A Singular Value Thresholding Algorithm for Matrix Completion

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A Singular Value Thresholding Algorithm for Matrix Completion (2008)

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by Jian-Feng Cai , Emmanuel J. Candès , Zuowei Shen

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555 - 22 self

BibTeX

@MISC{Cai08asingular,
    author = {Jian-Feng Cai and Emmanuel J. Candès and Zuowei Shen},
    title = {A Singular Value Thresholding Algorithm for Matrix Completion },
    year = {2008}
}

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Abstract

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 {X k, 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 that the soft-thresholding operation is applied to a sparse matrix; the second is that the rank of the iterates {X k} is empirically nondecreasing. Both these facts allow the algorithm to make use of very minimal storage space and keep the computational cost of each iteration low. On

Keyphrases

matrix completion singular value thresholding algorithm soft-thresholding operation many important application minimal storage space unknown entry interior point method remarkable feature large matrix large problem rank minimization problem famous netflix problem optimal solution novel algorithm convex relaxation minimum nuclear norm simple first-order computational cost singular value small subset off-the-shelf algorithm low rank low-rank matrix completion problem convex constraint easy-to-implement algorithm sparse matrix

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