Diego Cifuentes

Diego Cifuentes

Given an affine space of matrices $\mathcal{L}$ and a matrix $Θ\in \mathcal{L}$, consider the problem of computing the closest rank deficient matrix to $Θ$ on $\mathcal{L}$ with respect to the Frobenius norm. This is a nonconvex problem with several applications in control theory, computer algebra, and computer vision. We introduce a novel semidefinite programming (SDP) relaxation, and prove that it always gives the global minimizer of the nonconvex problem in the low noise regime, i.e., when $Θ$ is close to be rank deficient. Our SDP is the first convex relaxation for this problem with provable guarantees. We evaluate the performance of our SDP relaxation in examples from system identification, approximate GCD, triangulation, and camera resectioning. Our relaxation reliably obtains the global minimizer under non-adversarial noise, and its noise tolerance is significantly better than state of the art methods.

Diego Cifuentes, Ankur Moitra

The Burer-Monteiro method is one of the most widely used techniques for solving large-scale semidefinite programs (SDP). The basic idea is to solve a nonconvex program in $Y$, where $Y$ is an $n \times p$ matrix such that $X = Y Y^T$. In this paper, we show that this method can solve SDPs in polynomial time in a smoothed analysis setting. More precisely, we consider an SDP whose domain satisfies some compactness and smoothness assumptions, and slightly perturb the cost matrix and the constraints. We show that if $p \gtrsim \sqrt{2(1+η)m}$, where $m$ is the number of constraints and $η>0$ is any fixed constant, then the Burer-Monteiro method can solve SDPs to any desired accuracy in polynomial time, in the setting of smooth analysis. Our bound on $p$ approaches the celebrated Barvinok-Pataki bound in the limit as $η$ goes to zero, beneath which it is known that the nonconvex program can be suboptimal. Previous analyses were unable to give polynomial time guarantees for the Burer-Monteiro method, since they either assumed that the criticality conditions are satisfied exactly, or ignored the nontrivial problem of computing an approximately feasible solution. We address the first problem through a novel connection with tubular neighborhoods of algebraic varieties. For the feasibility problem we consider a least squares formulation, and provide the first guarantees that do not rely on the restricted isometry property.