Matrix completion by singular value thresholding: Sharp bounds

We consider the matrix completion problem where the aim is to esti-mate a large data matrix for which only a relatively small random subset of its entries is observed. Quite popular approaches to matrix completion problem are iterative thresholding methods. In spite of their empirical success, the t...

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Veröffentlicht in:Electronic journal of statistics Jg. 9; H. 2; S. 2348 - 2369
1. Verfasser: Klopp, Olga
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Shaker Heights, OH : Institute of Mathematical Statistics 01.01.2015
Schlagworte:
Mathematics
Statistics
low rank matrix estimation
60B20
15A83
matrix completion
62H12
minimax opti-mality AMS 2000 subject classification: 62J99
ISSN:1935-7524, 1935-7524
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Zusammenfassung:We consider the matrix completion problem where the aim is to esti-mate a large data matrix for which only a relatively small random subset of its entries is observed. Quite popular approaches to matrix completion problem are iterative thresholding methods. In spite of their empirical success, the theoretical guarantees of such iterative thresholding methods are poorly understood. The goal of this paper is to provide strong theo-retical guarantees, similar to those obtained for nuclear-norm penalization methods and one step thresholding methods, for an iterative thresholding algorithm which is a modification of the softImpute algorithm. An im-portant consequence of our result is the exact minimax optimal rates of convergence for matrix completion problem which were known until know only up to a logarithmic factor.
ISSN:1935-7524
1935-7524
DOI:10.1214/15-EJS1076