  The matrix completion problem consists in reconstructing a matrix from a
sample of entries, possibly observed with noise. A popular class of estimator,
known as nuclear norm penalized estimators, are based on minimizing the sum of
a data fitting term and a nuclear norm penalization. Here, we investigate the
case where the noise distribution belongs to the exponential family and is
sub-exponential. Our framework alllows for a general sampling scheme. We first
consider an estimator defined as the minimizer of the sum of a log-likelihood
term and a nuclear norm penalization and prove an upper bound on the Frobenius
prediction risk. The rate obtained improves on previous works on matrix
completion for exponential family. When the sampling distribution is known, we
propose another estimator and prove an oracle inequality w.r.t. the
Kullback-Leibler prediction risk, which translates immediatly into an upper
bound on the Frobenius prediction risk. Finally, we show that all the rates
obtained are minimax optimal up to a logarithmic factor.
