Non-Convex Matrix Completion and Related Problems via Strong Duality

This work studies the strong duality of non-convex matrix factorization problems: we show that under certain dual conditions, these problems and the dual have the same optimum. This has been well understood for convex optimization, but little was known for non-convex problems. We propose a novel analytical framework and prove that under certain dual conditions, the optimal solution of the matrix factorization program is the same as that of its bi-dual and thus the global optimality of the non-convex program can be achieved by solving its bi-dual which is convex. These dual conditions are satisfied by a wide class of matrix factorization problems, although matrix factorization is hard to solve in full generality. This analytical framework may be of independent interest to non-convex optimization more broadly. We apply our framework to two prototypical matrix factorization problems: matrix completion and robust Principal Component Analysis. These are examples of efficiently recovering a hidden matrix given limited reliable observations. Our framework shows that exact recoverability and strong duality hold with nearly-optimal sample complexity for the two problems.