The Generalized Persistent Nerve Theorem
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In this paper a parameterized generalization of a good cover filtration is introduced called an ϵ-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 ϵ apart are trivial. Assuming that one has an ϵ-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 ϵ 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.
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