A Singular Value Thresholding Algorithm for Matrix Completion

This paper introduces a novel algorithm to approximate the matrix with minimum nuclear norm among all matrices obeying a set of convex constraints. This problem may be understood as the convex relaxation of a rank minimization problem, and arises in many important applications as in the task of recovering a large matrix from a small subset of its entries (the famous Netflix problem). Off-the-shelf algorithms such as interior point methods are not directly amenable to large problems of this kind with over a million unknown entries. This paper develops a simple first-order and easy-to-implement algorithm that is extremely efficient at addressing problems in which the optimal solution has low rank. The algorithm is iterative and produces a sequence of matrices {Xk, Y k} and at each step, mainly performs a soft-thresholding operation on the singular values of the matrix Y k. There are two remarkable features making this attractive for low-rank matrix completion problems. The first is t...

• Type: Journal
• Year: 2010
• Where: SIAMJO
• Authors: Jian-Feng Cai, Emmanuel J. Candès, Zuowei Shen
• Keywords: Algorithms, Matrix, Nuclear Norm, SIAMJO 2010