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 generally not 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. Together with recent results from Alexandrov geometry, this allows for the first derivation of a finite sample convergence rate for sets of persistence diagrams that exhibit flat groupings.

翻译：Fr\'echet 平均值是一个重要的统计摘要和数据核心度的衡量标准;它已经被确定和研究为持久性同系物,由持久性图表所捕捉。然而,持久性图表空间的复杂几何学表明,Fr\'echet对特定一组持久性图表的平均值通常并不独特,这就禁止对人口手段的经验性手段提供理论保障。在本文中,我们为一组在持久性点(称为分组)之间显示多相匹配的持久性图表得出了一个差异表达方式。此外,我们提出了一个分组条件,我们称之为平坦性:一组持久性图表显示的扁形组合产生独特的Fr\'echet 值。这与Alexandrov 几何测的最新结果一起,使得首次得出显示单一组合的持久性图表的有限样本趋同率。