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Cup Product Persistence and Its Efficient Computation

Authors Tamal K. Dey , Abhishek Rathod

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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 As Get 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

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