Open Access
April 2011 Estimation of high-dimensional low-rank matrices
Angelika Rohde, Alexandre B. Tsybakov
Ann. Statist. 39(2): 887-930 (April 2011). DOI: 10.1214/10-AOS860


Suppose that we observe entries or, more generally, linear combinations of entries of an unknown m×T-matrix A corrupted by noise. We are particularly interested in the high-dimensional setting where the number mT of unknown entries can be much larger than the sample size N. Motivated by several applications, we consider estimation of matrix A under the assumption that it has small rank. This can be viewed as dimension reduction or sparsity assumption. In order to shrink toward a low-rank representation, we investigate penalized least squares estimators with a Schatten-p quasi-norm penalty term, p≤1. We study these estimators under two possible assumptions—a modified version of the restricted isometry condition and a uniform bound on the ratio “empirical norm induced by the sampling operator/Frobenius norm.” The main results are stated as nonasymptotic upper bounds on the prediction risk and on the Schatten-q risk of the estimators, where q∈[p, 2]. The rates that we obtain for the prediction risk are of the form rm/N (for m=T), up to logarithmic factors, where r is the rank of A. The particular examples of multi-task learning and matrix completion are worked out in detail. The proofs are based on tools from the theory of empirical processes. As a by-product, we derive bounds for the kth entropy numbers of the quasi-convex Schatten class embeddings SpM↪S2M, p<1, which are of independent interest.


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Angelika Rohde. Alexandre B. Tsybakov. "Estimation of high-dimensional low-rank matrices." Ann. Statist. 39 (2) 887 - 930, April 2011.


Published: April 2011
First available in Project Euclid: 9 March 2011

zbMATH: 1215.62056
MathSciNet: MR2816342
Digital Object Identifier: 10.1214/10-AOS860

Primary: 62F10 , 62G05

Keywords: empirical process , High-dimensional low-rank matrices , penalized least-squares estimator , quasi-convex Schatten class embeddings , Schatten norm , sparse recovery

Rights: Copyright © 2011 Institute of Mathematical Statistics

Vol.39 • No. 2 • April 2011
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