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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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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.

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

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