# Persistent Intersection Homology for the Analysis of Discrete Data

## TL;DR

This paper introduces persistent intersection homology as a novel topological tool for analyzing complex discrete data sets, especially those with singularities or multiple manifolds, enhancing data analysis and visualization.

## Contribution

It extends persistent homology to intersection homology, addressing singularities in data and providing strategies for effective analysis and visualization.

## Key findings

- Persistent intersection homology can handle data with singularities.
- Strategies are proposed to approximate data sets with singularities.
- The method improves feature extraction in complex data sets.

## Abstract

Topological data analysis is becoming increasingly relevant to support the analysis of unstructured data sets. A common assumption in data analysis is that the data set is a sample---not necessarily a uniform one---of some high-dimensional manifold. In such cases, persistent homology can be successfully employed to extract features, remove noise, and compare data sets. The underlying problems in some application domains, however, turn out to represent multiple manifolds with different dimensions. Algebraic topology typically analyzes such problems using intersection homology, an extension of homology that is capable of handling configurations with singularities. In this paper, we describe how the persistent variant of intersection homology can be used to assist data analysis in visualization. We point out potential pitfalls in approximating data sets with singularities and give strategies for resolving them.