  This paper deals with the trace regression model where $n$ entries or linear
combinations of entries of an unknown $m_1\times m_2$ matrix $A_0$ corrupted by
noise are observed. We propose a new nuclear norm penalized estimator of $A_0$
and establish a general sharp oracle inequality for this estimator for
arbitrary values of $n,m_1,m_2$ under the condition of isometry in expectation.
Then this method is applied to the matrix completion problem. In this case, the
estimator admits a simple explicit form and we prove that it satisfies oracle
inequalities with faster rates of convergence than in the previous works. They
are valid, in particular, in the high-dimensional setting $m_1m_2\gg n$. We
show that the obtained rates are optimal up to logarithmic factors in a minimax
sense and also derive, for any fixed matrix $A_0$, a non-minimax lower bound on
the rate of convergence of our estimator, which coincides with the upper bound
up to a constant factor. Finally, we show that our procedure provides an exact
recovery of the rank of $A_0$ with probability close to 1. We also discuss the
statistical learning setting where there is no underlying model determined by
$A_0$ and the aim is to find the best trace regression model approximating the
data.
