Fast Jacobi-type algorithm for computing distances between linear dynamical systems

The alignment distance is a novel metric between linear dynamical systems that has been shown to be very useful in many applications in computer vision. However, since the computation of the alignment distance requires solving a minimization problem on the orthogonal group, it is important to develop computationally efficient algorithms for solving this problem. In this paper, we present a fast and accurate Jacobi-type algorithm that solves this problem. Each step of the algorithm is equivalent to finding the roots of a quartic polynomial. We show that this rooting may be done efficiently and accurately using a careful implementation of Ferrari´s classical closed-form solution for quartic polynomials. For linear systems with orders that commonly arise in computer vision scenarios, our algorithm is roughly twenty times faster than a fast Riemannian gradient descent algorithm implementation and has comparable accuracy.