The Generalized Persistent Nerve Theorem

Abstract

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.

Spatial representability of neuronal activity

“…The latter approach is especially valuable as it allows extracting stable topological information from spiking data that may be generated from the maps with multiply connected firing fields or encumbered by other inherent irregularities, in spirit with the general ideas of topological persistence [68][69][70][71][72][73]. It should also be mentioned that mathematical discussions of the persistent nerve theorem, alternative to ours and more formal, have began to appear [115,116]; however at this point our studies are independent.…”

Rips complexes as nerves and a Functorial Dowker-Nerve Diagram

“…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].…”