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The Localized Union-Of-Balls Bifiltration

Authors Michael Kerber , Matthias Söls

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ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Algebraic topology
Keywords
  • Topological Data Analysis
  • Multi-Parameter Persistence
  • Persistent Local Homology

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Abstract

We propose an extension of the classical union-of-balls filtration of persistent homology: fixing a point q, we focus our attention to a ball centered at q whose radius is controlled by a second scale parameter. We discuss an absolute variant, where the union is just restricted to the q-ball, and a relative variant where the homology of the q-ball relative to its boundary is considered. Interestingly, these natural constructions lead to bifiltered simplicial complexes which are not k-critical for any finite k. Nevertheless, we demonstrate that these bifiltrations can be computed exactly and efficiently, and we provide a prototypical implementation using the CGAL library. We also argue that some of the recent algorithmic advances for 2-parameter persistence (which usually assume k-criticality for some finite k) carry over to the ∞-critical case.

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Michael Kerber and Matthias Söls. The Localized Union-Of-Balls Bifiltration. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 45:1-45:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.SoCG.2023.45

Author Details

Michael Kerber
  • Institute of Geometry, Technische Universität Graz, Austria
Matthias Söls
  • Institute of Geometry, Technische Universität Graz, Austria

Funding

The authors acknowledge the support of the Austrian Science Fund (FWF).
  • Kerber, Michael: Austrian Science Fund (FWF) grant P 33765-N.
  • Söls, Matthias: Austrian Science Fund (FWF): grant W1230.

Acknowledgements

The authors thank Anton Gfrerrer and Thomas Pock for helpful discussions.

Supplementary Materials

References

  1. Mahmuda Ahmed, Brittany Terese Fasy, and Carola Wenk. Local persistent homology based distance between maps. In Proceedings of the 22nd ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems (ACM SIGSPATIAL GIS 2014), pages 43-52, 2014. URL: https://doi.org/10.1145/2666310.2666390.
  2. Ángel Javier Alonso, Michael Kerber, and Siddharth Pritam. Filtration-domination in bifiltered graphs. In 2023 Proceedings of the Symposium on Algorithm Engineering and Experiments (ALENEX), pages 27-38, 2023. URL: https://doi.org/10.1137/1.9781611977561.ch3.
  3. Franz Aurenhammer, Rolf Klein, and Der-Tsai Lee. Voronoi Diagrams and Delaunay Triangulations. World Scientific, 2013. URL: https://doi.org/10.1142/8685.
  4. Ulrich Bauer, Michael Kerber, Fabian Roll, and Alexander Rolle. A Unified View on the Functorial Nerve Theorem and its Variations, 2022. URL: https://doi.org/10.48550/ARXIV.2203.03571.
  5. 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 (FOCS 2007), pages 536-546, 2007. URL: https://doi.org/10.1109/FOCS.2007.45.
  6. Paul Bendich, Ellen Gasparovic, John Harer, Rauf Izmailov, and Linda Ness. Multi-scale local shape analysis and feature selection in machine learning applications. In 2015 International Joint Conference on Neural Networks (IJCNN), pages 1-8, 2015. URL: https://doi.org/10.1109/IJCNN.2015.7280428.
  7. Paul Bendich, Bei Wang, and Sayan Mukherjee. Local Homology Transfer and Stratification Learning. In Proceedings of the 2012 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1355-1370, 2012. URL: https://doi.org/10.1137/1.9781611973099.107.
  8. Eric Berberich, Michael Hemmer, and Michael Kerber. A generic algebraic kernel for non-linear geometric applications. In Proceedings of the Twenty-Seventh Annual Symposium on Computational Geometry (SoCG 2011), pages 179-186, 2011. URL: https://doi.org/10.1145/1998196.1998224.
  9. Håvard Bakke Bjerkevik and Michael Kerber. Asymptotic Improvements on the Exact Matching Distance for 2-parameter Persistence, 2021. URL: https://doi.org/10.48550/ARXIV.2111.10303.
  10. Andrew J. Blumberg and Michael Lesnick. Stability of 2-Parameter Persistent Homology. Foundations of Computational Mathematics, 2022. URL: https://doi.org/10.1007/s10208-022-09576-6.
  11. Magnus Bakke Botnan, Steffen Oppermann, and Steve Oudot. Signed Barcodes for Multi-Parameter Persistence via Rank Decompositions. In 38th International Symposium on Computational Geometry (SoCG 2022), pages 19:1-19:18, 2022. URL: https://doi.org/10.4230/LIPIcs.SoCG.2022.19.
  12. Hervé Brönnimann, Andreas Fabri, Geert-Jan Giezeman, Susan Hert, Michael Hoffmann, Lutz Kettner, Sylvain Pion, and Stefan Schirra. 2D and 3D linear geometry kernel. In CGAL User and Reference Manual. CGAL Editorial Board, 5.5.1 edition, 2022. URL: https://doc.cgal.org/5.5.1/Manual/packages.html#PkgKernel23.
  13. Gunnar E. Carlsson and Afra Zomorodian. The Theory of Multidimensional Persistence. Discrete & Computational Geometry, 42:71-93, 2009. URL: https://doi.org/10.1007/s00454-009-9176-0.
  14. Frédéric Chazal, David Cohen-Steiner, Marc Glisse, Leonidas J. Guibas, and Steve Oudot. Proximity of Persistence Modules and Their Diagrams. In Proceedings of the Twenty-Fifth Annual Symposium on Computational Geometry (SoCG 2009), pages 237-246, 2009. URL: https://doi.org/10.1145/1542362.1542407.
  15. Frédéric Chazal and Steve Oudot. Towards Persistence-Based Reconstruction in Euclidean Spaces. In Proceedings of the Twenty-Fourth Annual Symposium on Computational Geometry (SoCG 2008), pages 232-241, 2008. URL: https://doi.org/10.1145/1377676.1377719.
  16. David Cohen-Steiner, Herbert Edelsbrunner, and John Harer. Extending Persistence Using Poincaré and Lefschetz Duality. Foundations of Computational Mathematics, 9:79-103, 2009. URL: https://doi.org/10.1007/s10208-008-9027-z.
  17. David Cohen-Steiner, Herbert Edelsbrunner, and Dmitriy Morozov. Vines and Vineyards by Updating Persistence in Linear Time. In Proceedings of the Twenty-Second Annual Symposium on Computational Geometry (SoCG 2006), pages 119-126, 2006. URL: https://doi.org/10.1145/1137856.1137877.
  18. René Corbet, Michael Kerber, Michael Lesnick, and Georg Osang. Computing the Multicover Bifiltration. Discrete & Computational Geometry, 2023. URL: https://doi.org/10.1007/s00454-022-00476-8.
  19. René Corbet, Ulderico Fugacci, Michael Kerber, Claudia Landi, and Bei Wang. A kernel for multi-parameter persistent homology. Computers & Graphics: X, 2, 2019. URL: https://doi.org/10.1016/j.cagx.2019.100005.
  20. Mark de Berg, Otfried Cheong, Marc J. van Kreveld, and Mark H. Overmars. Computational Geometry: Algorithms and Applications, 3rd Edition. Springer, 2008. Google Scholar
  21. Tamal Dey, Fengtao Fan, and Yusu Wang. Dimension Detection with Local Homology. In Proceedings of the 26th Canadian Conference on Computational Geometry (CCCG 2014), 2014. URL: http://www.cccg.ca/proceedings/2014/papers/paper40.pdf.
  22. Tamal K. Dey and Cheng Xin. Generalized persistence algorithm for decomposing multiparameter persistence modules. Journal of Applied and Computational Topology, 6:271-322, 2022. URL: https://doi.org/10.1007/s41468-022-00087-5.
  23. Herbert Edelsbrunner and John Harer. Computational Topology: An Introduction. American Mathematical Society, 2010. Google Scholar
  24. Herbert Edelsbrunner and Georg Osang. The Multi-Cover Persistence of Euclidean Balls. Discrete & Computational Geometry, 65:1296-1313, 2021. URL: https://doi.org/10.1007/s00454-021-00281-9.
  25. Brittany Terese Fasy and Bei Wang. Exploring persistent local homology in topological data analysis. In 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 6430-6434, 2016. URL: https://doi.org/10.1109/ICASSP.2016.7472915.
  26. Ulderico Fugacci, Michael Kerber, and Alexander Rolle. Compression for 2-parameter persistent homology. Computational Geometry, 109, 2023. URL: https://doi.org/10.1016/j.comgeo.2022.101940.
  27. Allen Hatcher. Algebraic Topology. Cambridge University Press, 2005. Google Scholar
  28. Clément Jamin, Sylvain Pion, and Monique Teillaud. 3D triangulations. In CGAL User and Reference Manual. CGAL Editorial Board, 5.5.1 edition, 2022. URL: https://doc.cgal.org/5.5.1/Manual/packages.html#PkgTriangulation3.
  29. Michael Kerber, Michael Lesnick, and Steve Oudot. Exact Computation of the Matching Distance on 2-Parameter Persistence Modules. In 35th International Symposium on Computational Geometry (SoCG 2019), pages 46:1-46:15, 2019. URL: https://doi.org/10.4230/LIPIcs.SoCG.2019.46.
  30. Michael Kerber and Arnur Nigmetov. Efficient Approximation of the Matching Distance for 2-Parameter Persistence. In 36th International Symposium on Computational Geometry (SoCG 2020), pages 53:1-53:16, 2020. URL: https://doi.org/10.4230/LIPIcs.SoCG.2020.53.
  31. Michael Lesnick. The Theory of the Interleaving Distance on Multidimensional Persistence Modules. Foundations of Computational Mathematics, 15:613-650, 2015. URL: https://doi.org/10.1007/s10208-015-9255-y.
  32. Michael Lesnick and Matthew Wright. Interactive Visualization of 2-D Persistence Modules, 2015. URL: https://doi.org/10.48550/ARXIV.1512.00180.
  33. Michael Lesnick and Matthew Wright. Computing Minimal Presentations and Bigraded Betti Numbers of 2-Parameter Persistent Homology. SIAM Journal on Applied Algebra and Geometry, 6:267-298, 2022. URL: https://doi.org/10.1137/20M1388425.
  34. Ezra Miller. Data structures for real multiparameter persistence modules, 2017. URL: https://doi.org/10.48550/ARXIV.1709.08155.
  35. Ezra Miller. Homological algebra of modules over posets, 2020. URL: https://doi.org/10.48550/ARXIV.2008.00063.
  36. Alexander Rolle. The Degree-Rips Complexes of an Annulus with Outliers. In 38th International Symposium on Computational Geometry (SoCG 2022), pages 58:1-58:14, 2022. URL: https://doi.org/10.4230/LIPIcs.SoCG.2022.58.
  37. Alexander Rolle and Luis Scoccola. Stable and consistent density-based clustering, 2020. URL: https://doi.org/10.48550/ARXIV.2005.09048.
  38. Primoz Skraba and Bei Wang. Approximating Local Homology from Samples. In Proceedings of the 2014 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 174-192, 2014. URL: https://doi.org/10.1137/1.9781611973402.13.
  39. Bernadette J. Stolz. Outlier-robust subsampling techniques for persistent homology, 2021. URL: https://doi.org/10.48550/ARXIV.2103.14743.
  40. Julius von Rohrscheidt and Bastian Rieck. TOAST: Topological Algorithm for Singularity Tracking, 2022. URL: https://doi.org/10.48550/ARXIV.2210.00069.
  41. Bei Wang, Brian Summa, Valerio Pascucci, and Mikael Vejdemo-Johansson. Branching and Circular Features in High Dimensional Data. IEEE Transactions on Visualization and Computer Graphics, 17:1902-11, 2011. URL: https://doi.org/10.1109/TVCG.2011.177.
  42. Matthew Wheeler, Jose Bouza, and Peter Bubenik. Activation Landscapes as a Topological Summary of Neural Network Performance. In 2021 IEEE International Conference on Big Data (Big Data), pages 3865-3870, 2021. URL: https://doi.org/10.1109/BigData52589.2021.9671368.
  43. Mariette Yvinec. 2D triangulations. In CGAL User and Reference Manual. CGAL Editorial Board, 5.5.1 edition, 2022. URL: https://doc.cgal.org/5.5.1/Manual/packages.html#PkgTriangulation2.
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