The geometric median as well as the Frechet mean of points in an Hadamard
space are important in both theory and applications. Surprisingly, no
algorithms for their computation are hitherto known. To address this issue, we
use a split version of the proximal point algorithm for minimizing a sum of
convex functions and prove that this algorithm produces a sequence converging
to a minimizer of the objective function, which extends a recent result of D.
Bertsekas (2001) into Hadamard spaces. The method is quite robust and not only
does it yield algorithms for the median and the mean, but it also applies to
various other optimization problems. We moreover show that another algorithm
for computing the Frechet mean can be derived from the law of large numbers due
to K.-T. Sturm (2002). In applications, computing medians and means is probably
most needed in tree space, which is an instance of an Hadamard space, invented
by Billera, Holmes, and Vogtmann (2001) as a tool for averaging phylogenetic
trees. It turns out, however, that it can be also used to model numerous other
tree-like structures. Since there now exists a polynomial-time algorithm for
computing geodesics in tree space due to M. Owen and S. Provan (2011), we
obtain efficient algorithms for computing medians and means, which can be
directly used in practice.

提出一种计算哈达玛空间中几何中心和 Frechet 平均的算法，并从树空间等应用中证明其实用性。

在 Hadamard 空间中计算中位数和平均值

Computing medians and means in Hadamard spaces

Given a distribution $\rho$ on persistence diagrams and observations
$X_1,...X_n \stackrel{iid}{\sim} \rho$ we introduce an algorithm in this paper
that estimates a Fr\'echet mean from the set of diagrams $X_1,...X_n$. If the
underlying measure $\rho$ is a combination of Dirac masses $\rho = \frac{1}{m}
\sum_{i=1}^m \delta_{Z_i}$ then we prove the algorithm converges to a local
minimum and a law of large numbers result for a Fr\'echet mean computed by the
algorithm given observations drawn iid from $\rho$. We illustrate the
convergence of an empirical mean computed by the algorithm to a population mean
by simulations from Gaussian random fields.

该论文介绍了一种可以从任意分布的坚持图像计算出 Frechet mean 的算法，并证明了当分布是 Dirac masses 的组合时，算法收敛于局部最小值，并给出了一个取自该分布的观测序列的 Frechet mean 的大数定理结果。同时，作者还用高斯随机场的模拟表明算法计算出的经验 mean 会收敛到总体 mean。