ESTIMATION OF HIGH - DIMENSIONAL LOW - RANK MATRICES 1 By Angelika Rohde and Alexandre

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 numbermT 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 multitask 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 S p →֒ S 2 , p < 1, which are of independent interest.