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

Abstract

A novel metric between linear dynamical systems, the alignment distance, was recently introduced, with promising results in many computer vision tasks. The computation of the alignment distance requires a minimization over the orthogonal group. 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 closedform 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.

Alignment Distances on Systems of Bags

“…For the sake of simplicity, we treat these two cases separately. The authors of [28] propose to compute the alignment distance of classical LDSs by writing matrices in SO(n) as products of Givens rotations, i.e. as n p=1 n q=p+1…”

“…for which closed-form solution formulas exist [28] or Newton type methods can be applied. If the global maximum is not unique, the candidate with the smallest value for |s| must be selected.…”

The alignment distance on Spaces of Linear Dynamical Systems

“…The use of the Frobenius norm makes this distance as computationally friendly as one could hope for. In [19], a fast Jacobi-type algorithm for finding the alignment distance (4) based ond F is proposed. Such an algorithm is a local minimization algorithm over O(n), however, an interesting experimental observation made in [19] is that this algorithm is more likely to find the global minimizer than the gradient algorithm.…”

“…In [19], a fast Jacobi-type algorithm for finding the alignment distance (4) based ond F is proposed. Such an algorithm is a local minimization algorithm over O(n), however, an interesting experimental observation made in [19] is that this algorithm is more likely to find the global minimizer than the gradient algorithm. Another feature ofd F is that it is a product distance, i.e., it is the sum of three separate terms coming from the product structure of the ambient space L m,n,p ⊃ Σ m,n,p .…”

“…This is a theoretical paper entirely devoted to discrete-time deterministic LDSs, and due to space limitation the proofs of most stated results will appear elsewhere. Some computational aspects and the case of stochastic LDSs have been addressed in [19] and [2], respectively.…”