Matrix completion by singular value thresholding: Sharp bounds

We consider the matrix completion problem where the aim is to estimate 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 theoretical 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 important consequence of our result is the exact minimax optimal rates of convergence for matrix completion problem which were know until now only up to a logarithmic factor.