 
Summary: 11 May 2010 Technical report UCLINMA2009.023v2
A GradientDescent Method for Curve Fitting on
Riemannian Manifolds
Chafik Samir
P.A. Absil
Anuj Srivastava
Eric Klassen§
Abstract
Given data points p0, . . . , pN on a closed submanifold M of Rn
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 analytical
expressions involving parallel transport and covariant integral along curves. Illustrations are
given in Rn
