论文标题
在$ c^0 $ - 透明的同源和树木
On $C^0$-persistent homology and trees
论文作者
论文摘要
在本文中，我们给出了一棵树的度量构造，该树可以正确识别$ \ mathbb {r} $ - $ x $上的$ \ mathbb {r} $的连接组件，并表明可以从这棵树中检索$ h_0 $ - persistent图。我们重新审视了Schweinhart先前提出的同源维度的概念，并就$ x $的上盒维度为后者提供了一些界限，从而部分回答了同一作者的问题。我们证明了Wasserstein稳定性定理的定量版本有效，适用于常规的$ x $和$α$-Hölder函数，并讨论了该理论对随机字段的某些应用以及其Superlevel集合的拓扑。
In this paper we give a metric construction of a tree which correctly identifies connected components of superlevel sets of $\mathbb{R}$-valued continuous functions $f$ on $X$ and show that it is possible to retrieve the $H_0$-persistent diagram from this tree. We revisit the notion of homological dimension previously introduced by Schweinhart and give some bounds for the latter in terms of the upper-box dimension of $X$, thereby partially answering a question of the same author. We prove a quantitative version of the Wasserstein stability theorem valid for regular enough $X$ and $α$-Hölder functions and discuss some applications of this theory to random fields and the topology of their superlevel sets.
