  Many results have been proved for various nuclear norm penalized estimators
of the uniform sampling matrix completion problem. However, most of these
estimators are not robust: in most of the cases the quadratic loss function and
its modifications are used. We consider robust nuclear norm penalized
estimators using two well-known robust loss functions: the absolute value loss
and the Huber loss. Under several conditions on the sparsity of the problem
(i.e. the rank of the parameter matrix) and on the regularity of the risk
function sharp and non-sharp oracle inequalities for these estimators are shown
to hold with high probability. As a consequence, the asymptotic behavior of the
estimators is derived. Similar error bounds are obtained under the assumption
of weak sparsity, i.e. the case where the matrix is assumed to be only
approximately low-rank. In all our results we consider a high-dimensional
setting. In this case, this means that we assume $n\leq pq$. Finally, various
simulations confirm our theoretical results.
