A Geodesic-Based Riemannian Gradient Approach to Averaging on the Lorentz Group

Abstract

Abstract:In this paper, we propose an efficient algorithm to solve the averaging problem on the Lorentz group O(n, k). Firstly, we introduce the geometric structures of O(n, k) endowed with a Riemannian metric where geodesic could be written in closed form. Then, the algorithm is presented based on the Riemannian-steepest-descent approach. Finally, we compare the above algorithm with the Euclidean gradient algorithm and the extended Hamiltonian algorithm. Numerical experiments show that the geodesic-based Riemannian-steepest-descent algorithm performs the best in terms of the convergence rate.

Rendezvous control for a double-integrator multi-particle system on Lorentz group SO (1,1)

“…In this section, we briefly review the Riemannian structure of the Lorentz group SO(1, 1) and provide some definitions and properties which will be necessary to follow discussions in later sections. We want to refer References 39,41,42 for an introduction to the Lorentz group SO(1, 1).…”

“…The Riemannian metric and the corresponding covariant derivative along a curve on SO(1, 1) are given as follows. 42 Definition 2. For each A ∈ SO(1, 1), the Riemannian metric on T A SO(1, 1) is given by, for V, W ∈ T A SO(1, 1),…”

Empirical Means on Pseudo-Orthogonal Groups

“…A Riemannian setting for the metrization of the pseudo-orthogonal group was proposed and studied in [12]. We believe that both Riemannian and pseudo-Riemannian metrizations are worth investigating as they lead to quite different analytic results.…”

“…Since, in general, an empirical mean of a collection of mathematical objects living in a metric space is the closest point in the space to all points in the collection, the problem of defining and computing an empirical mean may be formulated in terms of minimization of a sum-of-squared distance problem. Function minimization over smooth matrix manifolds has received considerable attention due to its broad application range [8][9][10][11][12][13]. In general, smooth manifolds in function optimization serve to conveniently represent non-linear constraints.…”