GLOBAL OPTIMALITY IN LOW-RANK MATRIX OPTIMIZATION

This paper considers the minimization of a general objective function f(X) over the set of non-square n x m matrices where the optimal solution X ? is low-rank. To reduce the computational burden, we factorize the variable X into a product of two smaller matrices and optimize over these two matrices instead of X. We analyze the global geometry for a general and yet well-conditioned objective function f(X) whose restricted strong convexity and restricted strong smoothness constants are comparable. In particular, we show that the reformulated objective function has no spurious local minima and obeys the strict saddle property. These geometric properties imply that a number of iterative optimization algorithms (such as gradient descent) can provably solve the factored problem with global convergence.