# Distributions of Matching Distances in Topological Data Analysis

## TL;DR

This paper explores the behavior of matching distances in two-parameter persistent homology, revealing how geometric differences in data influence topological similarity measures, and provides foundational insights for analyzing complex data structures.

## Contribution

It introduces key results on matching distances in two-parameter persistence modules, addressing a less-studied area in topological data analysis with practical implications.

## Key findings

- Matching distance varies with geometric differences in point clouds
- Results serve as a foundation for analyzing complex data structures
- Provides insights into two-parameter persistent homology behavior

## Abstract

In topological data analysis, we want to discern topological and geometric structure of data, and to understand whether or not certain features of data are significant as opposed to simply random noise. While progress has been made on statistical techniques for single-parameter persistence, the case of two-parameter persistence, which is highly desirable for real-world applications, has been less studied. This paper provides an accessible introduction to two-parameter persistent homology and presents results about matching distance between 2-D persistence modules obtained from families of point clouds. Results include observations of how differences in geometric structure of point clouds affect the matching distance between persistence modules. We offer these results as a starting point for the investigation of more complex data.