On $C^0$-persistent homology and trees

TL;DR

This paper constructs a metric tree to identify connected components of superlevel sets of continuous functions, relates it to persistent homology, and explores stability and applications to random fields.

Contribution

It introduces a metric tree construction that captures $H_0$-persistent diagrams and provides bounds on homological dimension using upper-box dimension.

Findings

The tree correctly identifies connected components of superlevel sets.

The $H_0$-persistent diagram can be retrieved from the constructed tree.

A quantitative Wasserstein stability theorem is established for regular spaces and Hölder functions.

Abstract

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 $\alpha$-H\"older functions and discuss some applications of this theory to random fields and the topology of their superlevel sets.