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

1, 1,2,* and 3
1
School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China
2
Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing 100081, China
3
Department of Mathematics, Duke University, Durham, NC 27708, USA
*
Author to whom correspondence should be addressed.
Entropy 2017, 19(12), 698; https://doi.org/10.3390/e19120698
Received: 19 November 2017 / Revised: 16 December 2017 / Accepted: 17 December 2017 / Published: 20 December 2017
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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. View Full-Text
Keywords: Lorentz group; geodesic; average; Riemannian-steepest-descent algorithm; extended Hamiltonian algorithm Lorentz group; geodesic; average; Riemannian-steepest-descent algorithm; extended Hamiltonian algorithm
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Wang, J.; Sun, H.; Li, D. A Geodesic-Based Riemannian Gradient Approach to Averaging on the Lorentz Group. Entropy 2017, 19, 698.

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