Density Sensitive Bifiltered Dowker Complexes via Total Weight

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 Čech 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 Čech 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.

40 pages, 12 figures. Expanded results and improved exposition. Code available at https://github.com/nihell/pyDowker/