measurements, we propose a truncated power method to replace the spectral initialization for non-convex sparse phase retrieval algorithms. We prove that $m=\Omega(\bar{s} s\log n)$ measurements, where $\bar{s}$ is the stable sparsity of the underlying signal, are sufficient to produce a desired initial guess. When the underlying signal contains only very few significant components, the sample complexity of the proposed algorithm is $m=\Omega(s\log n)$ and optimal. Numerical experiments illustrate that the proposed method is more sample-efficient than state-of-the-art algorithms.

翻译：我们研究稀疏相位恢复问题，即从$m$个仅含幅度测量值中恢复一个$s$-稀疏、长度为$n$的信号。两阶段非凸方法近年来在该问题中备受关注。尽管存在非凸性，许多两阶段算法在适当初始化后能线性收敛到潜在解。然而，在样本复杂度方面，这些算法的瓶颈通常来自初始化阶段。尽管精化阶段通常仅需$m=\Omega(s\log n)$个测量值，但初始化阶段广泛使用的谱初始化方法需要$m=\Omega(s^2\log n)$个测量值才能产生理想的初始估计，这导致总样本复杂度在阶数上高于必要水平。为减少测量数量，我们提出一种截断幂方法替代非凸稀疏相位恢复算法中的谱初始化。我们证明：当$m=\Omega(\bar{s} s\log n)$个测量值（其中$\bar{s}$为潜在信号的稳定稀疏度）时，足以产生理想的初始估计。当潜在信号仅包含极少数显著分量时，所提算法的样本复杂度为$m=\Omega(s\log n)$且达到最优。数值实验表明，与最先进算法相比，所提方法具有更高的样本效率。