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On $\ell_p$-Vietoris-Rips complexes

Sergei O. Ivanov, Xiaomeng Xu

TL;DR

This paper unifies the ell_p-Vietoris-Rips theory across all p in [1,∞] through distance-matrix norms, connecting VR simplicial sets and complexes with blurred magnitude homology. It develops a general ν-VR framework, proving a stability theorem for persistent homology, a manifold-level homotopy equivalence at small scales for geodesic spaces, and invariance under metric completion, while showing that zero-scale limit homology is independent of p and commutes with filtered colimits. The work also introduces pro-simplicial-set perspectives, demonstrates commutation with filtered colimits for strict VR, and extends classical results (e.g., Hausmann) to the ν-setting, including explicit computations on spheres and circles. Together, these results enhance theoretical guarantees for topological data analysis using generalized VR constructions and provide a cohesive bridge between VR theory and magnitude homology via a broad, axiomatic norm-based framework.

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.

On $\ell_p$-Vietoris-Rips complexes

TL;DR

This paper unifies the ell_p-Vietoris-Rips theory across all p in [1,∞] through distance-matrix norms, connecting VR simplicial sets and complexes with blurred magnitude homology. It develops a general ν-VR framework, proving a stability theorem for persistent homology, a manifold-level homotopy equivalence at small scales for geodesic spaces, and invariance under metric completion, while showing that zero-scale limit homology is independent of p and commutes with filtered colimits. The work also introduces pro-simplicial-set perspectives, demonstrates commutation with filtered colimits for strict VR, and extends classical results (e.g., Hausmann) to the ν-setting, including explicit computations on spheres and circles. Together, these results enhance theoretical guarantees for topological data analysis using generalized VR constructions and provide a cohesive bridge between VR theory and magnitude homology via a broad, axiomatic norm-based framework.

Abstract

We study the concepts of the -Vietoris-Rips simplicial set and the -Vietoris-Rips complex of a metric space, where This theory unifies two established theories: for this is the classical theory of Vietoris-Rips complexes, and for 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 "-Vietoris-Rips spaces" are homotopy equivalent to the manifold; (3) we demonstrate that the -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 -Vietoris-Rips spaces, as the scale parameter tends to zero, does not depend on ; and that the homology groups of the -Vietoris-Rips spaces commute with filtered colimits of metric spaces.

Paper Structure

This paper contains 28 sections, 51 theorems, 214 equations.

Table of Contents

  1. Introduction
  2. Definitions of the $\ell_p$-Vietoris-Rips spaces
  3. Distance matrix norms
  4. Stability theorem
  5. Riemannian manifolds
  6. Metric completions and shortly contractible spaces
  7. Pro-simplicial sets and limit homology
  8. Filtered colimits of metric spaces
  9. Acknowledgements
  10. Vietoris-Rips spaces
  11. Distance matrix norms
  12. Vietoris-Rips simplicial sets
  13. Relation to blurred magnitude homology
  14. Vietoris-Rips complexes
  15. Cyclic versions of the Vietoris-Rips spaces
  16. ...and 13 more sections

Key Result

Lemma 2.2

For any distance matrix norm $\nu,$ the collection of maps $\nu^{\sf sym}$ is a symmetric distance matrix norm.

Theorems & Definitions (104)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • ...and 94 more