Annals of Statistics

Estimation of high-dimensional low-rank matrices

Angelika Rohde and Alexandre B. Tsybakov

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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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Ann. Statist., Volume 39, Number 2 (2011), 887-930.

First available in Project Euclid: 9 March 2011

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Zentralblatt MATH identifier

Primary: 62G05: Estimation 62F10: Point estimation

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


Rohde, Angelika; Tsybakov, Alexandre B. Estimation of high-dimensional low-rank matrices. Ann. Statist. 39 (2011), no. 2, 887--930. doi:10.1214/10-AOS860.

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