Effective phase retrieval of sparse signals with convergence guarantee

Phase retrieval, as a representative nonlinear inverse problem, is of increasing interest recently for its broad applications in imaging and engineering. Prediction of the signal amounts to solving a nonconvex optimization problem, which is generally NP-hard to solve. Collecting more measurements or providing prior information of the unknown are two often-seen strategies to facilitate the problem. In this paper, we propose an efficient and effective algorithm to solve phase retrieval with certain prior of the signal, in particular the signal itself is sparse in the natural basis. By formulating phase retrieval (PR) problem in the splitting form, we propose ADMM (alternating direction method of multipliers) with convergence guarantee to tackle the resulting nonconvex problem. Even though the algorithm is not new and there also exist several works on the convergence of ADMM, we clarify that our formulation for phase retrieval is not a specific application example of their investigated models. Hence, we investigate the convergence of our algorithm and show that ADMM converges a stationary point when penalty parameter ρ is large enough. It is observed that the recovery performance degrades when the penalty parameter increases. For better performance, we propose a practical scheme of tuning the penalty parameter ρ. Demonstration of the superior recovery performance on sparse phase retrieval (SPR) is conducted and it shows that our method numerically infers near-exact solution without providing good initialization. Our proposed method distinguishes itself from other existing competitive algorithms in two aspects: (a) the initialization is much less crucial for the algorithmic success than other compared methods; (b) the sampling complexity for the phase transition of recovery is much markedly reduced than other existing methods.