Journal of Dynamical and Control Systems

, Volume 16, Issue 1, pp 121–148 | Cite as

Higher-order smoothing splines versus least squares problems on Riemannian manifolds

Article

Abstract

In this paper, we present a generalization of the classical least squares problem on Euclidean spaces, introduced by Lagrange, to more general Riemannian manifolds. Using the variational definition of Riemannian polynomials, we formulate a higher-order variational problem on a manifold equipped with a Riemannian metric, which depends on a smoothing parameter and gives rise to what we call smoothing geometric splines. These are curves with a certain degree of smoothness that best fit a given set of points at given instants of time and reduce to Riemannian polynomials when restricted to each subinterval.

We show that the Riemannian mean of the given points is achieved as a limiting process of the above. Also, when the Riemannian manifold is an Euclidean space, our approach generates, in the limit, the unique polynomial curve which is the solution of the classical least squares problem. These results support our belief that the approach presented in this paper is the natural generalization of the classical least squares problem to Riemannian manifolds.

Key words and phrases

Riemannian manifolds smoothing splines Lie groups least square problems geometric polynomials 

2000 Mathematics Subject Classification

65D07 65D10 49K15 53A35 49K15 47J30 53C22 

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Institute of Systems and RoboticsUniversity of CoimbraCoimbraPortugal
  2. 2.Department of MathematicsUniversity of Trás-os-Montes and Alto DouroVila RealPortugal
  3. 3.Department of MathematicsUniversity of CoimbraCoimbraPortugal
  4. 4.School of Science and TechnologyUniversity of New EnglandNew EnglandAustralia

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