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Persistence stability for geometric complexes

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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.

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Notes

  1. All vector spaces are taken to be over an arbitrary field \(\mathbf {k}\), fixed throughout this paper.

  2. We use simplicial homology with coefficients in the field \(\mathbf {k}\).

  3. We will usually drop the word ‘intrinsic’ unless we are contrasting it with ‘ambient’.

  4. See [13] chap.1, def 1.2 for a definition of the length of a curve in a metric space.

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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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Authors and Affiliations

  1. INRIA Saclay, Ile-de-France, Palaiseau, France

    Frédéric Chazal & Steve Oudot

  2. Pomona College, Claremont, CA, USA

    Vin de Silva

Authors
  1. Frédéric Chazal
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  2. Vin de Silva
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  3. Steve Oudot
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Corresponding author

Correspondence to Vin de Silva.

Additional information

The authors gratefully acknowledge the following funding sources for this work: Digiteo project C3TTA (including the Digiteo chair held by the second author); European project CG-Learning (EC contract No. 255827); ANR project GIGA (ANR-09-BLAN-0331-01); DARPA project Sensor Topology and Minimal Planning ‘SToMP’ (HR0011-07-1-0002). The second author is a 2013 Simons Fellow, and is supported in part by the Institute for Mathematics and its Applications with funds provided by the National Science Foundation.

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Chazal, F., de Silva, V. & Oudot, S. Persistence stability for geometric complexes. Geom Dedicata 173, 193–214 (2014). https://doi.org/10.1007/s10711-013-9937-z

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