Reeb Graph of Sample Thickenings

TL;DR

This paper develops a framework for approximating the Reeb graph of an unknown space from sampled data, extending existing topological reconstruction results to Reeb graphs and providing an algorithm for practical computation.

Contribution

It generalizes interleaving distances for Reeb graphs and adapts reconstruction results to broader topological spaces, including an algorithm for sample thickenings.

Findings

Reeb graph of a sample thickening approximates the true Reeb graph under certain conditions.

Many results for constructible spaces extend to general topological spaces.

An algorithm for computing Reeb graphs from sample thickenings is provided.

Abstract

We consider the Reeb graph of a thickening of points sampled from an unknown space. Our main contribution is a framework to transfer reconstruction results similar to the well-known work of Niyogi, Smale, and Weinberger to the setting of Reeb graphs. To this end, we first generalize and study the interleaving distances for Reeb graphs. We find that many of the results previously established for constructible spaces also hold for general topological spaces. We use this to show that under certain conditions for topological spaces with real-valued Lipschitz maps, the Reeb graph of a sample thickening approximates the Reeb graph of the underlying space. Finally, we provide an algorithm for computing the Reeb graph of a sample thickening.