Newton’s method, zeroes of vector fields, and the Riemannian center of mass. (English) Zbl 1062.53025
Summary: 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 \(\mu\) 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; see [the author, On the convergence of some Procrustean averaging algorithms, Preprint (2003).
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 \(Q\).
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 \(Q\).
MSC:
| 53C20 | Global Riemannian geometry, including pinching |
| 60E05 | Probability distributions: general theory |
| 62H05 | Characterization and structure theory for multivariate probability distributions; copulas |
| 65J15 | Numerical solutions to equations with nonlinear operators |
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