The Generalized Persistent Nerve Theorem

TL;DR

This paper generalizes the Persistent Nerve Theorem by introducing {}-good covers, providing tight bounds on the bottleneck distance between nerve and space filtrations, with applications to non-convex spaces.

Contribution

It introduces the concept of {}-good covers, extends the Persistent Nerve Theorem, and develops a new symmetrization technique to improve interleaving bounds.

Findings

Provides a tight linear bound on bottleneck distance based on {} and homology dimension.

Introduces a computable chain map from nerve to space filtration.

Improves interleaving constant by a factor of 2 through symmetrization.

Abstract

In this paper a parameterized generalization of a good cover filtration is introduced called an {\epsilon}-good cover, defined as a cover filtration in which the reduced homology groups of the image of the inclusions between the intersections of the cover filtration at two scales {\epsilon} apart are trivial. Assuming that one has an {\epsilon}-good cover filtration of a finite simplicial filtration, we prove a tight bound on the bottleneck distance between the persistence diagrams of the nerve filtration and the simplicial filtration that is linear with respect to {\epsilon} and the homology dimension. This bound is the result of a computable chain map from the nerve filtration to the space filtration's chain complexes at a further scale. Quantitative guarantees for covers that are not good are useful for when one is working a non-convex metric space, or one has more simplicial covers that are not the result of triangulations of metric balls. The Persistent Nerve Lemma is also a corollary of our theorem as good covers are 0-good covers. Lastly, a technique is introduced that symmetrizes the asymmetric interleaving used to prove the bound by shifting the nerve filtration's persistence module, improving the interleaving constant by a factor of 2.