On $\varepsilon$ Approximations of Persistence Diagrams

TL;DR

This paper develops a theoretical framework for approximating persistent homology with arbitrary accuracy, focusing on filtrations from sub-level sets of functions on CW-complexes, with implementation details for hypercubes.

Contribution

It introduces a new algorithmic approach for $ ext{ε}$-approximations of persistence diagrams and analyzes error bounds, advancing computational methods in topological data analysis.

Findings

Framework for $ ext{ε}$-approximations of persistence diagrams

Error bounds for approximation quality

Implementation strategies for hypercube domains

Abstract

Biological and physical systems often exhibit distinct structures at different spatial/temporal scales. Persistent homology is an algebraic tool that provides a mathematical framework for analyzing the multi-scale structures frequently observed in nature. In this paper a theoretical framework for the algorithmic computation of an arbitrarily good approximation of the persistent homology is developed. We study the filtrations generated by sub-level sets of a function $f : X \to \mathbb{R}$, where $X$ is a CW-complex. In the special case $X = [0,1]^N$, $N \in \mathbb{N}$ we discuss implementation of the proposed algorithms. We also investigate a priori and a posteriori bounds of the approximation error introduced by our method.