Iterative rank-one matrix completion via singular value decomposition and nuclear norm regularization

Abstract

Matrix completion is widely used in many fields. In existing matrix completion methods, such as rank minimization and matrix factorization, the hyperparameters must be learned. However, hyperparameter tuning is a time-consuming and tedious process. In this paper, we propose a novel matrix completion method called IMC, i.e., iterative rank-one matrix completion via singular value decomposition (SVD) and nuclear norm regularization. First, we construct a rank-one matrix completion model using nuclear norm regularization. Then, the variables to be optimized in the model are divided into several blocks. Finally, the blocks are iteratively optimized one by one until convergence. For the optimization of each block, we propose an efficient solution scheme based on SVD, in which only the maximum singular value and leading singular vectors of a sparse matrix must be calculated. Further, a nonparametric singular value penalty function is designed to ensure a low-rank completion matrix. In addition, the optimization of each block uses only the values inside the observed entries; hence, no errors are accumulated. Test results shows that the proposed method converges rapidly and outperforms some state-of-the-art methods when applied to grayscale image restoration, recommendation systems, and vote networks.