Sparse Dowker nerves

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.