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December 2018 Robust low-rank matrix estimation
Andreas Elsener, Sara van de Geer
Ann. Statist. 46(6B): 3481-3509 (December 2018). DOI: 10.1214/17-AOS1666

Abstract

Many results have been proved for various nuclear norm penalized estimators of the uniform sampling matrix completion problem. However, most of these estimators are not robust: in most of the cases the quadratic loss function and its modifications are used. We consider robust nuclear norm penalized estimators using two well-known robust loss functions: the absolute value loss and the Huber loss. Under several conditions on the sparsity of the problem (i.e., the rank of the parameter matrix) and on the regularity of the risk function sharp and nonsharp oracle inequalities for these estimators are shown to hold with high probability. As a consequence, the asymptotic behavior of the estimators is derived. Similar error bounds are obtained under the assumption of weak sparsity, that is, the case where the matrix is assumed to be only approximately low-rank. In all of our results, we consider a high-dimensional setting. In this case, this means that we assume $n\leq pq$. Finally, various simulations confirm our theoretical results.

Citation

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Andreas Elsener. Sara van de Geer. "Robust low-rank matrix estimation." Ann. Statist. 46 (6B) 3481 - 3509, December 2018. https://doi.org/10.1214/17-AOS1666

Information

Received: 1 May 2016; Revised: 1 October 2017; Published: December 2018
First available in Project Euclid: 11 September 2018

zbMATH: 1412.62068
MathSciNet: MR3852659
Digital Object Identifier: 10.1214/17-AOS1666

Subjects:
Primary: 62F30, 62J05
Secondary: 62H12

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.46 • No. 6B • December 2018
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