Document Open Access Logo

Cup Product Persistence and Its Efficient Computation

Authors Tamal K. Dey , Abhishek Rathod

PDF
Thumbnail PDF

File

LIPIcs.SoCG.2024.50.pdf
  • Filesize: 0.84 MB
  • 15 pages

Document Identifiers

Author Details

Tamal K. Dey
  • Department of Computer Science, Purdue University, West Lafayette, IN, USA
Abhishek Rathod
  • Department of Computer Science, Ben Gurion University, Beersheba, Israel

Funding

This work is partially supported by the NSF grant CCF 2049010, DMS 2301360 and ERC 101039913.

Acknowledgements

We wish to acknowledge helpful initial discussions with Ulrich Bauer and Fabian Lenzen.

Cite AsGet BibTex

Tamal K. Dey and Abhishek Rathod. Cup Product Persistence and Its Efficient Computation. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 50:1-50:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SoCG.2024.50

Abstract

It is well-known that the cohomology ring has a richer structure than homology groups. However, until recently, the use of cohomology in persistence setting has been limited to speeding up of barcode computations. Some of the recently introduced invariants, namely, persistent cup-length, persistent cup modules and persistent Steenrod modules, to some extent, fill this gap. When added to the standard persistence barcode, they lead to invariants that are more discriminative than the standard persistence barcode. In this work, we devise an O(d n⁴) algorithm for computing the persistent k-cup modules for all k ∈ {2, … , d}, where d denotes the dimension of the filtered complex, and n denotes its size. Moreover, we note that since the persistent cup length can be obtained as a byproduct of our computations, this leads to a faster algorithm for computing it for d ≥ 3. Finally, we introduce a new stable invariant called partition modules of cup product that is more discriminative than persistent cup modules and devise an O(c(d)n⁴) algorithm for computing it, where c(d) is subexponential in d.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Algebraic topology
Keywords
  • Persistent cohomology
  • cup product
  • image persistence
  • persistent cup module

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Ulrich Bauer. Ripser: efficient computation of Vietoris-Rips persistence barcodes. Journal of Applied and Computational Topology, 5(3):391-423, 2021. Google Scholar
  2. Ulrich Bauer, Michael Kerber, and Jan Reininghaus. Clear and compress: Computing persistent homology in chunks. In Topological methods in data analysis and visualization III, pages 103-117. Springer, 2014. Google Scholar
  3. Ulrich Bauer, Michael Kerber, Jan Reininghaus, and Hubert Wagner. Phat - persistent homology algorithms toolbox. Journal of Symbolic Computation, 78:76-90, 2017. Google Scholar
  4. Ulrich Bauer and Michael Lesnick. Induced matchings of barcodes and the algebraic stability of persistence. In Proceedings of the thirtieth annual symposium on Computational geometry, pages 355-364, 2014. Google Scholar
  5. Ulrich Bauer, Talha Bin Masood, Barbara Giunti, Guillaume Houry, Michael Kerber, and Abhishek Rathod. Keeping it sparse: Computing persistent homology revisited. arXiv preprint, 2022. URL: https://arxiv.org/abs/2211.09075.
  6. Ulrich Bauer and Maximilian Schmahl. Efficient Computation of Image Persistence. In Erin W. Chambers and Joachim Gudmundsson, editors, 39th International Symposium on Computational Geometry (SoCG 2023), volume 258 of Leibniz International Proceedings in Informatics (LIPIcs), pages 14:1-14:14, Dagstuhl, Germany, 2023. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.SoCG.2023.14.
  7. Francisco Belchí and Anastasios Stefanou. A-infinity persistent homology estimates detailed topology from point cloud datasets. Discrete & Computational Geometry, pages 1-24, 2021. Google Scholar
  8. Jean-Daniel Boissonnat, Tamal K Dey, and Clément Maria. The compressed annotation matrix: An efficient data structure for computing persistent cohomology. In European Symposium on Algorithms, pages 695-706. Springer, 2013. Google Scholar
  9. Peter Brooksbank, E O’Brien, and James Wilson. Testing isomorphism of graded algebras. Transactions of the American Mathematical Society, 372(11):8067-8090, 2019. Google Scholar
  10. Frédéric Chazal, Vin De Silva, Marc Glisse, and Steve Oudot. The structure and stability of persistence modules, volume 10. Springer, 2016. Google Scholar
  11. Frédéric Chazal, Vin De Silva, and Steve Oudot. Persistence stability for geometric complexes. Geometriae Dedicata, 173(1):193-214, 2014. Google Scholar
  12. David Cohen-Steiner, Herbert Edelsbrunner, and John Harer. Stability of persistence diagrams. Discrete Comput. Geom., 37(1):103-120, January 2007. URL: https://doi.org/10.1007/s00454-006-1276-5.
  13. David Cohen-Steiner, Herbert Edelsbrunner, John Harer, and Dmitriy Morozov. Persistent homology for kernels, images, and cokernels. In Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1011-1020. SIAM, 2009. Google Scholar
  14. Marco Contessoto, Facundo Mémoli, Anastasios Stefanou, and Ling Zhou. Persistent cup-length. In 38th International Symposium on Computational Geometry, SoCG 2022, June 7-10, 2022, Berlin, Germany, volume 224 of LIPIcs, pages 31:1-31:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. Google Scholar
  15. Marco Contessoto, Facundo Mémoli, Anastasios Stefanou, and Ling Zhou. Persistent cup-length, 2021. URL: https://doi.org/10.48550/ARXIV.2107.01553.
  16. Wladimir de Azevedo Pribitkin. Simple upper bounds for partition functions. The Ramanujan Journal, 18(1):113-119, 2009. Google Scholar
  17. Vin De Silva, Dmitriy Morozov, and Mikael Vejdemo-Johansson. Dualities in persistent (co) homology. Inverse Problems, 27(12):124003, 2011. Google Scholar
  18. Tamal K. Dey and Abhishek Rathod. Cup product persistence and its efficient computation, 2024. URL: https://arxiv.org/abs/2212.01633.
  19. Tamal K. Dey and Yusu Wang. Computational Topology for Data Analysis. Cambridge University Press, 2022. Google Scholar
  20. Herbert Edelsbrunner and John Harer. Computational Topology: An Introduction. Applied Mathematics. American Mathematical Society, 2010. Google Scholar
  21. Hitesh Gakhar and Jose A. Perea. Sliding window persistence of quasiperiodic functions. Journal of Applied and Computational Topology, 8(1):55-92, 2024. Google Scholar
  22. Allen Hatcher. Algebraic topology. Cambridge University Press, Cambridge, 2002. Google Scholar
  23. Jean-Claude Hausmann. Mod two homology and cohomology, volume 10. Springer, 2014. Google Scholar
  24. Estanislao Herscovich. A higher homotopic extension of persistent (co)homology. Journal of Homotopy and Related Structures, 13(3):599-633, 2018. Google Scholar
  25. Umberto Lupo, Anibal M. Medina-Mardones, and Guillaume Tauzin. Persistence Steenrod modules. Journal of Applied and Computational Topology, 6(4):475-502, 2022. Google Scholar
  26. Clément Maria, Jean-Daniel Boissonnat, Marc Glisse, and Mariette Yvinec. The Gudhi library: Simplicial complexes and persistent homology. In Hoon Hong and Chee Yap, editors, Mathematical Software - ICMS 2014, pages 167-174, Berlin, Heidelberg, 2014. Springer Berlin Heidelberg. Google Scholar
  27. William S. Massey. Higher order linking numbers. Journal of Knot Theory and Its Ramifications, 7:393-414, 1998. Google Scholar
  28. Facundo Mémoli, Anastasios Stefanou, and Ling Zhou. Persistent cup product structures and related invariants. Journal of Applied and Computational Topology, 2023. Google Scholar
  29. Robert E. Mosher and Martin C. Tangora. Cohomology operations and applications in homotopy theory. Courier Corporation, 2008. Google Scholar
  30. Luis Polanco. Applications of persistent cohomology to dimensionality reduction and classification problems. Phd thesis, Michigan State University, 2022. URL: https://doi.org/doi:10.25335/exk0-fs44.
  31. Norman E Steenrod. Products of cocycles and extensions of mappings. Annals of Mathematics, pages 290-320, 1947. Google Scholar
  32. Christopher J. Tralie and Jose A. Perea. (quasi)periodicity quantification in video data, using topology. SIAM Journal on Imaging Sciences, 11(2):1049-1077, 2018. Google Scholar
  33. Andrew Yarmola. Persistence and computation of the cup product. Undergraduate honors thesis, Stanford University, 2010. Google Scholar
  34. Afra Zomorodian and Gunnar Carlsson. Computing persistent homology. In Proceedings of the twentieth annual symposium on Computational geometry, pages 347-356, 2004. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact