Relative persistent homology

TL;DR

This paper introduces a new method for efficiently computing relative persistent homology of point clouds in low-dimensional Euclidean spaces using the relative Delaunay Čech complex, which is constructed from the Delaunay complex in higher dimensions.

Contribution

It proposes the relative Delaunay Čech complex, enabling efficient computation of relative persistent homology for larger point clouds in low dimensions.

Findings

The relative Delaunay Čech complex accurately computes relative persistent homology.

The method improves computational feasibility for larger datasets.

It extends persistent homology techniques to relative settings in low-dimensional spaces.

Abstract

The alpha complex efficiently computes persistent homology of a point cloud $X$ in Euclidean space when the dimension $d$ is low. Given a subset $A$ of $X$, relative persistent homology can be computed as the persistent homology of the relative \v{C}ech complex. But this is not computationally feasible for larger point clouds. The aim of this note is to present a method for efficient computation of relative persistent homology in low dimensional Euclidean space. We introduce the relative Delaunay \v{C}ech complex whose homology is the relative persistent homology. It can be constructed from the Delaunay complex of an embedding of the point clouds in $(d+1)$-dimensional Euclidean space.