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The Degree-Rips Complexes of an Annulus with Outliers

Author Alexander Rolle

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Alexander Rolle
  • Department of Mathematics, Technische Universität München, Germany

Acknowledgements

I would like to thank Michael Lesnick for helpful conversations about robustness of degree-Rips, and Luis Scoccola and Fabian Roll for various helpful conversations about topics related to this paper. I would also like to thank the reviewers for their constructive comments.

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Alexander Rolle. The Degree-Rips Complexes of an Annulus with Outliers. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 58:1-58:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.SoCG.2022.58

Abstract

The degree-Rips bifiltration is the most computable of the parameter-free, density-sensitive bifiltrations in topological data analysis. It is known that this construction is stable to small perturbations of the input data, but its robustness to outliers is not well understood. In recent work, Blumberg-Lesnick prove a result in this direction using the Prokhorov distance and homotopy interleavings. Based on experimental evaluation, they argue that a more refined approach is desirable, and suggest the framework of homology inference. Motivated by these experiments, we consider a probability measure that is uniform with high density on an annulus, and uniform with low density on the disc inside the annulus. We compute the degree-Rips complexes of this probability space up to homotopy type, using the Adamaszek-Adams computation of the Vietoris-Rips complexes of the circle. These degree-Rips complexes are the limit objects for the Blumberg-Lesnick experiments. We argue that the homology inference approach has strong explanatory power in this case, and suggest studying the limit objects directly as a strategy for further work.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • multi-parameter persistent homology
  • stability
  • homology inference

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References

  1. Michał Adamaszek and Henry Adams. The Vietoris-Rips complexes of a circle. Pac. J. Math, 290(1):1-40, 2017. URL: https://doi.org/10.2140/pjm.2017.290.1.
  2. 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.
  3. Paul Bendich, David Cohen-Steiner, Herbert Edelsbrunner, John Harer, and Dmitriy Morozov. Inferring local homology from sampled stratified spaces. In 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS2007), October 20-23, 2007, Providence, RI, USA, Proceedings, pages 536-546. IEEE Computer Society, 2007. URL: https://doi.org/10.1109/FOCS.2007.33.
  4. Andrew J. Blumberg and Michael Lesnick. Stability of 2-parameter persistent homology, 2020. URL: http://arxiv.org/abs/2010.09628.
  5. Omer Bobrowski, Sayan Mukherjee, and Jonathan E. Taylor. Topological consistency via kernel estimation. Bernoulli, 23(1):288-328, 2017. URL: https://doi.org/10.3150/15-BEJ744.
  6. Jean-Daniel Boissonnat, Frédéric Chazal, and Mariette Yvinec. Geometric and Topological Inference. Cambridge Texts in Applied Mathematics. Cambridge University Press, 2018. URL: https://doi.org/10.1017/9781108297806.
  7. Ricardo J. G. B. Campello, Davoud Moulavi, and Jörg Sander. Density-based clustering based on hierarchical density estimates. In Advances in Knowledge Discovery and Data Mining, volume 7819 of Lecture Notes in Computer Science, pages 160-172. Springer, 2013. Google Scholar
  8. Kamalika Chaudhuri and Sanjoy Dasgupta. Rates of convergence for the cluster tree. In J. D. Lafferty, C. K. I. Williams, J. Shawe-Taylor, R. S. Zemel, and A. Culotta, editors, Advances in Neural Information Processing Systems 23, pages 343-351. Curran Associates, Inc., 2010. Google Scholar
  9. Frédéric Chazal, Leonidas J. Guibas, Steve Y. Oudot, and Primoz Skraba. Persistence-based clustering in Riemannian manifolds. J. ACM, 60(6), November 2013. URL: https://doi.org/10.1145/2535927.
  10. Antonio Cuevas, Manuel Febrero, and Ricardo Fraiman. Estimating the number of clusters. Canadian Journal of Statistics, 28:367-382, 2000. Google Scholar
  11. Martin Ester, Hans-Peter Kriegel, Jörg Sander, and Xiaowei Xu. A density-based algorithm for discovering clusters in large spatial databases with noise. In KDD'96: Proceedings of the Second International Conference on Knowledge Discovery and Data Mining, pages 226-231. AAAI Press, 1996. Google Scholar
  12. Jean-Claude Hausmann. On the Vietoris-Rips complexes and a cohomology theory for metric spaces. In Prospects in topology. Proceedings of a conference in honor of William Browder, Princeton, NJ, USA, March 1994, pages 175-188. Princeton, NJ: Princeton University Press, 1995. Google Scholar
  13. J. F. Jardine. Stable components and layers. Canad. Math. Bull., pages 1-15, 2019. URL: https://doi.org/10.4153/S000843951900064X.
  14. J. F. Jardine. Persistent homotopy theory, 2020. URL: http://arxiv.org/abs/2002.10013.
  15. Michael Lesnick and Matthew Wright. Interactive visualization of 2-D persistence modules, 2015. URL: http://arxiv.org/abs/1512.00180.
  16. Leland McInnes and John Healy. Accelerated hierarchical density based clustering. In 2017 IEEE International Conference on Data Mining Workshops (ICDMW), volume 00, pages 33-42, November 2018. URL: https://doi.org/10.1109/ICDMW.2017.12.
  17. Partha Niyogi, Stephen Smale, and Shmuel Weinberger. Finding the homology of submanifolds with high confidence from random samples. Discrete Comput. Geom., 39:419-441, 2008. URL: https://doi.org/10.1007/s00454-008-9053-2.
  18. Steve Y. Oudot and Donald R. Sheehy. Zigzag zoology: Rips zigzags for homology inference. Found Comput Math, 15(5):1151-1186, 2015. URL: https://doi.org/10.1007/s10208-014-9219-7.
  19. Alessandro Rinaldo and Larry Wasserman. Generalized density clustering. Ann. Statist., 38(5):2678-2722, October 2010. URL: https://doi.org/10.1214/10-AOS797.
  20. Alexander Rolle and Luis Scoccola. Stable and consistent density-based clustering, 2020. URL: http://arxiv.org/abs/2005.09048.
  21. Luis Scoccola. Locally persistent categories and metric properties of interleaving distances. Electronic Thesis and Dissertation Repository. https://ir.lib.uwo.ca/etd/7119, 2020. Google Scholar
  22. The RIVET Developers. RIVET. 1.1.0, 2020. URL: https://github.com/rivetTDA/rivet/.
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