A Proximal Operator for Multispectral Phase Retrieval Problems

Abstract

Proximal algorithms have gained popularity in recent years in large-scale and distributed optimization problems. One such problem is the phase retrieval problem, for which proximal operators have been proposed recently. The phase retrieval problem commonly refers to the task of recovering a target signal based on the magnitude of linear projections of that signal onto known vectors, usually under the presence of noise. A more general problem is the multispectral phase retrieval problem, where sums of these magnitudes are observed instead. In this paper we study the proximal operator for this problem, which appears in applications like X-ray solution scattering. We show that despite its non-convexity, all local minimizers are global minimizers, guaranteeing the optimality of simple descent techniques. An efficient linear time exact Newton method is proposed based on the structure of the problem's Hessian. Initialization criteria are discussed and the computational performance of the proposed algorithm is compared to that of traditional descent methods. The studied proximal operator can be used in a distributed and parallel scenarios using an ADMM scheme and allows for exploiting the spectral characteristics of the problem's measurement matrices, known in many physical sensing applications, in a way that is not possible with non-splitted optimization algorithms. The dependency of the proximal operator on the rank of these matrices, instead of their dimension, can greatly reduce the memory and computation requirements for problems of moderate to large size (N>10000) when these measurement matrices admit a low-rank representation.