PhaseLin: Linear Phase Retrieval

Abstract: Phase retrieval deals with the recovery of complex- or real-valued signals from magnitude measurements. As shown recently, the method PhaseMax enables phase retrieval via convex optimization and without lifting the problem to a higher dimension. To succeed, PhaseMax requires an initial guess of the solution, which can be calculated via spectral initializers. In this paper, we show that with the availability of an initial guess, phase retrieval can be carried out with an ever simpler, linear procedure. Our algorithm, called PhaseLin, is the linear estimator that minimizes the mean squared error (MSE) when applied to the magnitude measurements. The linear nature of PhaseLin enables an exact and nonasymptotic MSE analysis for arbitrary measurement matrices. We furthermore demonstrate that by iteratively using PhaseLin, one arrives at an efficient phase retrieval algorithm that performs on par with existing convex and nonconvex methods on synthetic and real-world data.