Homotopy Reconstruction via the Cech Complex and the Vietoris-Rips Complex

Authors Jisu Kim, Jaehyeok Shin, Frédéric Chazal, Alessandro Rinaldo, Larry Wasserman



PDF
Thumbnail PDF

File

LIPIcs.SoCG.2020.54.pdf
  • Filesize: 0.66 MB
  • 19 pages

Document Identifiers

Author Details

Jisu Kim
  • Inria Saclay - Île-de-France, Palaiseau, France
Jaehyeok Shin
  • Department of Statistics & Data Science, Carnegie Mellon University, Pittsburgh, PA, USA
Frédéric Chazal
  • Inria Saclay - Île-de-France, Palaiseau, France
Alessandro Rinaldo
  • Department of Statistics & Data Science, Carnegie Mellon University, Pittsburgh, PA, USA
Larry Wasserman
  • Department of Statistics & Data Science, Machine Learning Department, Carnegie Mellon University, Pittsburgh, PA, USA

Acknowledgements

We want to thank André Lieutier and Henry Adams for the thoughtful discussions and comments.

Cite AsGet BibTex

Jisu Kim, Jaehyeok Shin, Frédéric Chazal, Alessandro Rinaldo, and Larry Wasserman. Homotopy Reconstruction via the Cech Complex and the Vietoris-Rips Complex. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 54:1-54:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.SoCG.2020.54

Abstract

We derive conditions under which the reconstruction of a target space is topologically correct via the Čech complex or the Vietoris-Rips complex obtained from possibly noisy point cloud data. We provide two novel theoretical results. First, we describe sufficient conditions under which any non-empty intersection of finitely many Euclidean balls intersected with a positive reach set is contractible, so that the Nerve theorem applies for the restricted Čech complex. Second, we demonstrate the homotopy equivalence of a positive μ-reach set and its offsets. Applying these results to the restricted Čech complex and using the interleaving relations with the Čech complex (or the Vietoris-Rips complex), we formulate conditions guaranteeing that the target space is homotopy equivalent to the Čech complex (or the Vietoris-Rips complex), in terms of the μ-reach. Our results sharpen existing results.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Algebraic topology
  • Theory of computation → Computational geometry
Keywords
  • Computational topology
  • Homotopy reconstruction
  • Homotopy Equivalence
  • Vietoris-Rips complex
  • Čech complex
  • Reach
  • μ-reach
  • Nerve Theorem
  • Offset
  • Double offset
  • Consistency

Metrics

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

References

  1. Eddie Aamari, Jisu Kim, Frédéric Chazal, Bertrand Michel, Alessandro Rinaldo, and Larry Wasserman. Estimating the reach of a manifold. Electronic Journal of Statistics, 13(1):1359-1399, 2019. URL: https://doi.org/10.1214/19-EJS1551.
  2. Michał Adamaszek and Henry Adams. The Vietoris-Rips complexes of a circle. Pacific Journal of Mathematics, 290(1):1-40, 2017. URL: https://doi.org/10.2140/pjm.2017.290.1.
  3. Michał Adamaszek, Henry Adams, and Florian Frick. Metric reconstruction via optimal transport. SIAM Journal on Applied Algebra and Geometry, 2(4):597-619, 2018. URL: https://doi.org/10.1137/17M1148025.
  4. Henry Adams and Joshua Mirth. Metric thickenings of euclidean submanifolds. Topology and its Applications, 254:69-84, 2019. URL: https://doi.org/10.1016/j.topol.2018.12.014.
  5. Paul Alexandroff. Über den allgemeinen Dimensionsbegriff und seine Beziehungen zur elementaren geometrischen Anschauung. Mathematische Annalen, 98(1):617-635, 1928. URL: https://doi.org/10.1007/BF01451612.
  6. Dominique Attali, André Lieutier, and David Salinas. Vietoris–rips complexes also provide topologically correct reconstructions of sampled shapes. Computational Geometry, 46(4):448-465, 2013. 27th Annual Symposium on Computational Geometry (SoCG 2011). URL: https://doi.org/10.1016/j.comgeo.2012.02.009.
  7. Anders Björner. Topological Methods, page 1819–1872. MIT Press, Cambridge, MA, USA, 1996. Google Scholar
  8. Frédéric Chazal, David Cohen-Steiner, and André Lieutier. A sampling theory for compact sets in euclidean space. Discrete & Computational Geometry, 41(3):461-479, 2009. URL: https://doi.org/10.1007/s00454-009-9144-8.
  9. Frédéric Chazal, David Cohen-Steiner, André Lieutier, and Boris Thibert. Shape smoothing using double offsets. In Proceedings of the 2007 ACM Symposium on Solid and Physical Modeling, SPM ’07, page 183–192, New York, NY, USA, 2007. Association for Computing Machinery. URL: https://doi.org/10.1145/1236246.1236273.
  10. Frédéric Chazal, Brittany Fasy, Fabrizio Lecci, Bertr, Michel, Aless, ro Rinaldo, and Larry Wasserman. Robust topological inference: Distance to a measure and kernel distance. Journal of Machine Learning Research, 18(159):1-40, 2018. URL: http://jmlr.org/papers/v18/15-484.html.
  11. Frédéric Chazal and André Lieutier. Weak feature size and persistent homology: Computing homology of solids in ℝⁿ from noisy data samples. In Proceedings of the Twenty-First Annual Symposium on Computational Geometry, SCG ’05, page 255–262, New York, NY, USA, 2005. Association for Computing Machinery. URL: https://doi.org/10.1145/1064092.1064132.
  12. Frédéric Chazal and André Lieutier. Smooth manifold reconstruction from noisy and non-uniform approximation with guarantees. Computational Geometry, 40(2):156-170, 2008. URL: https://doi.org/10.1016/j.comgeo.2007.07.001.
  13. Frédéric Chazal and Steve Yann Oudot. Towards persistence-based reconstruction in euclidean spaces. In Proceedings of the Twenty-Fourth Annual Symposium on Computational Geometry, SCG ’08, page 232–241, New York, NY, USA, 2008. Association for Computing Machinery. URL: https://doi.org/10.1145/1377676.1377719.
  14. Antonio Cuevas. Set estimation: another bridge between statistics and geometry. Boletín de Estadística e Investigación Operativa. BEIO, 25(2):71-85, 2009. Google Scholar
  15. Antonio Cuevas and Alberto Rodríguez-Casal. On boundary estimation. Advances in Applied Probability, 36(2):340-354, 2004. URL: https://doi.org/10.1239/aap/1086957575.
  16. Vin de Silva and Robert Ghrist. Coverage in sensor networks via persistent homology. Algebraic & Geometric Topology, 7:339-358, 2007. URL: https://doi.org/10.2140/agt.2007.7.339.
  17. Tamal K. Dey. Curve and Surface Reconstruction: Algorithms with Mathematical Analysis. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, 2006. URL: https://doi.org/10.1017/CBO9780511546860.002.
  18. Jürgen Eckhoff. Chapter 2.1 - Helly, Radon, and Carathéodory type theorems. In P.M. GRUBER and J.M. WILLS, editors, Handbook of Convex Geometry, pages 389-448. North-Holland, Amsterdam, 1993. URL: https://doi.org/10.1016/B978-0-444-89596-7.50017-1.
  19. Herbert Edelsbrunner and John Harer. Computational Topology: An Introduction. American Mathematical Society, 2010. URL: http://www.ams.org/bookstore-getitem/item=MBK-69.
  20. Herbert Edelsbrunner and Ernst P. Mücke. Three-dimensional alpha shapes. ACM Transactions on Graphics, 13(1):43-72, January 1994. URL: https://doi.org/10.1145/174462.156635.
  21. Herbert Federer. Curvature measures. Transactions of the American Mathematical Society, 93:418-491, 1959. URL: https://doi.org/10.2307/1993504.
  22. Karsten Grove. Critical point theory for distance functions. In Differential geometry: Riemannian geometry (Los Angeles, CA, 1990), volume 54 of Proc. Sympos. Pure Math., pages 357-385. Amer. Math. Soc., Providence, RI, 1993. Google Scholar
  23. Allen Hatcher. Algebraic topology. Cambridge University Press, Cambridge, 2002. Google Scholar
  24. Jean-Claude Hausmann. On the Vietoris-Rips complexes and a cohomology theory for metric spaces. In Prospects in topology (Princeton, NJ, 1994), volume 138 of Annals of Mathematics Studies, pages 175-188. Princeton Univ. Press, Princeton, NJ, 1995. Google Scholar
  25. Janko Latschev. Vietoris-Rips complexes of metric spaces near a closed Riemannian manifold. Archiv der Mathematik, 77(6):522-528, 2001. URL: https://doi.org/10.1007/PL00000526.
  26. John M. Lee. Introduction to smooth manifolds, volume 218 of Graduate Texts in Mathematics. Springer, New York, second edition, 2013. Google Scholar
  27. Partha Niyogi, Stephen Smale, and Shmuel Weinberger. Finding the homology of submanifolds with high confidence from random samples. Discrete & Computational Geometry, 39(1-3):419-441, 2008. URL: https://doi.org/10.1007/s00454-008-9053-2.
  28. Vanessa Robins, James D. Meiss, and Elizabeth Bradley. Computing connectedness: disconnectedness and discreteness. Physica D: Nonlinear Phenomena, 139(3):276-300, 2000. URL: https://doi.org/10.1016/S0167-2789(99)00228-6.
  29. Amit Singer and Hau-Tieng Wu. Vector diffusion maps and the connection laplacian. Communications on Pure and Applied Mathematics, 65(8):1067-1144, 2012. URL: https://doi.org/10.1002/cpa.21395.
  30. Jean-Luc Starck, Vicent Martínez, D. Donoho, Ofer Levi, Philippe Querre, and Enn Saar. Analysis of the spatial distribution of galaxies by multiscale methods. EURASIP Journal on Applied Signal Processing, 2005(15):2455-2469, 2005. URL: https://doi.org/10.1155/ASP.2005.2455.
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