An Approximate Nerve Theorem

Govc, Dejan; Škraba, Primož

doi:10.1007/s10208-017-9368-6

Cited by 20 publications

(19 citation statements)

References 30 publications

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“…A functorial version of the Nerve Lemma appears in [7] and later in [8] for pairs of finite good open covers of paracompact spaces. Approximate homological versions were obtained in [14] and [5]. On the other hand, a functorial version of the Dowker duality was proved in [8].…”

Section: Functorial Dowker-nerve Diagrammentioning

confidence: 99%

Rips complexes as nerves and a Functorial Dowker-Nerve Diagram

Virk

2019

Preprint

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Using ideas of the Dowker duality we prove that the Rips complex at scale r is homotopy equivalent to the nerve of a cover consisting of sets of prescribed diameter. We then develop a functorial version of the Nerve theorem coupled with the Dowker duality, which is presented as a Functorial Dowker-Nerve Diagram. These results are incorporated into a systematic theory of filtrations arising from covers. As a result we provide a general framework for reconstruction of spaces by Rips complexes, a short proof of the reconstruction result of Hausmann, and completely classify reconstruction scales for metric graphs. Furthermore we introduce a new extraction method for homology of a space based on nested Rips complexes at a single scale, which requires no conditions on neighboring scales nor the Euclidean structure of the ambient space.

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Section: Functorial Dowker-nerve Diagrammentioning

confidence: 99%

Rips complexes as nerves and a Functorial Dowker-Nerve Diagram

Virk

2019

Preprint

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“…In September 2016, we submitted our results for presentation at the 26th Fall Workshop on Computational Geometry, which notably implied the Persistent Nerve Lemma as a corollary, for our space assumptions, as originally desired. Soon afterwards, Govc and Skraba updated their arXiv submission relaxing their cover filtration assumption, and their paper has recently been accepted to a journal [17].…”

Section: History Of the Problemmentioning

confidence: 99%

The Generalized Persistent Nerve Theorem

Cavanna,

Sheehy

2018

Preprint

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The Nerve Theorem equates the homotopy type of a suitably covered topological space with that of a combinatorial simplicial complex called a nerve. After filtering a space one can compute the filtration's persistent homology. In persistence theory the Nerve Theorem has the Persistent Nerve Lemma as an analogue which equates the persistent homology of a filtration of spaces and that of the filtration of nerves corresponding to a filtration of covers assuming each cover is good. As nerves are discrete, their geometric realizations can serve as proxies for topological spaces and filtrations in algorithmic settings, e.g. the Čech and Rips complexes are nerves so one can compute their homology over various scales rather than that of a well-sampled space.In this paper we introduce a parameterized generalization of a good cover filtration called an ε-good cover. It is 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 in question. Quantitative guarantees for covers that are not good are useful for when one has a cover of a non-convex metric space, or one has more generally constructed simplicial covers that are not the result of triangulations of metric balls. These guarantees also aid in situations when constructing a good cover filtration is computationally intensive, but a smaller ε-good cover's construction is feasible.Other notable contributions are the introduction of an interleaving between the nerve and covered space's chain complexes up to chain homotopy and the constructive nature of the interleaving that ultimately provides the bound on the bottleneck distance, The Persistent Nerve Lemma is a direct corollary of our main theorem as good covers are 0-good covers. Furthermore, we symmetrize the asymmetric interleaving used to prove the bound by shifting the nerve filtration, improving the interleaving distance by a factor of 2.

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“…If U is an open cover of a compact space X such that every non-empty intersection of finitely many sets in U is contractible, then X is homotopy equivalent to the nerve of U. the notion of good cover. In [9], the hypothesis of being a good cover is relaxed. Then using persistent homology, results about the reconstruction of the homology of the original space are obtained.…”

Section: Introductionmentioning

confidence: 99%

Computational approximations of compact metric spaces

Chocano,

Morón,

del Portal

2021

Preprint

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Given a compact metric space X, we associate to it an inverse sequence of finite T 0 topological spaces. The inverse limit of this inverse sequence contains a homeomorphic copy of X that is a strong deformation retract. We provide a method to approximate the homology groups of X and other algebraic invariants. Finally, we study computational aspects and the implementation of this method.Example 1.2. We consider the topological space X ⊆ R 2 given in Figure 1. Let U denote the open cover given by U 1 , U 2 , U 3 , U 4 and all the possible intersections of them. It is clear that N (U) has the same homotopy type of X, see Figure 1. This approach has a drawback. It is not always easy to find an open cover satisfying the hypothesis of the nerve theorem. Recently, interesting results have been obtained modifying

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An Approximate Nerve Theorem

Cited by 20 publications

References 30 publications

Rips complexes as nerves and a Functorial Dowker-Nerve Diagram

Rips complexes as nerves and a Functorial Dowker-Nerve Diagram

The Generalized Persistent Nerve Theorem

Computational approximations of compact metric spaces

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