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

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

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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

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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.

Cite As Get 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

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

Funding

  • Kim, Jisu: Partially supported by Samsung Scholarship.

Acknowledgements

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

Supplementary Materials

References

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