Generalized Low-Rank Matrix Completion via Nonconvex Schatten $p$-Norm Minimization

Abstract

In this paper, we present a generalized low-rank matrix completion (LRMC) model for topological interference management (TIM), thereby maximizing the achievable degrees of freedom (DoFs) only based on the network connectivity information. Unfortunately, contemporary convex relaxation approaches, e.g, nuclear norm minimization, fail to return low-rank solutions, due to the poor structures in the generalized low-rank model. Most existing nonconvex approaches, however, often need the optimal rank as prior information, which is unavailable in our setting. We thus propose a novel nonconvex relaxation approach with the nonconvex Schatten $p$ -norm to provide a tight approximation for the rank function. A smooth function is formulated to approximate the nonsmooth and nonconvex objective, then an Iteratively Reweighted Least Squares (IRLS- $p$ ) method is employed to handle the nonconvexity of the model, which iteratively minimizes the weighted Frobenius norm models of smoothed subproblems while driving the smoothing parameter to 0. We further improve the efficiency by proposing an Iteratively Adaptively Reweighted Least Squares (IARLS- $p$ ) algorithm, which uses an adaptively updating strategy for the smoothing parameters in each iteration. Numerical results exhibit the ability of the proposed algorithm to find low-rank solutions, that is, it can achieve higher DoFs in most cases.