Approximating Local Homology from Samples

TL;DR

This paper introduces a method to approximate local homology using Vietoris-Rips complexes, enabling high-dimensional applications like stratification learning with improved robustness and simplicity.

Contribution

It demonstrates that local homology persistence diagrams can be approximated via Vietoris-Rips complexes, overcoming high-dimensional reconstruction challenges.

Findings

Vietoris-Rips complexes effectively approximate local homology.

The approach is robust in high dimensions.

Enables practical applications like stratification learning.

Abstract

Recently, multi-scale notions of local homology (a variant of persistent homology) have been used to study the local structure of spaces around a given point from a point cloud sample. Current reconstruction guarantees rely on constructing embedded complexes which become difficult in high dimensions. We show that the persistence diagrams used for estimating local homology, can be approximated using families of Vietoris-Rips complexes, whose simple constructions are robust in any dimension. To the best of our knowledge, our results, for the first time, make applications based on local homology, such as stratification learning, feasible in high dimensions.