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Efficient Algorithms for Complexes of Persistence Modules with Applications

Authors Tamal K. Dey , Florian Russold, Shreyas N. Samaga

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Author Details

Tamal K. Dey
  • Department of Computer Science, Purdue University, West Lafayette, IN, USA
Florian Russold
  • Institute of Geometry, Graz University of Technology, Austria
Shreyas N. Samaga
  • Department of Computer Science, Purdue University, West Lafayette, IN, USA

Funding

This work is supported partially by NSF grants CCF 2049010 and 2301360 and by the Austrian Science Fund (FWF): W1230.

Cite AsGet BibTex

Tamal K. Dey, Florian Russold, and Shreyas N. Samaga. Efficient Algorithms for Complexes of Persistence Modules with Applications. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 51:1-51:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SoCG.2024.51

Abstract

We extend the persistence algorithm, viewed as an algorithm computing the homology of a complex of free persistence or graded modules, to complexes of modules that are not free. We replace persistence modules by their presentations and develop an efficient algorithm to compute the homology of a complex of presentations. To deal with inputs that are not given in terms of presentations, we give an efficient algorithm to compute a presentation of a morphism of persistence modules. This allows us to compute persistent (co)homology of instances giving rise to complexes of non-free modules. Our methods lead to a new efficient algorithm for computing the persistent homology of simplicial towers and they enable efficient algorithms to compute the persistent homology of cosheaves over simplicial towers and cohomology of persistent sheaves on simplicial complexes. We also show that we can compute the cohomology of persistent sheaves over arbitrary finite posets by reducing the computation to a computation over simplicial complexes.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Algebraic topology
Keywords
  • Persistent (co)homology
  • Persistence modules
  • Sheaves
  • Presentations

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References

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