On $\ell_p$-Vietoris-Rips complexes

TL;DR

This paper introduces and studies the $ ext{ell}_p$-Vietoris-Rips complexes, unifying classical and blurred magnitude homology theories, and proves stability, homotopy equivalence, and invariance results for these complexes.

Contribution

It generalizes Vietoris-Rips complexes to $ ext{ell}_p$-settings, establishing stability, homotopy invariance, and homology limit independence across different $p$ values.

Findings

Proves stability of the persistent homology for $ ext{ell}_p$-Vietoris-Rips complexes.

Shows homotopy equivalence to manifolds for small scales on compact Riemannian manifolds.

Demonstrates homology groups are independent of $p$ as scale tends to zero.

Abstract

We study the concepts of the $\ell_p$-Vietoris-Rips simplicial set and the $\ell_p$-Vietoris-Rips complex of a metric space, where $1\leq p \leq \infty.$ This theory unifies two established theories: for $p=\infty,$ this is the classical theory of Vietoris-Rips complexes, and for $p=1,$ this corresponds to the blurred magnitude homology theory. We prove several results that are known for the Vietoris-Rips complex in the general case: (1) we prove a stability theorem for the corresponding version of the persistent homology; (2) we show that, for a compact Riemannian manifold and a sufficiently small scale parameter, all the "$\ell_p$-Vietoris-Rips spaces" are homotopy equivalent to the manifold; (3) we demonstrate that the $\ell_p$-Vietoris-Rips spaces are invariant (up to homotopy) under taking the metric completion. Additionally, we show that the limit of the homology groups of the $\ell_p$-Vietoris-Rips spaces, as the scale parameter tends to zero, does not depend on $p$; and that the homology groups of the $\ell_p$-Vietoris-Rips spaces commute with filtered colimits of metric spaces.