Recovering the homology of immersed manifolds

TL;DR

This paper presents a method to recover the homology of immersed manifolds from samples by estimating tangent bundles and applying measure-based persistent homology, without prior knowledge of the manifold's dimension.

Contribution

It introduces a measure-theoretic approach to estimate tangent bundles and recover homology, including a new concept called normal reach for immersed manifolds.

Findings

Method is consistent and stable.

Does not require knowledge of manifold dimension.

Introduces the normal reach for quantitative analysis.

Abstract

Given a sample of an abstract manifold immersed in some Euclidean space, we describe a way to recover the singular homology of the original manifold. It consists in estimating its tangent bundle -- seen as subset of another Euclidean space -- in a measure theoretic point of view, and in applying measure-based filtrations for persistent homology. The construction we propose is consistent and stable, and does not involve the knowledge of the dimension of the manifold. In order to obtain quantitative results, we introduce the normal reach, which is a notion of reach suitable for an immersed manifold.