On ℓp-Vietoris-Rips complexes

2024-11-04

arXiv.org

AI Summary

This paper introduces the ℓₚ-Vietoris–Rips simplicial set and complex theory (1 ≤ p ≤ ∞), unifying the classical Vietoris–Rips complex (p = ∞) and fuzzy magnitude homology (p = 1). To address limitations of existing persistent homology frameworks, we develop a generalized persistent homology framework grounded in algebraic topology, metric geometry, and persistence theory, establishing the first systematic theory for the ℓₚ-family of complexes. Our main contributions are: (1) proving stability with respect to the p-Wasserstein distance; (2) establishing homotopy equivalence reconstruction for compact Riemannian manifolds at sufficiently small scales; (3) showing that zero-scale limit homology groups are independent of p and that the homology functor commutes with filtered colimits; and (4) verifying homological invariance under metric completion. These results provide a more flexible and robust geometric foundation for persistent homology, enhancing its applicability to diverse metric data structures.

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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 $1leq p leq infty.$ 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.

Problem

• Generalizing Vietoris-Rips complex theory
• Unifying blurred magnitude homology theory
• Studying stability and invariance properties

Innovation

• Generalizes Vietoris-Rips complexes
• Proves persistent homology stability
• Shows homotopy equivalence invariance