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Acta Applicandae Mathematicae

, Volume 124, Issue 1, pp 43–54 | Cite as

Stable Comparison of Multidimensional Persistent Homology Groups with Torsion

  • Patrizio Frosini
Article

Abstract

The present lack of a stable method to compare persistent homology groups with torsion is a relevant problem in current research about Persistent Homology and its applications in Pattern Recognition. In this paper we introduce a pseudo-distance d T that represents a possible solution to this problem. Indeed, d T is a pseudo-distance between multidimensional persistent homology groups with coefficients in an Abelian group, hence possibly having torsion. Our main theorem proves the stability of the new pseudo-distance with respect to the change of the filtering function, expressed both with respect to the max-norm and to the natural pseudo-distance between topological spaces endowed with ℝ n -valued filtering functions. Furthermore, we prove a result showing the relationship between d T and the matching distance in the 1-dimensional case, when the homology coefficients are taken in a field and hence the comparison can be made.

Keywords

Multidimensional persistent homology Shape comparison Matching distance Natural pseudo-distance 

Mathematics Subject Classification

55N35 68U05 

Notes

Acknowledgements

Research partially supported by DISTEF. The author thanks Sara, and all his friends in the “Coniglietti” Band.

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Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.ARCESUniversità di BolognaBolognaItaly
  2. 2.Dipartimento di MatematicaUniversità di BolognaBolognaItaly

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