Computing the Conformal Barycenter

The conformal barycenter of a point cloud on the sphere at infinity Poincare ball model hyperbolic space is analogue geometric median in Euclidean space. It was defined by Douady and Earle as part construction conformally natural way to extend homeomorphisms circle disk, it plays central role Millson Kapovich's configuration cyclic linkages with fixed edgelengths. In this paper we consider problem computing barycenter. Abikoff Ye have given an iterative algorithm for measures $\mathbb{S}^1$ which guaranteed converge. We analyze Riemannian versions Newton's method computed intrinsic geometry model. give Newton-Kantorovich (NK) conditions under show that step size converge quadratically any $\mathbb{S}^d$ (including infinite-dimensional spheres). For $n$ atoms finite dimensional obey NK conditions, explicit linear bound computation time required approximate error. prove our hold all but exponentially few atom measures. unique regularized line search will always (eventually superlinearly) Though do not hard bounds algorithm, experiments extremely efficient practice particular much faster than Abikoff-Ye iteration.