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Journal of Applied and Computational Topology

, Volume 3, Issue 4, pp 381–422 | Cite as

On the geometrical properties of the coherent matching distance in 2D persistent homology

  • Andrea Cerri
  • Marc Ethier
  • Patrizio FrosiniEmail author
Article
  • 30 Downloads

Abstract

In this paper we study a new metric for comparing Betti numbers functions in bidimensional persistent homology, based on coherent matchings, i.e. families of matchings that vary in a continuous way. We prove some new results about this metric, including a property of stability. In particular, we show that the computation of this distance is strongly related to suitable filtering functions associated with lines of slope 1, so underlining the key role of these lines in the study of bidimensional persistence. In order to prove these results, we introduce and study the concepts of extended Pareto grid for a normal filtering function as well as of transport of a matching. As a by-product, we obtain a theoretical framework for managing the phenomenon of monodromy in 2D persistent homology.

Keywords

Multidimensional persistence Jacobi set Extended Pareto grid Coherent transport of a matching Persistent monodromy group 

Mathematics Subject Classification

Primary 55N35 Secondary 57R19 65D18 68U05 

Notes

Acknowledgements

Work carried out under the auspices of INdAM-GNSAGA. M.E. has been partially supported by the Toposys project FP7-ICT-318493-STREP, as well as an ESF Short Visit Grant under the Applied and Computational Algebraic Topology networking programme. He would like to acknowledge his former employers, the Université de Saint-Boniface in Winnipeg, Canada, and Champlain–Saint-Lawrence College in Québec, Canada, for work on this paper done during his employment. A.C. has been partially supported by the FP7 Integrated Project IQmulus, FP7-ICT-2011–318787, and the H2020 Project Gravitate, H2020-REFLECTIVE-7-2014-665155. The authors are grateful to Claudia Landi for her valuable advice. This paper is dedicated to the memory of Ola Mouaffek Shihab Eddin, Naya Raslan and Abish Masih.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.FST - Fom Software TechnologyCattolica (RN)Italy
  2. 2.UER en sciences de l’éducationUniversité du Québec en Abitibi-TémiscamingueRouyn-NorandaCanada
  3. 3.Dipartimento di MatematicaUniversità di BolognaBolognaItaly

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