Comparing Metrics on Arbitrary Spaces using Topological Data Analysis

TL;DR

This paper explores how topological data analysis can be used to compare different metrics on data sets, demonstrated through visual point cloud data and abstract non-transitive dice spaces.

Contribution

It introduces a novel approach to metric comparison using topological data analysis on both visual and abstract data spaces.

Findings

Topological methods effectively distinguish different metrics.

Application to both visual and abstract data spaces demonstrates versatility.

Provides new tools for metric analysis in complex data structures.

Abstract

We use the notion of topological data analysis to compare metrics on data sets. We provide two different motivating examples for this. The first of these is a point cloud data set that has $\mathbb{R}^2$ as its ambient space, and is therefore very visual. the second deals with a very abstract space which arises through the study of non-transitive dice.