Estudo Geralhttps://estudogeral.sib.uc.ptThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Sat, 04 Apr 2020 11:58:43 GMT2020-04-04T11:58:43Z5011High order smoothing splines versus least squares problems on Riemannian manifoldshttp://hdl.handle.net/10316/11236Title: High order smoothing splines versus least squares problems on Riemannian manifolds
Authors: Machado, L.; Leite, F. Silva; Krakowski, K.
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 high 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.
Tue, 01 Jan 2008 00:00:00 GMThttp://hdl.handle.net/10316/112362008-01-01T00:00:00Z