PDMM: A Novel Primal-Dual Majorization-Minimization Algorithm for Poisson Phase-Retrieval Problem

In this paper, we introduce a novel iterative algorithm for the problem of phase-retrieval where the measurements consist of only the magnitude of linear function of the unknown signal, and the noise in the measurements follow Poisson distribution. The proposed algorithm is based on the principle of majorization-minimization (MM); however, the application of MM here is very novel and distinct from the way MM has been usually used to solve optimization problems in the literature. More precisely, we reformulate the original minimization problem into a saddle point problem by invoking Fenchel dual representation of the log(.) term in the Poisson likelihood function. We then propose tighter surrogate functions over both primal and dual variables resulting in a double-loop MM algorithm, which we have named as Primal-Dual Majorization-Minimization (PDMM) algorithm. The iterative steps of the resulting algorithm are simple to implement and involve only computing matrix vector products. We also extend our algorithm to handle various l(1) regularized Poisson phase-retrieval problems (which exploit sparsity). The proposed algorithm is compared with previously proposed algorithms such as wirtinger flow (WF), MM (conventional), and alternating direction methods of multipliers (ADMM) for the Poisson data model. The simulation results under different experimental settings show that PDMM is faster than the competing methods, and its performance in recovering the original signal is at par with the state-of-the-art algorithms.

Automated summary

In this paper, a novel iterative algorithm for phase-retrieval problems is introduced, which deals with measurements consisting of only the magnitude of a linear function of the unknown signal and follows Poisson distribution. The proposed algorithm is based on the majorization-minimization (MM) principle, but with a novel application that differs from traditional optimization problem solving methods. The algorithm reformulates the original minimization problem into a saddle point problem by utilizing the Fenchel dual representation. It then proposes tighter surrogate functions and creates a double-loop MM algorithm called Primal-Dual Majorization-Minimization (PDMM). The simulation results show that PDMM is faster than competing methods and achieves similar performance in signal recovery compared to state-of-the-art algorithms.