Summary: Tech. report UCL-INMA-2009.023
A Gradient-Descent Method for Curve Fitting on
August 28, 2009
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 sum-of-squares 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 steepest-descent algorithm
on a set of curves on M. The steepest-descent direction, defined in the sense of the first-order
and second-order Palais metric, respectively, is shown to admit simple formulas.
Keywords: curve fitting, steepest-descent, Sobolev space, Palais metric, geodesic distance,
energy minimization, splines, piecewise geodesic, smoothing, Karcher mean.