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

On the geometrical properties of the coherent matching distance in 2D persistent homology 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Applied and Computational Topology Springer Journals

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

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Publisher
Springer Journals
Copyright
Copyright © 2019 by Springer Nature Switzerland AG
Subject
Mathematics; Algebraic Topology; Computational Science and Engineering; Mathematical and Computational Biology
ISSN
2367-1726
eISSN
2367-1734
DOI
10.1007/s41468-019-00041-y
Publisher site
See Article on Publisher Site

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.

Journal

Journal of Applied and Computational Topology – Springer Journals

Published: Sep 7, 2019

References

  • Multidimensional size functions for shape comparison
    Biasotti, S; Cerri, A; Frosini, P; Giorgi, D; Landi, C
  • A new algorithm for computing the 2-dimensional matching distance between size functions
    Biasotti, S; Cerri, A; Frosini, P; Giorgi, D
  • Algebraic stability of zigzag persistence modules
    Botnan, MB; Lesnick, M
  • One-dimensional reduction of multidimensional persistent homology
    Cagliari, F; Fabio, B; Ferri, M
  • The theory of multidimensional persistence
    Carlsson, G; Zomorodian, A
  • Necessary conditions for discontinuities of multidimensional persistent Betti numbers
    Cerri, A; Frosini, P
  • A new approximation algorithm for the matching distance in multidimensional persistence
    Cerri, A; Frosini, P
  • Hausdorff stability of persistence spaces
    Cerri, A; Landi, C
  • Betti numbers in multidimensional persistent homology are stable functions
    Cerri, A; Fabio, B; Ferri, M; Frosini, P; Landi, C
  • Discrete Geometry for Computer Imagery
    Cerri, Andrea; Ethier, Marc; Frosini, Patrizio
  • Stability of persistence diagrams
    Cohen-Steiner, D; Edelsbrunner, H; Harer, JL
  • Natural pseudo-distance and optimal matching between reduced size functions
    d’Amico, M; Frosini, P; Landi, C
  • Topological persistence and simplification
    Edelsbrunner, H; Letscher, D; Zomorodian, A
  • Size functions and formal series
    Frosini, P; Landi, C
  • Size homotopy groups for computation of natural size distances
    Frosini, P; Mulazzani, M
  • The theory of the interleaving distance on multidimensional persistence modules
    Lesnick, M
  • Morse theory for two functions
    Wan, YH