We consider two statistical problems at the intersection of functional and
non-Euclidean data analysis: the determination of a Fr\'echet mean in the
Wasserstein space of multivariate distributions; and the optimal registration
of deformed random measures and point processes. We elucidate how the two
problems are linked, each being in a sense dual to the other. We first study
the finite sample version of the problem in the continuum. Exploiting the
tangent bundle structure of Wasserstein space, we deduce the Fr\'echet mean via
gradient descent. We show that this is equivalent to a Procrustes analysis for
the registration maps, thus only requiring successive solutions to pairwise
optimal coupling problems. We then study the population version of the problem,
focussing on inference and stability: in practice, the data are i.i.d.
realisations from a law on Wasserstein space, and indeed their observation is
discrete, where one observes a proxy finite sample or point process. We
construct regularised nonparametric estimators, and prove their consistency for
the population mean, and uniform consistency for the population Procrustes
registration maps.

本研究探讨了函数数据分析和非欧几里得数据分析领域中的两个统计学问题：在多元分布的 Wasserstein 空间中确定 Fréchet 均值，以及畸变随机测量和点过程的最佳注册。研究表明，这两个问题在某种意义上是相互关联的，并通过利用 Wasserstein 空间的切向丛结构，通过梯度下降方法推导出了 Fréchet 均值，并表明这等效于 Procrustes 分析；然后构建了建模估计器，并证明了它们对总体平均值和总体 Procrustes 注册图的一致性。

Wasserstein 空间中的 Fréchet 平均和 Procrustes 分析

Fréchet Means and Procrustes Analysis in Wasserstein Space

Spectral clustering is a broad class of clustering procedures in which an
intractable combinatorial optimization formulation of clustering is "relaxed"
into a tractable eigenvector problem, and in which the relaxed solution is
subsequently "rounded" into an approximate discrete solution to the original
problem. In this paper we present a novel margin-based perspective on multiway
spectral clustering. We show that the margin-based perspective illuminates both
the relaxation and rounding aspects of spectral clustering, providing a unified
analysis of existing algorithms and guiding the design of new algorithms. We
also present connections between spectral clustering and several other topics
in statistics, specifically minimum-variance clustering, Procrustes analysis
and Gaussian intrinsic autoregression.

本篇论文提出了一种基于边缘的多极谱聚类方法，并将其应用于谱聚类问题的松弛和圆整方面的统一分析，为现有算法的分析提供了指导，并介绍了谱聚类与多个统计学主题的联系，包括最小方差聚类、匹配分析和高斯内在自回归。