Matrix Completion via Non-convex Programming

We consider the matrix completion problem in the noisy setting. To achieve statistically efficient estimation of the unknown low-rank matrix, solving convex optimization problems with nuclear norm constraints has been both theoretically and empirically proved a successful strategy under certain regularity conditions. However, the bias induced by the nuclear norm penalty may compromise the estimation accuracy. To address this problem, following a parallel line of research in sparse regression models, we study the performance of a family of non-convex regularizers in the matrix completion problem. In particular, a fast first-order algorithm is proposed to solve the non-convex programming problem. We also describe a degree-of-freedom based reparametrization to “refine” the search along the solution path. Numerical experiments show that these non-convex methods outperform the traditional use of nuclear norm regularization in both simulated and real data sets.