Phase retrieval for sub-Gaussian measurements

Abstract

Generally, the phase retrieval problem can be viewed as the reconstruction of a function/signal from only the magnitude of linear measurements. These measurements can be, for example, the Fourier transform of the density function. Computationally the phase retrieval problem is very challenging. Many algorithms for phase retrieval are based on i.i.d. Gaussian random measurements. However, Gaussian random measurements remain one of the very few classes of measurements. In this paper, we develop an efficient phase retrieval algorithm for sub-Gaussian random frames. We provide a general condition for measurements and develop a modified spectral initialization. In the algorithm, we first obtain a good approximation of the solution through the initialization, and from there we use Wirtinger Flow to solve for the solution. We prove that the algorithm converges to the global minimizer linearly.