标题： 持久图的景观与侵蚀距离的等价性 显示英文标题

标题： Equivalence of Landscape and Erosion Distances for Persistence Diagrams

摘要： 本文建立了在拓扑数据分析中用于分析持久图的三种最突出度量之间的联系：瓶颈距离、Patel的侵蚀距离和Bubenik的景观距离。 我们的主要结果表明，侵蚀距离和景观距离相等，从而将前者的自然范畴理论解释与后者的计算方便结构联系起来。 证明利用了de Silva等人提出的带有流的范畴框架，并对持久景观的结构提供了额外的见解。 我们的等价结果被用来证明关于侵蚀距离几何性质的几个结果。 我们证明了侵蚀距离不是长度度量，并且其内在度量是瓶颈距离。 我们还证明了侵蚀距离不能粗略地嵌入到任何希尔伯特空间中，即使限制到来自0次持久同调的持久图也是如此。 此外，我们证明在该子空间上，侵蚀距离与瓶颈距离一致，因此我们的非嵌入定理推广了近期文献中的几个结果。 显示英文摘要

摘要： This paper establishes connections between three of the most prominent metrics used in the analysis of persistence diagrams in topological data analysis: the bottleneck distance, Patel's erosion distance, and Bubenik's landscape distance. Our main result shows that the erosion and landscape distances are equal, thereby bridging the former's natural category-theoretic interpretation with the latter's computationally convenient structure. The proof utilizes the category with a flow framework of de Silva et al., and leads to additional insights into the structure of persistence landscapes. Our equivalence result is applied to prove several results on the geometry of the erosion distance. We show that the erosion distance is not a length metric, and that its intrinsic metric is the bottleneck distance. We also show that the erosion distance does not coarsely embed into any Hilbert space, even when restricted to persistence diagrams arising from degree-0 persistent homology. Moreover, we show that erosion distance agrees with bottleneck distance on this subspace, so that our non-embeddability theorem generalizes several results in the recent literature.