Density Sensitive Bifiltered Dowker Complexes via Total Weight
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arXiv
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| Format: | Preprint |
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2024
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| author | Hellmer, Niklas Spaliński, Jan |
| author_facet | Hellmer, Niklas Spaliński, Jan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_15592 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Density Sensitive Bifiltered Dowker Complexes via Total Weight Hellmer, Niklas Spaliński, Jan Algebraic Topology 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. |
| title | Density Sensitive Bifiltered Dowker Complexes via Total Weight |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2405.15592 |