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Tight Bounds for the Learning of Homotopy à la Niyogi, Smale, and Weinberger for Subsets of Euclidean Spaces and of Riemannian Manifolds

Authors Dominique Attali, Hana Dal Poz Kouřimská , Christopher Fillmore , Ishika Ghosh , André Lieutier, Elizabeth Stephenson , Mathijs Wintraecken

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Author Details

Dominique Attali
  • Univ. Grenoble Alpes, CNRS, Grenoble INP, GIPSA-lab, Grenoble, France
Hana Dal Poz Kouřimská
  • IST Austria, Klosterneuburg, Austria
Christopher Fillmore
  • IST Austria, Klosterneuburg, Austria
Ishika Ghosh
  • IST Austria, Klosterneuburg, Austria
  • Michigan State University, East Lansing, MI, USA
André Lieutier
  • Aix-en-Provence, France
Elizabeth Stephenson
  • IST Austria, Klosterneuburg, Austria
Mathijs Wintraecken
  • Inria Sophia Antipolis, Université Côte d'Azur, Sophia Antipolis, France

Funding

This research has been supported by the European Research Council (ERC), grant No. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), grant No. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant No. I 02979-N35.
  • Wintraecken, Mathijs: Supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754411, the Austrian science fund (FWF) grant No. M-3073, and the welcome package from IDEX of the Université Côte d'Azur.

Acknowledgements

We thank Jean-Daniel Boissonnat, Herbert Edelsbrunner, and Mariette Yvinec for discussion.

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Dominique Attali, Hana Dal Poz Kouřimská, Christopher Fillmore, Ishika Ghosh, André Lieutier, Elizabeth Stephenson, and Mathijs Wintraecken. Tight Bounds for the Learning of Homotopy à la Niyogi, Smale, and Weinberger for Subsets of Euclidean Spaces and of Riemannian Manifolds. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 11:1-11:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SoCG.2024.11

Abstract

In this article we extend and strengthen the seminal work by Niyogi, Smale, and Weinberger on the learning of the homotopy type from a sample of an underlying space. In their work, Niyogi, Smale, and Weinberger studied samples of C² manifolds with positive reach embedded in ℝ^d. We extend their results in the following ways:  
- As the ambient space we consider both ℝ^d and Riemannian manifolds with lower bounded sectional curvature. 
- In both types of ambient spaces, we study sets of positive reach - a significantly more general setting than C² manifolds - as well as general manifolds of positive reach. 
- The sample P of a set (or a manifold) 𝒮 of positive reach may be noisy. We work with two one-sided Hausdorff distances - ε and δ - between P and 𝒮. We provide tight bounds in terms of ε and δ, that guarantee that there exists a parameter r such that the union of balls of radius r centred at the sample P deformation-retracts to 𝒮. We exhibit their tightness by an explicit construction. 
We carefully distinguish the roles of δ and ε. This is not only essential to achieve tight bounds, but also sensible in practical situations, since it allows one to adapt the bound according to sample density and the amount of noise present in the sample separately.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Homotopy
  • Inference
  • Sets of positive reach

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