  In the present paper, we consider the problem of matrix completion with
noise. Unlike previous works, we consider quite general sampling distribution
and we do not need to know or to estimate the variance of the noise. Two new
nuclear-norm penalized estimators are proposed, one of them of "square-root"
type. We analyse their performance under high-dimensional scaling and provide
non-asymptotic bounds on the Frobenius norm error. Up to a logarithmic factor,
these performance guarantees are minimax optimal in a number of circumstances.
