Geodesic regression on spheres from a numerical optimization viewpoint

Abstract

In the geodesic regression problem it is given a set of data points at known times and the goal is to find a geodesic that best fits the data. This problem corresponds to the generalization of the classical linear regression problem to curved spaces. Here we are interested in the geodesic regression problem on Euclidean spheres. Contrary to the Euclidean situation, the normal equations turn out to be highly nonlinear. To overcome this difficulty, we look at the geodesic regression problem in the unit n-sphere as an optimization problem in the Euclidean space ℝⁿ⁺¹ and use the MATLAB optimization toolbox to solve it numerically.

Keywords:

• geodesics
• geodesic distance
• normal equations
• constrained nonlinear optimization
• sequential quadratic programming

2000 AMS Subject Classifications:

• 53B21
• 58C05
• 65D10
• 65K05
• 90C30

Acknowledgements

This work is funded by FEDER funds through Operational Programme for Competitiveness Factors – COMPETE and National Funds through Foundation for Science and Technology (FCT) in Projects scope: FCOMP-01-0124-FEDER-022674 and PTDC/EEA-CRO/122812/2010.