Geometriae Dedicata

, Volume 173, Issue 1, pp 193–214 | Cite as

Persistence stability for geometric complexes

  • Frédéric Chazal
  • Vin de Silva
  • Steve Oudot
Original Paper


In this paper we study the properties of the homology of different geometric filtered complexes (such as Vietoris–Rips, Čech and witness complexes) built on top of totally bounded metric spaces. Using recent developments in the theory of topological persistence, we provide simple and natural proofs of the stability of the persistent homology of such complexes with respect to the Gromov–Hausdorff distance. We also exhibit a few noteworthy properties of the homology of the Rips and Čech complexes built on top of compact spaces.


Persistent homology Vietoris–Rips complex Cech complex  Gromov–Hausdorff distance 

Mathematics Subject Classification (2000)

51F99 55N99 



The authors are grateful to Steve Smale for fruitful discussions that motivated the results Sect. 5.2, and to J.-M. Droz for suggesting the idea of the proof of Proposition 5.9.


  1. 1.
    Attali, D., Lieutier, A., Salinas, D.: Vietoris–Rips complexes also provide topologically correct reconstructions of sampled shapes. In: Proceedings of the 27th Annual ACM Symposium on Computational geometry, SoCG ’11, pp. 491–500. ACM, New York, NY, USA (2011). doi: 10.1145/1998196.1998276
  2. 2.
    Bartholdi, L., Schick, T., Smale, N., Smale, S., Baker, A.W.: Hodge theory on metric spaces. Found. Comput. Math. 12(1), 1–48 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry, Graduate Studies in Mathematics, vol. 33. American Mathematical Society, Providence, RI (2001)Google Scholar
  4. 4.
    Chazal, F., Cohen-Steiner, D., Glisse, M., Guibas, L., Oudot, S.: Proximity of persistence modules and their diagrams. In: SCG, pp. 237–246 (2009). doi: 10.1145/1542362.1542407
  5. 5.
    Chazal, F., Cohen-Steiner, D., Guibas, L.J., Mémoli, F., Oudot, S.Y.: Gromov–Hausdorff stable signatures for shapes using persistence. Computer Graphics Forum (Proceedings of the SGP 2009) pp. 1393–1403 (2009)Google Scholar
  6. 6.
    Chazal, F., Oudot, S.Y.: Towards persistence-based reconstruction in euclidean spaces. In: Proceedings of the Twenty-Fourth Annual Symposium on Computational Geometry, SCG ’08, pp. 232–241. ACM, New York, NY, USA (2008). doi: 10.1145/1377676.1377719
  7. 7.
    Chazal, F., de Silva, V., Glisse, M., Oudot, S.: The structure and stability of persistence modules (2012). ArXiv:1207.3674 [math.AT]Google Scholar
  8. 8.
    Dowker, C.H.: Homology groups of relations. Ann. Math. 56(1), 84–95 (1952)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Droz, J.M.: A subset of Euclidean space with large Vietoris–Rips homology (2012). ArXiv:1210.4097 [math.GT]Google Scholar
  10. 10.
    Edelsbrunner, H., Harer, J.: Computational Topology: An Introduction. American Mathematical Society, Providence, RI (2010)Google Scholar
  11. 11.
    Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. Discret. Comput. Geom. 28, 511–533 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Ghys, E., de la Harpe, P.: Sur les groupes hyperboliques d’après Mikhael Gromov, vol. 83. Birkhäuser, Basel (1990)CrossRefzbMATHGoogle Scholar
  13. 13.
    Gromov, M.: Metric Structures for Riemannian and Non-Riemannian Spaces, 2nd edn. Birkhäuser, Basel (2007)zbMATHGoogle Scholar
  14. 14.
    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge, MA (2001).
  15. 15.
    Hausmann, J.C.: On the Vietoris–Rips complexes and a cohomology theory for metric spaces. Ann. Math. Stud. 138, 175–188 (1995)MathSciNetGoogle Scholar
  16. 16.
    Latschev, J.: Vietoris–Rips complexes of metric spaces near a closed Riemannian manifold. Archiv der Mathematik 77(6), 522–528 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Munkres, J.R.: Elements of Algebraic Topology. Westview Press, Boulder, CO (1984)zbMATHGoogle Scholar
  18. 18.
    Zomorodian, A., Carlsson, G.: Computing persistent homology. Discret. Comput. Geom. 33(2), 249–274 (2005)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.INRIA Saclay, Ile-de-FrancePalaiseauFrance
  2. 2.Pomona CollegeClaremontUSA

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