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 (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, square-root MC does not rely on the knowledge of the size of the noise.While solving square-root MC is a convex program, our statistical analysis of square-root MC hinges on its intimate connections to a nonconvex rank-constrained estimator.