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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

Abstract

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.

Keywords

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

Mathematics Subject Classification (2000)

51F99 55N99 

Notes

Acknowledgments

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.

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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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