Can a mathematician and an engineer be friends? - Critical points to low-rank Hankel matrix approximation

Rank-deficient Hankel matrices frequently arise in the context of systems and control theory and the combination with a least-squares approximation is used in system identification problems. In order to find the global best approximation, one can compute all critical points of the distance function and select the real minimum. We construct a simple polynomial system, a multiparameter eigenvalue problem, from which these points can be computed. We use basic tools from intersection theory to compute the number of solutions, recovering and extending previous results on the Euclidean Distance degree of secants of rational normal curves. We also study special metrics and data points for which the number of solution changes.

This is ongoing work with Sibren Lagauw, KU Leuven.