论文标题
计算保形的barycenter
Computing the conformal barycenter
论文作者
论文摘要
在尤克里德空间中点云的几何中位数的poincaré球模型的球体上的点云上的共形重点。它是由Douady和Earle定义的，是一种自然自然方法的构建的一部分，将圆的同态性格扩展到磁盘同构的同构，并且在Millson和Kapovich的循环链接模型中起着核心作用，该模型具有固定的EDGELENGEMS。 在本文中，我们考虑了计算保形barycenter的问题。 Abikoff和Ye给出了$ \ Mathbb {s}^1 $的措施的迭代算法，保证会收敛。我们分析了牛顿方法在庞加雷球模型的固有几何形状中计算的牛顿方法的riemannian版本。我们为牛顿 - 坎多维奇（NK）条件提供了纽顿的条件，在该条件下，牛顿的方法具有固定步长的方法，可以在任何$ \ mathbb {s}^d $（包括Infinite-Dimential-Dimensional-dimensional Spheres）上四处汇合到保形的Barycenter。对于符合NK条件的有限维球上，$ n $原子给出的度量，我们在将保形barycenter近似于固定误差所需的计算时间上给出了明确的线性绑定。我们证明我们的NK条件适用于所有$ N $ ATOM的措施，但所有这些条件都适用于所有的条件。对于所有具有独特的保形barycenter的措施，我们表明，使用线路搜索的牛顿的正则化方法将始终（最终是上线性地）汇合到保形barycenter。尽管我们对该算法没有困难的时间范围，但实验表明，它在实践中非常有效，尤其是比Abikoff-ye迭代更快。
The conformal barycenter of a point cloud on the sphere at infinity of the Poincaré ball model of hyperbolic space is a hyperbolic analogue of the geometric median of a point cloud in Euclidean space. It was defined by Douady and Earle as part of a construction of a conformally natural way to extend homeomorphisms of the circle to homeomorphisms of the disk, and it plays a central role in Millson and Kapovich's model of the configuration space of cyclic linkages with fixed edgelengths. In this paper we consider the problem of computing the conformal barycenter. Abikoff and Ye have given an iterative algorithm for measures on $\mathbb{S}^1$ which is guaranteed to converge. We analyze Riemannian versions of Newton's method computed in the intrinsic geometry of the Poincare ball model. We give Newton-Kantorovich (NK) conditions under which we show that Newton's method with fixed step size is guaranteed to converge quadratically to the conformal barycenter for measures on any $\mathbb{S}^d$ (including infinite-dimensional spheres). For measures given by $n$ atoms on a finite dimensional sphere which obey the NK conditions, we give an explicit linear bound on the computation time required to approximate the conformal barycenter to fixed error. We prove that our NK conditions hold for all but exponentially few $n$ atom measures. For all measures with a unique conformal barycenter we show that a regularized Newton's method with line search will always converge (eventually superlinearly) to the conformal barycenter. Though we do not have hard time bounds for this algorithm, experiments show that it is extremely efficient in practice and in particular much faster than the Abikoff-Ye iteration.
