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The Ultimate Frontier: An Optimality Construction for Homotopy Inference (Media Exposition)

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

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LIPIcs.SoCG.2024.87.pdf
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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Homotopy
  • Inference
  • Sets of positive reach

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Abstract

In our companion paper "Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of Euclidean spaces and of Riemannian manifolds" we gave optimal bounds (in terms of the two one-sided Hausdorff distances) on a sample P of an input shape 𝒮 (either manifold or general set with positive reach) such that one can infer the homotopy of 𝒮 from the union of balls with some radius centred at P, both in Euclidean space and in a Riemannian manifold of bounded curvature. The construction showing the optimality of the bounds is not straightforward. The purpose of this video is to visualize and thus elucidate said construction in the Euclidean setting.

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Dominique Attali, Hana Dal Poz Kouřimská, Christopher Fillmore, Ishika Ghosh, André Lieutier, Elizabeth Stephenson, and Mathijs Wintraecken. The Ultimate Frontier: An Optimality Construction for Homotopy Inference (Media Exposition). In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 87:1-87:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SoCG.2024.87

Author Details

Dominique Attali
  • Université 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.

Supplementary Materials

References

  1. Dominique Attali, Hana Dal Poz Kouřimská, Christopher Fillmore, Ishika Ghosh, André Lieutier, Elizabeth Stephenson, and Mathijs Wintraecken. Supplementary material: The ultimate frontier: An optimality construction for homotopy inference. HAL preprint, 2024. URL: https://hal.science/hal-04501285.
  2. 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. arXiv preprint, accepted for SoCG 2024, 2024. URL: https://arxiv.org/abs/2206.10485.
  3. Dominique Attali, André Lieutier, and David Salinas. Vietoris-rips complexes also provide topologically correct reconstructions of sampled shapes. Computational Geometry, 46(4):448-465, 2013. 27th Annual Symposium on Computational Geometry (SoCG 2011). URL: https://doi.org/10.1016/j.comgeo.2012.02.009.
  4. F. Chazal, D. Cohen-Steiner, and A. Lieutier. A sampling theory for compact sets in Euclidean space. Discrete and Computational Geometry, 41(3):461-479, 2009. Google Scholar
  5. Jisu Kim, Jaehyeok Shin, Frédéric Chazal, Alessandro Rinaldo, and Larry Wasserman. Homotopy Reconstruction via the Cech Complex and the Vietoris-Rips Complex. In Sergio Cabello and Danny Z. Chen, editors, 36th International Symposium on Computational Geometry (SoCG 2020), volume 164 of Leibniz International Proceedings in Informatics (LIPIcs), pages 54:1-54:19, Dagstuhl, Germany, 2020. Schloss Dagstuhl-Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.SoCG.2020.54.
  6. P. Niyogi, S. Smale, and S. Weinberger. Finding the homology of submanifolds with high confidence from random samples. Discrete & Computational Geometry, 39(1-3):419-441, 2008. Google Scholar
  7. Yuan Wang and Bei Wang. Topological inference of manifolds with boundary. Computational Geometry, 88:101606, 2020. URL: https://doi.org/10.1016/j.comgeo.2019.101606.
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