The Fr\'echet mean is an important statistical summary and measure of centrality of data; it has been defined and studied for persistent homology captured by persistence diagrams. However, the complicated geometry of the space of persistence diagrams implies that the Fr\'echet mean for a given set of persistence diagrams is not necessarily unique, which prohibits theoretical guarantees for empirical means with respect to population means. In this paper, we derive a variance expression for a set of persistence diagrams exhibiting a multi-matching between the persistence points known as a grouping. Moreover, we propose a condition for groupings, which we refer to as flatness: sets of persistence diagrams that exhibit flat groupings give rise to unique Fr\'echet means. We derive a finite sample convergence result for general groupings, which results in convergence for Fr\'echet means if the groupings are flat. Finally, we interpret flat groupings in a recently-proposed general framework of Fr\'echet means in Alexandrov geometry. Together with recent results from Alexandrov geometry, this allows for the first derivation of a finite sample convergence rate for sets of persistence diagrams and lays the ground for viability of the Fr\'echet mean as a practical statistical summary of persistent homology.

翻译：Fréchet均值是重要的统计摘要和数据中心性度量；它已被定义并应用于持久性图捕获的持续同调。然而，持久性图空间的复杂几何结构意味着给定持久性图集合的Fréchet均值不一定唯一，这阻碍了关于总体均值的经验均值的理论保证。本文针对表现出持久点间多重匹配（称为分组）的持久性图集合推导了方差表达式。此外，我们提出了一种称为平坦性的分组条件：表现出平坦分组的持久性图集合产生唯一的Fréchet均值。我们推导了一般分组的有限样本收敛结果，若分组是平坦的，则Fréchet均值收敛。最后，我们在最近提出的Alexandrov几何中Fréchet均值的一般框架下解释了平坦分组。结合Alexandrov几何的最新结果，这首次推导了持久性图集合的有限样本收敛速率，并为Fréchet均值作为持续同调实用统计摘要的可行性奠定了基础。