📝 SummarXiv

Abstract

Sparse phase retrieval, which aims to recover a $k$-sparse signal from $m$ phaseless measurements, poses a fundamental question regarding the minimal sample complexity required for success. While the optimal sample complexity is known to be $\Omega(k \log n)$, existing algorithms can only achieve it for signals under restrictive structural assumptions. This paper introduces a novel and robust initialization algorithm, termed \ac{gESP}, designed to overcome this limitation. Theoretically, we prove that gESP significantly expands the family of signals that can be recovered with the optimal sample complexity. Our analysis unifies existing results on previously studied signal models and surpasses them by establishing performance bounds for a more general class of signals. Extensive simulations validate our theoretical findings, demonstrating that gESP consistently outperforms state-of-the-art methods across diverse signal types, thereby pushing the boundaries of efficient and optimal sparse phase retrieval.