Optimal Tuning-Free Convex Relaxation for Noisy Matrix Completion

This paper is concerned with noisy matrix completion—the problem of recovering a low-rank matrix from partial and noisy entries. Under uniform sampling and incoherence assumptions, we prove that a tuning-free square-root matrix completion estimator (
${\mathtt {square{-}root~ MC}}$
) achieves optimal statistical performance for solving the noisy matrix completion problem. Similar to the square-root Lasso estimator in high-dimensional linear regression,
${\mathtt {square{-}root~ MC}}$
does not rely on the knowledge of the size of the noise. While solving
${\mathtt {square{-}root~ MC}}$
is a convex program, our statistical analysis of
${\mathtt {square{-}root~ MC}}$
hinges on its intimate connections to a nonconvex rank-constrained estimator.