We consider the matrix completion problem where the aim is toestimate 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.
KLOPP, O. (2015). Matrix completion by singular value thresholding : sharp bounds. The Electronic Journal of Statistics, 9(2), pp. 2348-2369.