For any fixed c ≥ 1, there exist finite metric spaces whose Vietoris-Rips filtration cannot be c-approximated by any finitely presented construction of linear size; for c < √2, exponential size is required.
arXiv:2405.15592 [math]
2 Pith papers cite this work. Polarity classification is still indexing.
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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.
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math.AT 2representative citing papers
Three new proofs of Dowker duality are presented using poset fiber lemmas, along with a generalization showing that homologies of simplicial complexes and relational complexes form a long exact sequence.
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Lower Bounds for Approximating the Vietoris-Rips Filtration
For any fixed c ≥ 1, there exist finite metric spaces whose Vietoris-Rips filtration cannot be c-approximated by any finitely presented construction of linear size; for c < √2, exponential size is required.
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Short, new proofs of Dowker duality
Three new proofs of Dowker duality are presented using poset fiber lemmas, along with a generalization showing that homologies of simplicial complexes and relational complexes form a long exact sequence.