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DOI: 10.4230/LIPIcs.SoCG.2024.75

Morse Theory for the k-NN Distance Function

Authors: Yohai Reani and Omer Bobrowski

Published in: LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Abstract

We study the k-th nearest neighbor distance function from a finite point-set in ℝ^d. We provide a Morse theoretic framework to analyze the sub-level set topology. In particular, we present a simple combinatorial-geometric characterization for critical points and their indices, along with detailed information about the possible changes in homology at the critical levels. We conclude by computing the expected number of critical points for a homogeneous Poisson process. Our results deliver significant insights and tools for the analysis of persistent homology in order-k Delaunay mosaics, and random k-fold coverage.

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Yohai Reani and Omer Bobrowski. Morse Theory for the k-NN Distance Function. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 75:1-75:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{reani_et_al:LIPIcs.SoCG.2024.75,
  author =	{Reani, Yohai and Bobrowski, Omer},
  title =	{{Morse Theory for the k-NN Distance Function}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{75:1--75:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.75},
  URN =		{urn:nbn:de:0030-drops-200202},
  doi =		{10.4230/LIPIcs.SoCG.2024.75},
  annote =	{Keywords: Applied topology, Morse theory, Distance function, k-nearest neighbor}
}

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Morse Theory for the k-NN Distance Function

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