  This work studies the strong duality of non-convex matrix factorization
problems: we show that under certain dual conditions, these problems and its
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 show that under certain dual conditions, the optimal
solution of the matrix factorization program is the same as 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 problems
are 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 (PCA). These are
examples of efficiently recovering a hidden matrix given limited reliable
observations of it. Our framework shows that exact recoverability and strong
duality hold with nearly-optimal sample complexity guarantees for matrix
completion and robust PCA.
