  We consider the matrix completion problem where the aim is to esti-mate a
large data matrix for which only a relatively small random subset of its
entries is observed. Quite popular approaches to matrix completion problem are
iterative thresholding methods. In spite of their empirical success, the
theoretical guarantees of such iterative thresholding methods are poorly
understood. The goal of this paper is to provide strong theo-retical
guarantees, similar to those obtained for nuclear-norm penalization methods and
one step thresholding methods, for an iterative thresholding algorithm which is
a modification of the softImpute algorithm. An im-portant consequence of our
result is the exact minimax optimal rates of convergence for matrix completion
problem which were known until know only up to a logarithmic factor.
