Burer-Monteiro factorizability of nuclear norm regularized optimization

Abstract: This paper studies the relationship between the nuclear norm-regularized minimization problem, which minimizes the sum of a $C^2$ function $h$ and a positive multiple of the nuclear norm, and its factorized problem obtained by the Burer-Monteiro technique. We first prove that every second-order stationary point of the factorized problem corresponds to an approximate stationary point of its non-factorized counterpart, and those rank-deficient ones correspond to global minimizers of the latter problem when $h$ is additionally convex, conforming with the observations in [2, 15]. Next, discarding the rank condition on the second-order stationary points but assuming the convexity and Lipschitz differentiability of $h$, we characterize, with respect to some natural problem parameters, when every second-order stationary point of the factorized problem is a global minimizer of the corresponding nuclear norm-regularized problem. More precisely, we subdivide the class of Lipschitz differentiable convex $C^2$ functions into subclasses according to those natural parameters and characterize when each subclass consists solely of functions $h$ such that every second-order stationary point of the associated factorized model is a global minimizer of the nuclear norm regularized model. In particular, explicit counterexamples are established when the characterizing condition on the said parameters is violated.