Sparse Dowker Nerves

TL;DR

This paper introduces sparse filtered simplicial complexes for persistent homology applicable to point clouds and networks, extending existing methods to arbitrary metric spaces and generalizing metric spaces and networks through Dowker dissimilarities.

Contribution

It extends sparse Čech complexes to arbitrary metric spaces and introduces Dowker dissimilarities, providing a unified framework for analyzing point clouds and networks.

Findings

Extended sparse Čech complexes to arbitrary metric spaces.

Formulated interleaving using strict 2-categories.

Introduced Dowker dissimilarities as a generalization of metrics and networks.

Abstract

We propose sparse versions of filtered simplicial complexes used to compte persistent homology of point clouds and of networks. In particular we extend a slight variation of the Sparse \v{C}ech Complex of Cavanna, Jahanseir and Sheehy from point clouds in Cartesian space 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.