Expectation of topological invariants

TL;DR

This paper investigates the expected topological invariants of complexes built from sample points on manifolds, showing convergence properties of Betti numbers and Euler characteristic as scale varies.

Contribution

It introduces a novel analysis of the expectation of topological invariants for Vietoris-Rips and Čech complexes on Riemannian manifolds, including convergence results.

Findings

Betti number and Euler characteristic are Lipschitz functions of scale

Betti curve converges to the manifold's Betti number within an interval

Provides bounds and conditions for topological invariant convergence

Abstract

In this paper, we study the expectation values of topological invariants of the Vietoris-Rips complex and \v{C}ech complex for a finite set of sample points on a Riemannian manifold. We show that the Betti number and Euler characteristic of the complexes are Lipschitz functions of the scale parameter and that there is an interval such that the Betti curve converges to the Betti number of the underlying manifold.