Geometric Approaches on Persistent Homology

TL;DR

This paper introduces geometric concepts like the width of homology classes to interpret persistent homology, providing bounds and geometric insights into persistence diagrams, especially for graph filtrations.

Contribution

It presents new geometric notions such as the width of homology classes and applies them to analyze persistence diagrams in graph filtrations.

Findings

Provides geometric interpretations of persistence diagrams.

Offers explicit bounds for homology class lifespan in graph filtrations.

Introduces the concept of width for homology classes.

Abstract

We introduce several geometric notions, including the width of a homology class, to the theory of persistent homology. These ideas provide geometric interpretations of persistence diagrams. Indeed, we give quantitative and geometric descriptions of the "life span" or "persistence" of a homology class. As a case study, we analyze the power filtration on unweighted graphs, and provide explicit bounds for the life spans of homology classes in persistence diagrams in all dimensions.