Finding the Homology of Manifolds using Ellipsoids

TL;DR

This paper introduces a method to determine the topological invariants of a manifold by using a union of ellipsoids centered at sample points, which simplifies the process and reduces the required sample density.

Contribution

It demonstrates that deformation retraction of ellipsoid unions preserves homotopy type, improving sample efficiency and suggesting elongated shapes for persistent homology analysis.

Findings

Ellipsoids can be used to recover the manifold's homology.

Thickening points to ellipsoids requires less data.

Elongated shapes enhance persistent homology barcodes.

Abstract

A standard problem in applied topology is how to discover topological invariants of data from a noisy point cloud that approximates it. We consider the case where a sample is drawn from a properly embedded C1-submanifold without boundary in a Euclidean space. We show that we can deformation retract the union of ellipsoids, centered at sample points and stretching in the tangent directions, to the manifold. Hence the homotopy type, and therefore also the homology type, of the manifold is the same as that of the nerve complex of the cover by ellipsoids. By thickening sample points to ellipsoids rather than balls, our results require a smaller sample density than comparable results in the literature. They also advocate using elongated shapes in the construction of barcodes in persistent homology.