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October 2011 Nuclear-norm penalization and optimal rates for noisy low-rank matrix completion
Vladimir Koltchinskii, Karim Lounici, Alexandre B. Tsybakov
Ann. Statist. 39(5): 2302-2329 (October 2011). DOI: 10.1214/11-AOS894

Abstract

This paper deals with the trace regression model where n entries or linear combinations of entries of an unknown m1 × m2 matrix A0 corrupted by noise are observed. We propose a new nuclear-norm penalized estimator of A0 and establish a general sharp oracle inequality for this estimator for arbitrary values of n, m1, m2 under the condition of isometry in expectation. Then this method is applied to the matrix completion problem. In this case, the estimator admits a simple explicit form, and we prove that it satisfies oracle inequalities with faster rates of convergence than in the previous works. They are valid, in particular, in the high-dimensional setting m1m2n. We show that the obtained rates are optimal up to logarithmic factors in a minimax sense and also derive, for any fixed matrix A0, a nonminimax lower bound on the rate of convergence of our estimator, which coincides with the upper bound up to a constant factor. Finally, we show that our procedure provides an exact recovery of the rank of A0 with probability close to 1. We also discuss the statistical learning setting where there is no underlying model determined by A0, and the aim is to find the best trace regression model approximating the data. As a by-product, we show that, under the restricted eigenvalue condition, the usual vector Lasso estimator satisfies a sharp oracle inequality (i.e., an oracle inequality with leading constant 1).

Citation

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Vladimir Koltchinskii. Karim Lounici. Alexandre B. Tsybakov. "Nuclear-norm penalization and optimal rates for noisy low-rank matrix completion." Ann. Statist. 39 (5) 2302 - 2329, October 2011. https://doi.org/10.1214/11-AOS894

Information

Published: October 2011
First available in Project Euclid: 30 November 2011

zbMATH: 1231.62097
MathSciNet: MR2906869
Digital Object Identifier: 10.1214/11-AOS894

Subjects:
Primary: 62H12, 62J99
Secondary: 60B20, 60G15

Rights: Copyright © 2011 Institute of Mathematical Statistics

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Vol.39 • No. 5 • October 2011
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