Extended successive convex approximation for phase retrieval with dictionary learning

Phase retrieval aims at recovering unknown signals from magnitude measurements of linear mixtures. In this paper, we consider the phase retrieval with dictionary learning problem, which includes another prior information that the signal admits a sparse representation over an unknown dictionary. The task is to jointly estimate the dictionary and the sparse representation from magnitude-only measurements. To this end, we study two complementary formulations and develop efficient parallel algorithms by extending the successive convex approximation framework using a smooth majorization. The first algorithm is termed compact-SCAphase and is preferable in the case of moderately diverse mixture models with a low number of mixing components. It adopts a compact formulation that avoids auxiliary variables. The proposed algorithm is highly scalable and has reduced parameter tuning cost. The second algorithm, referred to as SCAphase, uses auxiliary variables and is favorable in the case of highly diverse mixture models. It also renders simple incorporation of additional side constraints. The performance of both methods is evaluated when applied to blind channel estimation from subband magnitude measurements in a multi-antenna random access network. Simulation results show the efficiency of the proposed techniques compared to state-of-the-art methods.