 
Summary: Tech. report UCLINMA2009.023
A GradientDescent Method for Curve Fitting on
Riemannian Manifolds
Chafik Samir
P.A. Absil
Anuj Srivastava
Eric Klassen§
August 28, 2009
Abstract
Given data points p0, . . . , pN on a manifold M and time instants 0 = t0 < t1 < . . . <
tN = 1, we consider the problem of finding a curve on M that best approximates the data
points at the given instants while being as "regular" as possible. Specifically, is expressed
as the curve that minimizes the weighted sum of a sumofsquares term penalizing the lack of
fitting to the data points and a regularity term defined, in the first case as the mean squared
velocity of the curve, and in the second case as the mean squared acceleration of the curve.
In both cases, the optimization task is carried out by means of a steepestdescent algorithm
on a set of curves on M. The steepestdescent direction, defined in the sense of the firstorder
and secondorder Palais metric, respectively, is shown to admit simple formulas.
Keywords: curve fitting, steepestdescent, Sobolev space, Palais metric, geodesic distance,
energy minimization, splines, piecewise geodesic, smoothing, Karcher mean.
