# Same But Different: Distance Correlations Between Topological Summaries

## TL;DR

This paper explores how distance correlation can compare different topological summaries of data, highlighting its application in analyzing complex data structures across various metric spaces.

## Contribution

It introduces the use of distance correlation for comparing topological summaries in different metric spaces, providing a non-parametric statistical tool for topological data analysis.

## Key findings

- Distance correlation effectively compares topological summaries across different metric spaces.
- Different topological summaries can yield varying statistical conclusions.
- The method is applicable to various data models and scalar measures.

## Abstract

Persistent homology allows us to create topological summaries of complex data. In order to analyse these statistically, we need to choose a topological summary and a relevant metric space in which this topological summary exists. While different summaries may contain the same information (as they come from the same persistence module), they can lead to different statistical conclusions since they lie in different metric spaces. The best choice of metric will often be application-specific. In this paper we discuss distance correlation, which is a non-parametric tool for comparing data sets that can lie in completely different metric spaces. In particular we calculate the distance correlation between different choices of topological summaries. We compare some different topological summaries for a variety of random models of underlying data via the distance correlation between the samples. We also give examples of performing distance correlation between topological summaries and other scalar measures of interest, such as a paired random variable or a parameter of the random model used to generate the underlying data. This article is meant to be expository in style, and will include the definitions of standard statistical quantities in order to be accessible to non-statisticians.