A Gradient-Descent Method for Curve Fitting on Riemannian Manifolds

Given data points p ₀,…,p _N on a closed submanifold M of ℝ ⁿ and time instants 0=t ₀<t ₁<⋅⋅⋅<t _N =1, we consider the problem of finding a curve γ on M that best approximates the data points at the given instants while being as “regular” as possible. Specifically, γ 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 analytical expressions involving parallel transport and covariant integral along curves. Illustrations are given in ℝ ⁿ and on the unit sphere.