A Singular Value Thresholding Algorithm for Matrix Completion

13 years 2 months ago

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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...

Jian-Feng Cai, Emmanuel J. Candès, Zuowei S

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