Morse Theory for the k-NN Distance Function

Authors Yohai Reani , Omer Bobrowski



PDF
Thumbnail PDF

File

LIPIcs.SoCG.2024.75.pdf
  • Filesize: 2.2 MB
  • 16 pages

Document Identifiers

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

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 As Get 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

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. S. Agarwal, Jongwoo Lim, L. Zelnik-Manor, P. Perona, D. Kriegman, and S. Belongie. Beyond pairwise clustering. In 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05), volume 2, pages 838-845 vol. 2, June 2005. Google Scholar
  2. AA Agrachev, D Pallaschke, and S Scholtes. On morse theory for piecewise smooth functions. Journal of Dynamical and Control Systems, 3(4):449-469, 1997. Google Scholar
  3. Siva Athreya, Rahul Roy, and Anish Sarkar. On the coverage of space by random sets. Advances in Applied Probability, 36(1):1-18, 2004. Google Scholar
  4. Omer Bobrowski. Homological connectivity in random čech complexes. Probability Theory and Related Fields, 183(3-4):715-788, 2022. Google Scholar
  5. Omer Bobrowski and Robert J. Adler. Distance functions, critical points, and the topology of random čech complexes. Homology, Homotopy and Applications, 16(2):311-344, 2014. Google Scholar
  6. Omer Bobrowski, Matthias Schulte, and D Yogeshwaran. Poisson process approximation under stabilization and palm coupling. Annales Henri Lebesgue, 5:1489-1534, 2022. Google Scholar
  7. Sung Nok Chiu, Dietrich Stoyan, Wilfrid S. Kendall, and Joseph Mecke. Stochastic geometry and its applications. John Wiley & Sons, 2013. Google Scholar
  8. Frank H Clarke. Generalized gradients and applications. Transactions of the American Mathematical Society, 205:247-262, 1975. Google Scholar
  9. Antonio Cuevas and Alberto Rodríguez-Casal. On boundary estimation. Advances in Applied Probability, pages 340-354, 2004. Google Scholar
  10. Sahibsingh A Dudani. The distance-weighted k-nearest-neighbor rule. IEEE Transactions on Systems, Man, and Cybernetics, 4:325-327, 1976. Google Scholar
  11. Herbert Edelsbrunner, John Harer, et al. Persistent homology - A survey. Contemporary mathematics, 453(26):257-282, 2008. Google Scholar
  12. Herbert Edelsbrunner and Anton Nikitenko. Poisson-delaunay mosaics of order k. Discrete & computational geometry, 62:865-878, 2019. Google Scholar
  13. Herbert Edelsbrunner, Anton Nikitenko, and Georg Osang. A step in the delaunay mosaic of order k. Journal of Geometry, 112:1-14, 2021. Google Scholar
  14. Herbert Edelsbrunner, Anton Nikitenko, and Matthias Reitzner. Expected sizes of PoissonendashDelaunay mosaics and their discrete Morse functions. Advances in Applied Probability, 49(3):745-767, 2017. Google Scholar
  15. Herbert Edelsbrunner and Georg Osang. The multi-cover persistence of euclidean balls. Discrete & Computational Geometry, 65:1296-1313, 2021. Google Scholar
  16. Herbert Edelsbrunner and Raimund Seidel. Voronoi diagrams and arrangements. In Proceedings of the first annual symposium on Computational geometry, pages 251-262, 1985. Google Scholar
  17. Leopold Flatto and Donald J. Newman. Random coverings. Acta Mathematica, 138(1):241-264, 1977. Google Scholar
  18. Vladimir Gershkovich and Hyam Rubinstein. Morse theory for Min-type functions. The Asian Journal of Mathematics, 1(4):696-715, 1997. Google Scholar
  19. Martin Haenggi, Jeffrey G. Andrews, François Baccelli, Olivier Dousse, and Massimo Franceschetti. Stochastic geometry and random graphs for the analysis and design of wireless networks. IEEE Journal on Selected Areas in Communications, 27(7):1029-1046, 2009. Google Scholar
  20. Peter Hall. On the coverage of k-dimensional space by k-dimensional spheres. The Annals of Probability, 13(3):991-1002, 1985. Google Scholar
  21. Allen Hatcher. Algebraic topology. Cambridge University Press, Cambridge, 2002. Google Scholar
  22. Svante Janson. Random coverings in several dimensions. Acta Mathematica, 156(1):83-118, 1986. Google Scholar
  23. Der-Tsai Lee. On k-nearest neighbor voronoi diagrams in the plane. IEEE transactions on computers, 100(6):478-487, 1982. Google Scholar
  24. Roger E Miles. Isotropic random simplices. Advances in Applied Probability, 3(2):353-382, 1971. Google Scholar
  25. John Willard Milnor. Morse theory. Princeton university press, 1963. Google Scholar
  26. Patrick AP Moran and S Fazekas de St Groth. Random circles on a sphere. Biometrika, pages 389-396, 1962. Google Scholar
  27. Mathew D Penrose. Random euclidean coverage from within. Probability Theory and Related Fields, 185(3-4):747-814, 2023. Google Scholar
  28. Yohai Reani and Omer Bobrowski. A coupled alpha complex. Journal of Computational Geometry, 14(1):221-256, 2023. Google Scholar
  29. Rolf Schneider and Wolfgang Weil. Stochastic and integral geometry. Springer Science & Business Media, 2008. Google Scholar
  30. Michael Ian Shamos and Dan Hoey. Closest-point problems. In 16th Annual Symposium on Foundations of Computer Science (sfcs 1975), pages 151-162. IEEE, 1975. Google Scholar
  31. Donald R Sheehy. A multicover nerve for geometric inference. In CCCG, pages 309-314, 2012. Google Scholar
  32. Bang Wang. Coverage problems in sensor networks: A survey. ACM Computing Surveys (CSUR), 43(4):1-53, 2011. Google Scholar
  33. Anatoly Zhigljavsky and Antanasz Zilinskas. Stochastic global optimization, volume 9. Springer Science & Business Media, 2007. Google Scholar
  34. Afra Zomorodian and Gunnar Carlsson. Computing persistent homology. Discrete & Computational Geometry, 33(2):249-274, 2004. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail