Newton's method, zeroes of vector fields, and the Riemannian center of mass

Details Newton's method, zeroes of vector fields, and the Riemannian center of mass Newton's method, zeroes of vector fields, and the Riemannian center of mass

2003-11-02

arxiv.org/abs/math/0311011v1

Mathematics

Differential Geometry

43 pages, 1 figure

Scientific paper

We present an iterative technique for finding zeroes of vector fields on Riemannian manifolds. As a special case we obtain a ``nonlinear averaging algorithm'' that computes the centroid of a mass distribution supported in a set of small enough diameter D in a Riemannian manifold M. We estimate the convergence rate of our general algorithm and the more special Riemannian averaging algorithm. The algorithm is also used to provide a constructive proof of Karcher's theorem on the existence and local uniqueness of the center of mass, under a somewhat stronger requirement than Karcher's on D. Another corollary of our results is a proof of convergence, for a fairly large open set of initial conditions, of the ``GPA algorithm'' used in statistics to average points in a shape-space, and a quantitative explanation of why the GPA algorithm converges rapidly in practice. We also show that a mass distribution in M with support Q has a unique center of mass in a (suitably defined) convex hull of Q.

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