Phase retrieval with background information

Abstract Phase retrieval, which is actually an inverse problem without standard uniqueness guarantee, has been the subject of intense algorithmic development in various aspects. Under different settings, it is necessary to design different efficient models and solvers, and, of course, provide the theoretical supports. In this paper, we consider the scenario that background information of the signal is available; and then, a model called phase retrieval with background information is proposed. With sufficient prior background information, we construct a loss function and apply the projected gradient descent method to search the ground truth. Theoretically, we first provide the uniqueness guarantee for the proposed model, and then prove that each stationary point admits the global optimum with probability 1. Next we present the convergence result of the corresponding projected gradient descent. Numerical simulations on 1D and 2D signals demonstrate the efficiency and robustness of the proposed model and algorithm.