Determining homology of an unknown space from a sample

TL;DR

This paper presents a method to determine the homology of an unknown space using only a sample of points and the intrinsic Čech complex, without needing the ambient Euclidean space.

Contribution

It introduces conditions under which the homology of the unknown space can be inferred from the persistent homology of the intrinsic Čech complex.

Findings

Homology of the space can be recovered from sample data.

Persistent homology of the intrinsic Čech complex reflects the true homology.

Conditions for accurate homology inference are established.

Abstract

The homology of an unknown subspace of Euclidean space can be determined from the intrinsic \v{C}ech complex of a sample of points in the subspace, without reference to the ambient Euclidean space. More precisely, given a subspace $X$ of Euclidean space and a sample $A$ of points in $X$, we give conditions for the homology of $X$ to be isomorphic to a certain persistent homology group of the intrinsic \v{C}ech complex.