Higher-order smoothing splines versus least squares problems on Riemannian manifolds. (English) Zbl 1203.65028

Machado, L.; Silva Leite, F.; Krakowski, K.

J. Dyn. Control Syst. 16, No. 1, 121-148 (2010).

Summary: 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.

MSC:

65D07	Numerical computation using splines
65D10	Numerical smoothing, curve fitting
49K15	Optimality conditions for problems involving ordinary differential equations
53A35	Non-Euclidean differential geometry
47J30	Variational methods involving nonlinear operators
53C22	Geodesics in global differential geometry
41A15	Spline approximation
49N60	Regularity of solutions in optimal control

Keywords:

Riemannian manifolds; smoothing splines; Lie groups; least square problems; geometric polynomials

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