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

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.