Paper ID sheet UCL-INMA-2009.023

Title

A gradient-descent method for curve fitting on Riemannian manifolds

Authors

Chafik Samir, P.-A. Absil, Anuj Srivastava, Eric Klassen

Abstract

Given data points $p_0,\ldots,p_N$ on a manifold $\mathcal{M}$ and time instants $0=t_0 \lt t_1 \lt \ldots \lt t_N=1$, we consider the problem of finding a curve $\gamma$ on $\mathcal{M}$ that best approximates the data points at the given instants while being as ``regular'' as possible. Specifically, $\gamma$ 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 $\mathcal{M}$. The steepest-descent direction, defined in the sense of the first-order and second-order Palais metric, respectively, is shown to admit simple formulas.

Key words

curve fitting; steepest-descent; Sobolev space; Palais metric; geodesic distance; energy minimization; splines; piecewise geodesic; smoothing; Karcher mean

Status

Foundations of Computational Mathematics, 12(1), pp. 49-73, 2012

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• Publisher's version: doi:10.1007/s10208-011-9091-7
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BibTeX entry

@article {SamAbsSriKla2012,
   author = {Samir, Chafik and Absil, P.-A. and Srivastava, Anuj and Klassen, Eric},
   title = {A Gradient-Descent Method for Curve Fitting on {Riemannian} Manifolds},
   journal = {Foundations of Computational Mathematics},
   publisher = {Springer New York},
   issn = {1615-3375},
   pages = {49--73},
   volume = {12},
   issue = {1},
   url = {http://dx.doi.org/10.1007/s10208-011-9091-7},
   doi = {10.1007/s10208-011-9091-7},
   year = {2012}
}