论文标题
Fréchet持久图的独特性的几何条件
A Geometric Condition for Uniqueness of Fréchet Means of Persistence Diagrams
论文作者
论文摘要
Fréchet平均值是数据中心性的重要统计摘要和度量。它已被定义和研究，以通过持久图捕获的持续同源性。但是，持久图的复杂几何形状意味着，对于给定的一组持久图，fréchet的平均值并不一定是独一无二的，这禁止对人口手段的经验手段的理论保证。在本文中，我们得出了一组持久图的方差表达式，该图表在称为分组的持久性点之间表现出多匹配。此外，我们提出了分组的条件，我们称之为平坦。我们证明，表现出扁平分组的一组持久图产生了独特的fréchet手段。我们得出了一般分组的有限样品收敛结果，这导致Fréchet的收敛意味着如果分组是平坦的。然后，我们在最近在Alexandrov几何形状中的Fréchet手段的一般框架中解释了扁平分组。最后，我们表明，对于流动性数据，可以将持久图截断以构造平面分组。
The Fréchet 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 Fréchet 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; we prove that sets of persistence diagrams that exhibit flat groupings give rise to unique Fréchet means. We derive a finite sample convergence result for general groupings, which results in convergence for Fréchet means if the groupings are flat. We then interpret flat groupings in a recently-proposed general framework of Fréchet means in Alexandrov geometry. Finally, we show that for manifold-valued data, the persistence diagrams can be truncated to construct flat groupings.
