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Sparse Higher Order Čech Filtrations

Authors Mickaël Buchet, Bianca B. Dornelas, Michael Kerber

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ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Computational complexity and cryptography
Keywords
  • Sparsification
  • k-fold cover
  • Higher order Čech complexes

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Abstract

For a finite set of balls of radius r, the k-fold cover is the space covered by at least k balls. Fixing the ball centers and varying the radius, we obtain a nested sequence of spaces that is called the k-fold filtration of the centers. For k = 1, the construction is the union-of-balls filtration that is popular in topological data analysis. For larger k, it yields a cleaner shape reconstruction in the presence of outliers. We contribute a sparsification algorithm to approximate the topology of the k-fold filtration. Our method is a combination and adaptation of several techniques from the well-studied case k = 1, resulting in a sparsification of linear size that can be computed in expected near-linear time with respect to the number of input points.

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Mickaël Buchet, Bianca B. Dornelas, and Michael Kerber. Sparse Higher Order Čech Filtrations. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 20:1-20:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.SoCG.2023.20

Author Details

Mickaël Buchet
  • Institute of Geometry, TU Graz, Austria
Bianca B. Dornelas
  • Institute of Geometry, TU Graz, Austria
Michael Kerber
  • Institute of Geometry, TU Graz, Austria

Funding

This research has been supported by the Austrian Science Fund (FWF), grant numbers P 33765-N and W1230.

Acknowledgements

We thank Alexander Rolle for his valuable input and encouragement of this work.

References

  1. Hirokazu Anai, Frédéric Chazal, Marc Glisse, Yuichi Ike, Hiroya Inakoshi, Raphaël Tinarrage, and Yuhei Umeda. Dtm-based filtrations. In Gill Barequet and Yusu Wang, editors, 35th International Symposium on Computational Geometry, SoCG 2019, June 18-21, 2019, Portland, Oregon, USA, volume 129 of LIPIcs, pages 58:1-58:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.SoCG.2019.58.
  2. Ulrich Bauer, Michael Kerber, Fabian Roll, and Alexander Rolle. A unified view on the functorial nerve theorem and its variations, 2022. URL: https://arxiv.org/abs/2203.03571.
  3. Nello Blaser and Morten Brun. Sparse nerves in practice. In Andreas Holzinger, Peter Kieseberg, A M. Tjoa, and Edgar R. Weippl, editors, Machine Learning and Knowledge Extraction - CD-MAKE 2019, Canterbury, UK, August 26-29, 2019, Proceedings, volume 11713 of Lecture Notes in Computer Science, pages 272-284. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-29726-8_17.
  4. Andrew J. Blumberg and Michael Lesnick. Stability of 2-parameter persistent homology. CoRR, 2020. URL: https://arxiv.org/abs/2010.09628.
  5. Thomas Bonis, Maks Ovsjanikov, Steve Oudot, and Frédéric Chazal. Persistence-based pooling for shape pose recognition. In Alexandra Bac and Jean-Luc Mari, editors, Computational Topology in Image Context - 6th International Workshop, CTIC 2016, Marseille, France, June 15-17, 2016, Proceedings, volume 9667 of Lecture Notes in Computer Science, pages 19-29. Springer, 2016. URL: https://doi.org/10.1007/978-3-319-39441-1_3.
  6. Magnus B. Botnan and Gard Spreemann. Approximating persistent homology in euclidean space through collapses. Applicable Algebra in Engineering, Communication and Computing, 26(1-2):73-101, January 2015. URL: https://doi.org/10.1007/s00200-014-0247-y.
  7. Bernhard Brehm and Hanne Hardering. Sparips. CoRR, 2018. URL: https://arxiv.org/abs/1807.09982.
  8. Mickaël Buchet, Frédéric Chazal, Steve Y. Oudot, and Donald R. Sheehy. Efficient and robust persistent homology for measures. Comput. Geom., 58:70-96, 2016. URL: https://doi.org/10.1016/j.comgeo.2016.07.001.
  9. Mickaël Buchet, Bianca B. Dornelas, and Michael Kerber. Sparse higher order Čech filtrations, 2023. URL: https://arxiv.org/abs/2303.06666.
  10. Gunnar Carlsson. Topology and data. Bulletin of the American Mathematical Society, 46(2):255-308, 2009. URL: https://doi.org/10.1090/S0273-0979-09-01249-X.
  11. Nicholas J. Cavanna, Mahmoodreza Jahanseir, and Donald R. Sheehy. A geometric perspective on sparse filtrations. In Proceedings of the 27th Canadian Conference on Computational Geometry, CCCG 2015, Kingston, Ontario, Canada, August 10-12, 2015. Queen’s University, Ontario, Canada, 2015. URL: http://research.cs.queensu.ca/cccg2015/CCCG15-papers/01.pdf.
  12. Frédéric Chazal, David Cohen-Steiner, Marc Glisse, Leonidas J. Guibas, and Steve Oudot. Proximity of persistence modules and their diagrams. In John Hershberger and Efi Fogel, editors, Proceedings of the 25th ACM Symposium on Computational Geometry, Aarhus, Denmark, June 8-10, 2009, pages 237-246. ACM, 2009. URL: https://doi.org/10.1145/1542362.1542407.
  13. Frédéric Chazal, David Cohen-Steiner, and Quentin Mérigot. Geometric inference for probability measures. Found. Comput. Math., 11(6):733-751, 2011. URL: https://doi.org/10.1007/s10208-011-9098-0.
  14. Aruni Choudhary, Michael Kerber, and Sharath Raghvendra. Improved topological approximations by digitization. In Timothy M. Chan, editor, Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 2675-2688. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975482.166.
  15. Aruni Choudhary, Michael Kerber, and Sharath Raghvendra. Polynomial-sized topological approximations using the permutahedron. Discret. Comput. Geom., 61(1):42-80, 2019. URL: https://doi.org/10.1007/s00454-017-9951-2.
  16. Aruni Choudhary, Michael Kerber, and Sharath Raghvendra. Improved approximate rips filtrations with shifted integer lattices and cubical complexes. J. Appl. Comput. Topol., 5(3):425-458, 2021. URL: https://doi.org/10.1007/s41468-021-00072-4.
  17. René Corbet, Michael Kerber, Michael Lesnick, and Georg Osang. Computing the multicover bifiltration. Discret. Comput. Geom., 2023. URL: https://doi.org/10.1007/s00454-022-00476-8.
  18. Mark de Berg, Otfried Cheong, Marc J. van Kreveld, and Mark H. Overmars. Computational geometry: algorithms and applications, 3rd Edition. Springer, 2008. URL: https://www.worldcat.org/oclc/227584184.
  19. Tamal K. Dey, Fengtao Fan, and Yusu Wang. Computing topological persistence for simplicial maps. In Siu-Wing Cheng and Olivier Devillers, editors, 30th International Symposium on Computational Geometry, SoCG 2014, Kyoto, Japan, June 08 - 11, 2014, page 345. ACM, 2014. URL: https://doi.org/10.1145/2582112.2582165.
  20. Tamal K. Dey, Dayu Shi, and Yusu Wang. Simba: An efficient tool for approximating rips-filtration persistence via simplicial batch collapse. ACM J. Exp. Algorithmics, 24(1):1.5:1-1.5:16, 2019. URL: https://doi.org/10.1145/3284360.
  21. Martin E. Dyer. A class of convex programs with applications to computational geometry. In David Avis, editor, Proceedings of the Eighth Annual Symposium on Computational Geometry, Berlin, Germany, June 10-12, 1992, pages 9-15. ACM, 1992. URL: https://doi.org/10.1145/142675.142681.
  22. Herbert Edelsbrunner and John L. Harer. Computational topology. American Mathematical Society, Providence, RI, 2010. An introduction. URL: https://doi.org/10.1090/mbk/069.
  23. Herbert Edelsbrunner, David Letscher, and Afra Zomorodian. Topological persistence and simplification. Discret. Comput. Geom., 28(4):511-533, 2002. URL: https://doi.org/10.1007/s00454-002-2885-2.
  24. Herbert Edelsbrunner and Georg Osang. A simple algorithm for higher-order delaunay mosaics and alpha shapes. CoRR, 2020. URL: https://arxiv.org/abs/2011.03617.
  25. Herbert Edelsbrunner and Georg Osang. The multi-cover persistence of euclidean balls. Discret. Comput. Geom., 65(4):1296-1313, 2021. URL: https://doi.org/10.1007/s00454-021-00281-9.
  26. Herbert Edelsbrunner and Raimund Seidel. Voronoi diagrams and arrangements. Discret. Comput. Geom., 1:25-44, 1986. URL: https://doi.org/10.1007/BF02187681.
  27. Kaspar Fischer. Smallest enclosing balls of balls: Combinatorial structure & algorithms. PhD thesis, Swiss Federal Institute of Technology, ETH Zürich, 2005. URL: https://people.inf.ethz.ch/emo/DoctThesisFiles/fischer05.pdf.
  28. Kaspar Fischer and Bernd Gärtner. The smallest enclosing ball of balls: combinatorial structure and algorithms. Int. J. Comput. Geom. Appl., 14(4-5):341-378, 2004. URL: https://doi.org/10.1142/S0218195904001500.
  29. Marcio Gameiro, Yasuaki Hiraoka, Shunsuke Izumi, Miroslav Kramar, Konstantin Mischaikow, and Vidit Nanda. A topological measurement of protein compressibility. Japan Journal of Industrial and Applied Mathematics, 32(1):1-17, 2015. URL: https://doi.org/10.1007/s13160-014-0153-5.
  30. Robert W. Ghrist. Elementary applied topology, volume 1. Createspace Seattle, WA, 2014. Google Scholar
  31. Teofilo F. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoretical Computer Science, 38:293-306, 1985. URL: https://doi.org/10.1016/0304-3975(85)90224-5.
  32. Leonidas J Guibas, Quentin Mérigot, and Dmitriy Morozov. Witnessed k-distance. Discret. Comput. Geom., 49(1):22-45, 2013. URL: https://doi.org/10.1007/s00454-012-9465-x.
  33. Sariel Har-Peled and Manor Mendel. Fast construction of nets in low-dimensional metrics and their applications. SIAM J. Comput., 35(5):1148-1184, 2006. URL: https://doi.org/10.1137/S0097539704446281.
  34. Yasuaki Hiraoka, Takenobu Nakamura, Akihiko Hirata, Emerson G. Escolar, Kaname Matsue, and Yasumasa Nishiura. Hierarchical structures of amorphous solids characterized by persistent homology. Proceedings of the National Academy of Sciences, 113(26):7035-7040, 2016. URL: https://doi.org/10.1073/pnas.1520877113.
  35. Lida Kanari, Pawel Dlotko, Martina Scolamiero, Ran Levi, Julian C. Shillcock, Kathryn Hess, and Henry Markram. A topological representation of branching neuronal morphologies. Neuroinformatics, 16(1):3-13, 2018. URL: https://doi.org/10.1007/s12021-017-9341-1.
  36. Saunders Mac Lane. Categories for the working mathematician., volume 5. New York, NY: Springer, 1998. URL: https://doi.org/10.1007/978-1-4757-4721-8.
  37. Clément Maria, Pawel Dlotko, Vincent Rouvreau, and Marc Glisse. Rips complex. In GUDHI User and Reference Manual. GUDHI Editorial Board, 3.5.0 edition, 2022. URL: https://gudhi.inria.fr/doc/3.5.0/group__rips__complex.html.
  38. Nimrod Megiddo. On the ball spanned by balls. Discret. Comput. Geom., 4:605-610, 1989. URL: https://doi.org/10.1007/BF02187750.
  39. David M. Mount and Eunhui Park. A dynamic data structure for approximate range searching. In David G. Kirkpatrick and Joseph S. B. Mitchell, editors, Proceedings of the 26th ACM Symposium on Computational Geometry, Snowbird, Utah, USA, June 13-16, 2010, pages 247-256. ACM, 2010. URL: https://doi.org/10.1145/1810959.1811002.
  40. James Munkres. Topology. Prentice Hall, 2ed edition, 2000. Google Scholar
  41. Julian B. Pérez, Sydney Hauke, Umberto Lupo, Matteo Caorsi, and Alberto Dassatti. Giotto-ph: A python library for high-performance computation of persistent homology of vietoris-rips filtrations. CoRR, 2021. URL: https://arxiv.org/abs/2107.05412.
  42. Jeff M. Phillips, Bei Wang, and Yan Zheng. Geometric inference on kernel density estimates. In Lars Arge and János Pach, editors, 31st International Symposium on Computational Geometry, SoCG 2015, June 22-25, 2015, Eindhoven, The Netherlands, volume 34 of LIPIcs, pages 857-871. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015. URL: https://doi.org/10.4230/LIPIcs.SOCG.2015.857.
  43. Raimund Seidel. On the number of faces in higher-dimensional voronoi diagrams. In D. Soule, editor, Proceedings of the Third Annual Symposium on Computational Geometry, Waterloo, Ontario, Canada, June 8-10, 1987, pages 181-185. ACM, 1987. URL: https://doi.org/10.1145/41958.41977.
  44. Donald R. Sheehy. A multicover nerve for geometric inference. In Proceedings of the 24th Canadian Conference on Computational Geometry, CCCG 2012, Charlottetown, Prince Edward Island, Canada, August 8-10, 2012, pages 309-314, 2012. URL: http://2012.cccg.ca/papers/paper52.pdf.
  45. Donald R. Sheehy. Linear-size approximations to the vietoris-rips filtration. Discret. Comput. Geom., 49(4):778-796, 2013. URL: https://doi.org/10.1007/s00454-013-9513-1.
  46. Donald R. Sheehy. A sparse delaunay filtration. In Kevin Buchin and Éric Colin de Verdière, editors, 37th International Symposium on Computational Geometry, SoCG 2021, June 7-11, 2021, Buffalo, NY, USA (Virtual Conference), volume 189 of LIPIcs, pages 58:1-58:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.SoCG.2021.58.
  47. Wilson A. Sutherland. Introduction to metric and topological spaces. Oxford Mathematics Series. Oxford University Press, 2 edition, 2009. Google Scholar
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