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Morse Theory for the k-NN Distance Function

Authors Yohai Reani , Omer Bobrowski

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Author Details

Yohai Reani
  • Viterbi Faculty of Electrical & Computer Engineering, Technion - Israel Institute of Technology, Haifa, Israel
Omer Bobrowski
  • School of Mathematical Sciences, Queen Mary University of London, UK
  • Viterbi Faculty of Electrical & Computer Engineering, Technion - Israel Institute of Technology, Haifa, Israel

Funding

  • Reani, Yohai: The author is partially supported by the Israel Science Foundation grant 2539/17.
  • Bobrowski, Omer: The author is partially supported by the Israel Science Foundation grant 1965/19, and by the EPSRC grant EP/Y008642/1.

Acknowledgements

The authors are grateful to Primoz Skraba, for his feedback and advice. We would also like to thank the anonymous referees for their useful comments and suggestions.

Cite AsGet BibTex

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)
https://doi.org/10.4230/LIPIcs.SoCG.2024.75

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Applied topology
  • Morse theory
  • Distance function
  • k-nearest neighbor

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