Abstract
We propose sparse versions of filtered simplicial complexes used to compute persistent homology of point clouds and of networks. In particular, we extend the Sparse Čech Complex of Cavanna et al. (A geometric perspective on sparse filtrations. CoRR, arXiv:1506.03797, 2015) from point clouds in convex metric spaces to point clouds in arbitrary metric spaces. Along the way we formulate interleaving in terms of strict 2-categories, and we introduce the concept of Dowker dissimilarities that can be considered as a common generalization of metric spaces and networks.
Highlights
This paper is the result of an attempt to obtain the interleaving guarantee for the sparse Cech complex of Cavanna et al (2015) without using the Nerve Theorem
The rationale for this was to generalize the result from convex metric spaces to arbitrary metric spaces
We show how Theorem 1 is a consequence of Theorem 2 and how the Sparse Cech complex (Cavanna et al 2015) fits into this context
Summary
This paper is the result of an attempt to obtain the interleaving guarantee for the sparse Cech complex of Cavanna et al (2015) without using the Nerve Theorem. The filtered clique complex of a finite weighted undirected simple graph G = (V , w), where w is a function w : G × G → [0, ∞] is an instance of a Dowker nerve: let P(V ) be the set of subsets of V and define. This is the first step in our proof of Theorem 2.
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