Estimation of high-dimensional low-rank matrices

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 quasinorm 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 ε [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 S^M_p →?S^M₂ ,p <1, which are of independent interest.