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Sparse Higher Order Čech Filtrations

Authors Mickaël Buchet, Bianca B. Dornelas, Michael Kerber

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ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Computational complexity and cryptography
Keywords
  • Sparsification
  • k-fold cover
  • Higher order Čech complexes

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Abstract

For a finite set of balls of radius r, the k-fold cover is the space covered by at least k balls. Fixing the ball centers and varying the radius, we obtain a nested sequence of spaces that is called the k-fold filtration of the centers. For k = 1, the construction is the union-of-balls filtration that is popular in topological data analysis. For larger k, it yields a cleaner shape reconstruction in the presence of outliers. We contribute a sparsification algorithm to approximate the topology of the k-fold filtration. Our method is a combination and adaptation of several techniques from the well-studied case k = 1, resulting in a sparsification of linear size that can be computed in expected near-linear time with respect to the number of input points.

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Mickaël Buchet, Bianca B. Dornelas, and Michael Kerber. Sparse Higher Order Čech Filtrations. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 20:1-20:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.SoCG.2023.20

Author Details

Mickaël Buchet
  • Institute of Geometry, TU Graz, Austria
Bianca B. Dornelas
  • Institute of Geometry, TU Graz, Austria
Michael Kerber
  • Institute of Geometry, TU Graz, Austria

Funding

This research has been supported by the Austrian Science Fund (FWF), grant numbers P 33765-N and W1230.

Acknowledgements

We thank Alexander Rolle for his valuable input and encouragement of this work.

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