Density Sensitive Bifiltered Dowker Complexes via Total Weight

TL;DR

This paper develops density-sensitive bifiltrations for Dowker complexes, integrating multicover filtrations and stability results, leading to more robust topological data analysis methods with practical algorithms.

Contribution

It introduces a novel density-sensitive bifiltration framework for Dowker complexes, combining multicover filtrations, Dowker duality, and stability theorems, with algorithms and applications.

Findings

Established a Dowker duality compatible with the new bifiltration.

Proved a density-sensitive stability theorem for probability measures.

Provided algorithms and computational examples demonstrating the approach.

Abstract

In this paper, we introduce new density-sensitive bifiltrations for data using the framework of Dowker complexes. Previously, Dowker complexes were studied to address directional or bivariate data whereas density-sensitive bifiltrations on \v{C}ech and Vietoris--Rips complexes were suggested to make them more robust, while increasing computational complexity. We combine these two lines of research, noting that the superlevels of the total weight function of a Dowker complex can be identified as an instance of Sheehy's multicover filtration. We prove a version of Dowker duality that is compatible with this filtration and show that it corresponds to the multicover nerve theorem. As a consequence, we find that the subdivision intrinsic \v{C}ech complex admits a smaller model. Moreover, regarding the total weight function as a counting measure, we generalize it to arbitrary measures and prove a density-sensitive stability theorem for the case of probability measures. As an application, we propose a robust landmark-based bifiltration which approximates the multicover bifiltration. Additionally, we provide an algorithm to calculate the appearances of simplices in our bifiltration and present computational examples.