  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 L1 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.
