A convex relaxation to compute the nearest structured rank deficient matrix

Given an affine space of matrices $\mathcal{L}$ and a matrix $\Theta\in \mathcal{L}$, consider the problem of computing the closest rank deficient matrix to $\Theta$ 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 $\Theta$ 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.