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Estimation of high-dimensional low rank matrices
A. Rohde and A.B. Tsybakov
(2009)
arxiv paper.
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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