## AbstractSuppose that we observe entries or, more generally, linear combi- nations 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 sam- ple 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 towards 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 opera- tor/Frobenius norm”. The main results are stated as non-asymptotic upper bounds on the prediction risk and on the Schatten-q risk of the estimators, where q belongs to [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 which are of independent interest.
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