Optimal sparse phase retrieval via a quasi-Bayesian approach

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.