Burer-Monteiro factorizability of nuclear norm regularized optimization

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, denoted by $f$, and its factorized problem obtained by the Burer-Monteiro technique. We are interested in deriving conditions that ensure every second-order stationary point of the factorized problem corresponds to a global minimizer of $f$, a property we call the $r$-factorizability of $f$ in this paper. Under suitable restricted isometry property (RIP) type assumptions on $h$, we prove the $r$-factorizability of $f$. Moreover, the RIP constant in our paper is tight, in the sense that we can construct concrete examples of $f$ that fail to be $r$-factorizable when the RIP constant is below the threshold. Our technique for constructing such examples is novel and may be of independent interest: specifically, we use a variant of the Von Neumann's trace inequality and relate the existence of such examples to the optimal value of a quadratic program involving the RIP constant, then we explicitly solve this optimization problem to detect all the possible counterexamples.