A DC Composite Optimization via Variable Smoothing for Robust Phase Retrieval with Nonconvex Loss Functions

Abstract: In this paper, we propose an optimization-based method for robust phase retrieval problem where the goal is to estimate an unknown signal from a quadratic measurement corrupted by outliers. To enhance the robustness of existing optimization models with the $\ell_1$ loss function, we propose a generalized model that can handle DC (Difference-of-Convex) loss functions beyond the $\ell_1$ loss. We view the cost function of the proposed model as a composition of a DC function with a smooth mapping, and develop a variable smoothing algorithm for minimizing such DC composite functions. At each step of our algorithm, we generate a smooth surrogate function by using the Moreau envelope of each (weakly) convex function in the DC function, and then perform the gradient descent update of the surrogate function. Unlike many existing algorithms for DC problems, the proposed algorithm does not require any inner loop. We also present a convergence analysis in terms of a DC composite critical point for the proposed algorithm. Our numerical experiment demonstrates that the proposed method with DC loss functions is more robust against outliers compared to existing methods with the $\ell_1$ loss.