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Matrix method for persistence modules on commutative ladders of finite type

  • Hideto Asashiba
  • Emerson G. EscolarEmail author
  • Yasuaki Hiraoka
  • Hiroshi Takeuchi
Original Paper Area 1

Abstract

The theory of persistence modules on the commutative ladders \(CL_n(\tau )\) provides an extension of persistent homology. However, an efficient algorithm to compute the generalized persistence diagrams is still lacking. In this work, we view a persistence module M on \(CL_n(\tau )\) as a morphism between zigzag modules, which can be expressed in a block matrix form. For the representation finite case (\(n\le 4\)), we provide an algorithm that uses certain permissible row and column operations to compute a normal form of the block matrix. In this form an indecomposable decomposition of M, and thus its persistence diagram, is obtained.

Keywords

Persistence modules Commutative ladders Computational topology Algorithms 

Mathematics Subject Classification

68W30 16G20 55N99 

Notes

Acknowledgements

On behalf of all authors, the corresponding author states that there is no conflict of interest. This work was partially supported by JST CREST Mathematics 15656429. H.T. is supported by JSPS KAKENHI Grant number JP16J03138.

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Copyright information

© The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  • Hideto Asashiba
    • 1
  • Emerson G. Escolar
    • 2
    Email author
  • Yasuaki Hiraoka
    • 2
    • 3
  • Hiroshi Takeuchi
    • 4
    • 5
  1. 1.Department of Mathematics, Faculty of ScienceShizuoka UniversityShizuokaJapan
  2. 2.Center for Advanced Intelligence Project, RIKENTokyoJapan
  3. 3.Kyoto University Institute for Advanced StudyKyoto UniversityKyotoJapan
  4. 4.Mathematics Department, Graduate School of ScienceTohoku UniversitySendaiJapan
  5. 5.Japan Society for the Promotion of ScienceTokyoJapan

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