Acta Applicandae Mathematicae

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

Stable Comparison of Multidimensional Persistent Homology Groups with Torsion

  • Patrizio Frosini


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.


Multidimensional persistent homology Shape comparison Matching distance Natural pseudo-distance 

Mathematics Subject Classification

55N35 68U05 



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


  1. 1.
    Biasotti, S., Cerri, A., Frosini, P., Giorgi, D., Landi, C.: Multidimensional size functions for shape comparison. J. Math. Imaging Vis. 32(2), 161–179 (2008) MathSciNetCrossRefGoogle Scholar
  2. 2.
    Cagliari, F., Di Fabio, B., Ferri, M.: One-dimensional reduction of multidimensional persistent homology. Proc. Am. Math. Soc. 138(8), 3003–3017 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Cagliari, F., Landi, C.: Finiteness of rank invariants of multidimensional persistent homology groups. Appl. Math. Lett. 24(4), 516–518 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Carlsson, G., Gurjeet, S., Zomorodian, A.: Computing multidimensional persistence. In: ISAAC ’09: Proceedings of the 20th International Symposium on Algorithms and Computation, pp. 730–739 (2009) CrossRefGoogle Scholar
  5. 5.
    Carlsson, G., Zomorodian, A., Collins, A., Guibas, L.J.: Persistence barcodes for shapes. Int. J. Shape Model. 11(2), 149–187 (2005) zbMATHCrossRefGoogle Scholar
  6. 6.
    Carlsson, G., Zomorodian, A.: The theory of multidimensional persistence. In: SCG ’07: Proceedings of the twenty-third annual symposium on Computational geometry, Gyeongju, South Korea, pp. 184–193 (2007) CrossRefGoogle Scholar
  7. 7.
    Carlsson, G., Zomorodian, A.: The theory of multidimensional persistence. Discrete Comput. Geom. 42(1), 71–93 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Cerri, A., Di Fabio, B., Ferri, M., Frosini, P., Landi, C.: Multidimensional persistent homology is stable. Technical report 2603, Università di Bologna (2009). Available at
  9. 9.
    Cerri, A., Ferri, M., Giorgi, D.: Retrieval of trademark images by means of size functions. Graph. Models 68(5), 451–471 (2006) CrossRefGoogle Scholar
  10. 10.
    Chazal, F., Cohen-Steiner, D., Glisse, M., Guibas, L.J., Oudot, S.Y.: Proximity of persistence modules and their diagrams. In: SCG ’09: Proceedings of the 25th Annual Symposium on Computational Geometry, Aarhus, Denmark, pp. 237–246 (2009) CrossRefGoogle Scholar
  11. 11.
    Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. Discrete Comput. Geom. 37(1), 103–120 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Cohen-Steiner, D., Edelsbrunner, H., Harer, J., Mileyko, Y.: Lipschitz functions have L p-stable persistence. Found. Comput. Math. 10(2), 127–139 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    d’Amico, M., Frosini, P., Landi, C.: Natural pseudo-distance and optimal matching between reduced size functions. Acta Appl. Math. 109(2), 527–554 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    de Silva, V., Ghrist, R.: Coverage in sensor networks via persistent homology. Algebr. Geom. Topol. 7, 339–358 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Donatini, P., Frosini, P.: Natural pseudodistances between closed manifolds. Forum Math. 16(5), 695–715 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Donatini, P., Frosini, P.: Natural pseudodistances between closed surfaces. J. Eur. Math. Soc. 9(2), 231–253 (2007) MathSciNetGoogle Scholar
  17. 17.
    Donatini, P., Frosini, P.: Natural pseudodistances between closed curves. Forum Math. 21(6), 981–999 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Edelsbrunner, H., Harer, J.: Persistent homology—a survey. Contemp. Math. 453, 257–282 (2008) MathSciNetCrossRefGoogle Scholar
  19. 19.
    Edelsbrunner, H., Harer, J.: Computational Topology: an Introduction. American Mathematical Society, Reading (2009) Google Scholar
  20. 20.
    Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. Discrete Comput. Geom. 28(4), 511–533 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Farber, M.: Invitation to Topological Robotics. European Mathematical Society, Zurich Lectures in Advanced Mathematics (2008) zbMATHCrossRefGoogle Scholar
  22. 22.
    Frosini, P., Mulazzani, M.: Size homotopy groups for computation of natural size distances. Bull. Belg. Math. Soc. 6(3), 455–464 (1999) MathSciNetzbMATHGoogle Scholar
  23. 23.
    Ghrist, R.: Barcodes: the persistent topology of data. Bull. Am. Math. Soc. (New Ser.) 45(1), 61–75 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Moroni, D., Salvetti, M., Salvetti, O.: Multi-scale representation and persistency for shape description. In: MDA ’08: Proceedings of the 3rd International Conference on Advances in Mass Data Analysis of Images and Signals in Medicine, Biotechnology, Chemistry and Food Industry, Leipzig, Germany, pp. 123–138 (2008) Google Scholar
  25. 25.
    Verri, A., Uras, C., Frosini, P., Ferri, M.: On the use of size functions for shape analysis. Biol. Cybern. 70, 99–107 (1993) zbMATHCrossRefGoogle Scholar
  26. 26.
    Zomorodian, A.: Topology for Computing. Cambridge Monographs on Applied and Computational Mathematics, vol. 16. Cambridge University Press, Cambridge (2005) zbMATHCrossRefGoogle Scholar

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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