The Global Optimization Geometry of Low-Rank Matrix Optimization

In this paper we characterize the optimization geometry of a matrix factorization problem where we aim to find n×r and m×r matrices U and V such that UV T approximates a given matrixX. We show that the objective function of the matrix factorization problem has no spurious local minima and obeys the strict saddle property not only for the exact-parameterization case where rank(X) = r, but also for the over-parameterization case where rank(X) < r and under-parameterization case where rank(X) > r. These geometric properties imply that a number of iterative optimization algorithms (such as gradient descent) converge to a global solution with random initialization. For the exact-parameterization case, we further show that the objective function satisfies the robust strict saddle property, ensuring global convergence of many local search algorithms in polynomial time. We extend the geometric analysis to the matrix sensing problem with the factorization approach and prove that this global optimization geometry is preserved as long as the measurement operator satisfies the standard restricted isometry property.