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A Sparse Multicover Bifiltration of Linear Size

Author Ángel Javier Alonso

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  • Mathematics of computing → Topology
  • Theory of computation → Computational geometry
Keywords
  • Multicover
  • Approximation
  • Sparsification
  • Multiparameter persistence

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Abstract

The k-cover of a point cloud X of ℝ^d at radius r is the set of all those points within distance r of at least k points of X. By varying r and k we obtain a two-parameter filtration known as the multicover bifiltration. This bifiltration has received attention recently due to being choice-free and robust to outliers. However, it is hard to compute: the smallest known equivalent simplicial bifiltration has O(|X|^{d+1}) simplices. In this paper we introduce a (1+ε)-approximation of the multicover bifiltration of linear size O(|X|), for fixed d and ε. The methods also apply to the subdivision Rips bifiltration on metric spaces of bounded doubling dimension yielding analogous results.

Cite As Get BibTex

Ángel Javier Alonso. A Sparse Multicover Bifiltration of Linear Size. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SoCG.2025.6

Author Details

Ángel Javier Alonso
  • Graz University of Technology, Austria

Funding

This research was funded in whole, or in part, by the Austrian Science Fund (FWF) 10.55776/P33765.

Acknowledgements

The author would like to thank his advisor Michael Kerber for helpful comments. He would also like to deeply thank an anonymous reviewer for many comments that helped to substantially improve the exposition. Part of the writing was carried out while the author was visiting Yasuaki Hiraoka’s group at the Kyoto University Institute of Advanced Study.

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