Phase Retrieval via Difference of Convex Programming

In this paper, we consider the convolutional phase retrieval problem, which is a crucial and challenging problem in signal processing and wireless communication. Our goal is to recover a signal from phaseless measurements. To address the challenge of phaseless measurements, we recast the recovery problem into a low-rank matrix optimization problem via matrix lifting. To further exactly detect the feasibility of resulting fixed-rank constraint, we propose a novel difference of convex functions (DC) representation for the rank function by exploiting the difference between trace norm and spectral norm, followed by presenting an efficient algorithm to solve the resulting DC programming. The simulation results demonstrate that our proposed algorithm outperforms the existing convex relaxation methods in terms of signal recovery sample complexity and the noise robustness.