Sparse phase retrieval aims to recover a $k$-sparse signal from $m$ phaseless measurements. While the theoretically optimal sample complexity for successful recovery is $Ω(k \log n)$, existing algorithms can only achieve this bound for signals with specific structural assumptions, leading to a notable gap between theory and practice. To bridge this gap, we introduce an efficient initialization algorithm, termed generalized Exponential Spectral Pursuit (gESP). We prove that gESP can significantly expand the family of signals that are guaranteed to be recovered with the optimal sample complexity, thereby extending the scope of theoretical optimality to a much broader class of signals. Extensive simulations validate our theoretical findings and demonstrate that gESP consistently outperforms the state-of-the-art methods across diverse signal types.

翻译：稀疏相位检索旨在从 $m$ 个无相位测量中恢复一个 $k$-稀疏信号。虽然理论上成功恢复的最优样本复杂度为 $\Omega(k \log n)$，但现有算法仅能在特定结构假设下实现该界，导致理论与实际之间存在显著差距。为弥合这一差距，我们提出了一种高效的初始化算法，称为广义指数谱追踪（gESP）。我们证明，gESP 能够显著扩展保证以最优样本复杂度恢复的信号族，从而将理论最优性范围拓展至更广泛的信号类型。大量仿真验证了我们的理论发现，并表明 gESP 在各类信号上均持续优于现有最优方法。