用贝叶斯方法解决稀疏信号相位恢复，理论最优且计算高效。
Optimal sparse phase retrieval via a quasi-Bayesian approach
- 用学生分布做连续收缩先验，实现稀疏性约束。
- 在亚指数噪声下达到最优收敛速度，与主流频率方法相当。
- 适合需要理论保障的高噪声稀疏信号恢复场景。
本文研究稀疏相位恢复这一基础逆问题，目标是在仅有变换幅度、相位信息不可得的情况下重构信号。利用真实信号的固有稀疏性，提出一种新型稀疏拟贝叶斯方法，并首次提供该方法的理论保证。具体地，采用缩放的学生分布作为连续收缩先验以强化稀疏性，并基于PAC-贝叶斯不等式框架进行分析。结果表明，所提出的贝叶斯估计器在亚指数噪声下达到极小极大最优收敛速率，与当前最先进的频率学派方法一致。为保证计算可行性，设计了高效的Langevin蒙特卡洛采样算法。数值实验显示，该方法性能可与现有频率学派技术相媲美，展现出在噪声环境下作为原理性替代方案的巨大潜力。
原文摘要 · Abstract (English)
This paper addresses the problem of sparse phase retrieval, a fundamental inverse problem in applied mathematics, physics, and engineering, where a signal need to be reconstructed using only the magnitude of its transformation while phase information remains inaccessible. Leveraging the inherent sparsity of many real-world signals, we introduce a novel sparse quasi-Bayesian approach and provide the first theoretical guarantees for such an approach. Specifically, we employ a scaled Student distribution as a continuous shrinkage prior to enforce sparsity and analyze the method using the PAC-Bayesian inequality framework. Our results establish that the proposed Bayesian estimator achieves minimax-optimal convergence rates under sub-exponential noise, matching those of state-of-the-art frequentist methods. To ensure computational feasibility, we develop an efficient Langevin Monte Carlo sampling algorithm. Through numerical experiments, we demonstrate that our method performs comparably to existing frequentist techniques, highlighting its potential as a principled alternative for sparse phase retrieval in noisy settings.
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