Matrix completion via max-norm constrained optimization

This paper studies matrix completion under a general sampling model using the max-norm as a convex relaxation for the rank of the matrix. The optimal rate of convergence is established for the Frobenius norm loss. It is shown that the max-norm constrained minimization method is rate-optimal and it yields a more stable approximate recovery guarantee, with respect to the sampling distributions, than previously used trace-norm based approaches. The computational effectiveness of this method is also studied, based on a first-order algorithm for solving convex programs involving a max-norm constraint.