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The distance function to a finite set is a topological Morse function
Published 22 Jul 2024 in math.DG and cs.CG | (2407.15578v1)
Abstract: In this short note, we show that the distance function to any finite set $X\subset \mathbb{R}n$ is a topological Morse function, regardless of whether $X$ is in general position. We also precisely characterize its topological critical points and their indices, and relate them to the differential critical points of the function.
- Stability and computation of medial axes-a state-of-the-art report. Mathematical foundations of scientific visualization, computer graphics, and massive data exploration, pages 109–125, 2009.
- Optimal reach estimation and metric learning. The Annals of Statistics, 51(3):1086–1108, 2023.
- Critical points of the distance function to a generic submanifold. arXiv preprint arXiv:2312.13147, 2023.
- Wasserstein convergence of Čech persistence diagrams for samplings of submanifolds. arXiv preprint arXiv:2406.14919, 2024.
- Estimating the reach of a manifold. Electronic Journal of Statistics, 13(1):1359 – 1399, 2019.
- Distance functions, critical points, and the topology of random Čech complexes. Homology, Homotopy and Applications, 16:311–344, 01 2014.
- The morse theory of Čech and delaunay complexes. Transactions of the American Mathematical Society, 369:1, 06 2016.
- Random Čech complexes on riemannian manifolds. Random Structures & Algorithms, 54(3):373–412, 2019.
- Wrapping Cycles in Delaunay Complexes: Bridging Persistent Homology and Discrete Morse Theory. In 40th International Symposium on Computational Geometry (SoCG 2024), volume 293 of Leibniz International Proceedings in Informatics (LIPIcs), pages 15:1–15:16, 2024.
- On the vanishing of homology in random Čech complexes. Random Structures and Algorithms, 07 2015.
- A sampling theory for compact sets in euclidean space. Discrete & Computational Geometry, 41:461–479, 06 2006.
- The structure and stability of persistence modules, volume 10. Springer, 2016.
- Topology guaranteeing manifold reconstruction using distance function to noisy data. volume 2006, pages 112–118, 06 2006.
- Frank H Clarke. Optimization and nonsmooth analysis. SIAM, 1990.
- G. Carlsson and M. Vejdemo-Johansson. Topological Data Analysis with Applications. Topological Data Analysis with Applications. Cambridge University Press, 2021.
- Random čech complexes on manifolds with boundary. Random Structures & Algorithms, 61:309 – 352, 2019.
- Computational topology: an introduction. American Mathematical Society, 2022.
- Morse theory for min-type functions. The Asian Journal of Mathematics, 1, 01 1997.
- Karsten Grove. Critical point theory for distance functions. Proceedings of Symposia in Pure Mathematics, pages 357–385, 1993.
- H.Th Jongen and D. Pallaschke. On linearization and continuous selections of functions. Optimization, 19(3):343–353, 1988.
- André Lieutier. Any open bounded subset of rn has the same homotopy type as its medial axis. Computer-Aided Design, 36(11):1029–1046, 2004.
- J.W. Milnor. Morse Theory. Annals of mathematics studies. Princeton University Press, 1963.
- Marston Morse. Topologically non-degenerate functions on a compact n-manifold M. Journal d’Analyse Mathématique, 7(1), December 1959.
- Finding the homology of submanifolds with high confidence from random samples. Discrete & Computational Geometry, 39:419–441, 2008.
- A roadmap for the computation of persistent homology. EPJ Data Science, 6:1–38, 2017.
- Morse theory for the k-nn distance function, 2024.
- Ralph Rockafellar. Convex Analysis. Princeton University Press, 1970.
- Dirk Siersma. Voronoi diagrams and morse theory of the distance function, 1996.
- Generalized morse theory of distance functions to surfaces for persistent homology. arXiv preprint arXiv:2306.14716, 2023.
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