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 recov...

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Bibliographic Details
Published inSIAM journal on optimization Vol. 20; no. 4; pp. 1956 - 1982
Main Authors Cai, Jian-Feng, Candès, Emmanuel J., Shen, Zuowei
Format Journal Article
LanguageEnglish
Published Philadelphia Society for Industrial and Applied Mathematics 01.01.2010
Subjects
Algorithms
Approximation
Convergence
Convex analysis
Mathematical analysis
Matrices
Matrix
Matrix methods
Norms
Optimization
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Summary: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). The algorithm is iterative, produces a sequence of matrices ..., and at each step mainly performs a soft-thresholding operation on the singular values of the matrix ... On the theoretical side, the paper provides a convergence analysis showing that the sequence of iterates converges. On the practical side, it provides numerical examples in which 1,000 x 1,000 matrices are recovered in less than a minute on a modest desktop computer. (ProQuest: ... denotes formulae/symbols omitted.)
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ISSN:1052-6234
1095-7189
DOI:10.1137/080738970