On DC Based Methods for Phase Retrieval

Abstract

In this paper, we develop a new computational approach which is based on minimizing the difference of two convex functions (DC) to solve a broader class of phase retrieval problems. The approach splits a standard nonlinear least squares minimizing function associated with the phase retrieval problem into the difference of two convex functions and then solves a sequence of convex minimization subproblems. For each subproblem, the Nesterov accelerated gradient descent algorithm or the Barzilai-Borwein (BB) algorithm is adopted. In addition, we apply the alternating projection method to improve the initial guess in [20] and make it much more closer to the true solution. In the setting of sparse phase retrieval, a standard ℓ 1 norm term is added to guarantee the sparsity, and the subproblem is solved approximately by a proximal gradient method with the shrinkage-threshold technique directly. Furthermore, a modified Attouch-Peypouquet technique is used to accelerate the iterative computation, which leads to more effective algorithms than the Wirtinger flow (WF) algorithm and the Gauss-Newton (GN) algorithm and etc. Indeed, DC based algorithms are able to recover the solution with high probability when the measurement number m ≈ 2n in the real case and m ≈ 3n in the complex case, where n is the dimension of the true solution. When m ≈ n, the ℓ 1-DC based algorithm is able to recover the sparse signals with high probability. Our main results show that the DC based methods converge to a critical point linearly. Our study is a deterministic analysis while the study for the Wirtinger flow (WF) algorithm and its variants, the Gauss-Newton (GN) algorithm, the trust region algorithm is based on the probability analysis. Finally, the paper discusses the existence and the number of distinct solutions for phase retrieval problem.