Foundations of Computational Mathematics

, Volume 15, Issue 3, pp 613–650 | Cite as

The Theory of the Interleaving Distance on Multidimensional Persistence Modules

Article

Abstract

In 2009, Chazal et al. introduced \(\epsilon \)-interleavings of persistence modules. \(\epsilon \)-interleavings induce a pseudometric \(d_\mathrm{I}\) on (isomorphism classes of) persistence modules, the interleaving distance. The definitions of \(\epsilon \)-interleavings and \(d_\mathrm{I}\) generalize readily to multidimensional persistence modules. In this paper, we develop the theory of multidimensional interleavings, with a view toward applications to topological data analysis. We present four main results. First, we show that on 1-D persistence modules, \(d_\mathrm{I}\) is equal to the bottleneck distance \(d_\mathrm{B}\). This result, which first appeared in an earlier preprint of this paper, has since appeared in several other places, and is now known as the isometry theorem. Second, we present a characterization of the \(\epsilon \)-interleaving relation on multidimensional persistence modules. This expresses transparently the sense in which two \(\epsilon \)-interleaved modules are algebraically similar. Third, using this characterization, we show that when we define our persistence modules over a prime field, \(d_\mathrm{I}\) satisfies a universality property. This universality result is the central result of the paper. It says that \(d_\mathrm{I}\) satisfies a stability property generalizing one which \(d_\mathrm{B}\) is known to satisfy, and that in addition, if \(d\) is any other pseudometric on multidimensional persistence modules satisfying the same stability property, then \(d\le d_\mathrm{I}\). We also show that a variant of this universality result holds for \(d_\mathrm{B}\), over arbitrary fields. Finally, we show that \(d_\mathrm{I}\) restricts to a metric on isomorphism classes of finitely presented multidimensional persistence modules.

Keywords

Multidimensional persistence Stability of persistent homology Persistence modules Interleavings Algebraic stability Isometry theorem 

Mathematics Subject Classification

55 68 

References

  1. 1.
    Aaron Adcock, Daniel Rubin, and Gunnar Carlsson. Classification of hepatic lesions using the matching metric. Computer Vision and Image Understanding, 121:36–42, 2014.CrossRefGoogle Scholar
  2. 2.
    Michael F Atiyah. On the krull-schmidt theorem with application to sheaves. Bulletin de la Société Mathématique de France, 84:307–317, 1956.MATHMathSciNetGoogle Scholar
  3. 3.
    U. Bauer and M. Lesnick. Induced matchings of barcodes and the algebraic stability of persistence. In Proceedings of the 2014 Annual Symposium on Computational Geometry, page 355. ACM, 2014. Extended version to appear in Discrete and Computational Geometry.Google Scholar
  4. 4.
    S. Biasotti, A. Cerri, P. Frosini, D. Giorgi, and C. Landi. Multidimensional size functions for shape comparison. Journal of Mathematical Imaging and Vision, 32(2):161–179, 2008.CrossRefMathSciNetGoogle Scholar
  5. 5.
    A. Blumberg and M. Lesnick. Universality of the homotopy interleaving distance. In preparation, 2015.Google Scholar
  6. 6.
    P. Bubenik, V. de Silva, and J. Scott. Metrics for generalized persistence modules. Foundations of Computational Mathematics, 2014. doi:10.1007/s10208-014-9229-5.
  7. 7.
    P. Bubenik and J. Scott. Categorification of persistent homology. Discrete & Computational Geometry, 51(3):600–627, 2014.CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    E. Carlsson, G. Carlsson, V. de Silva, and S. Fortune. An algebraic topological method for feature identification. International Journal of Computational Geometry and Applications, 16(4):291–314, 2006.CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    G. Carlsson. Topological pattern recognition for point cloud data. Acta Numerica, 23:289–368, May 2014.CrossRefMathSciNetGoogle Scholar
  10. 10.
    G. Carlsson, V. de Silva, and D. Morozov. Zigzag persistent homology and real-valued functions. In Proceedings of the 25th annual symposium on Computational geometry, pages 247–256. ACM, 2009.Google Scholar
  11. 11.
    G. Carlsson, T. Ishkhanov, V. de Silva, and A. Zomorodian. On the local behavior of spaces of natural images. International Journal of Computer Vision, 76(1):1–12, 2008.CrossRefGoogle Scholar
  12. 12.
    G. Carlsson and F. Mémoli. Multiparameter hierarchical clustering methods. Classification as a Tool for Research, pages 63–70, 2010.Google Scholar
  13. 13.
    G. Carlsson and A. Zomorodian. The theory of multidimensional persistence. Discrete and Computational Geometry, 42(1):71–93, 2009.CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    G. Carlsson, A. Zomorodian, A. Collins, and L. Guibas. Persistence barcodes for shapes. In Proceedings of the 2004 Eurographics/ACM SIGGRAPH symposium on Geometry processing, pages 124–135. ACM, 2004.Google Scholar
  15. 15.
    Andrea Cerri, Barbara Di Fabio, Massimo Ferri, Patrizio Frosini, and Claudia Landi. Betti numbers in multidimensional persistent homology are stable functions. Mathematical Methods in the Applied Sciences, 36(12):1543–1557, 2013.CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    F. Chazal, D. Cohen-Steiner, M. Glisse, L.J. Guibas, and S.Y. Oudot. Proximity of persistence modules and their diagrams. In Proceedings of the 25th annual symposium on Computational geometry, pages 237–246. ACM, 2009.Google Scholar
  17. 17.
    F. Chazal, D. Cohen-Steiner, L.J. Guibas, F. Mémoli, and S.Y. Oudot. Gromov-Hausdorff stable signatures for shapes using persistence. In Proceedings of the Symposium on Geometry Processing, pages 1393–1403. Eurographics Association, 2009.Google Scholar
  18. 18.
    F. Chazal, D. Cohen-Steiner, and Q. Mérigot. Geometric inference for probability measures. Foundations of Computational Mathematics, pages 1–19, 2011.Google Scholar
  19. 19.
    F. Chazal, W. Crawley-Boevey, and V. de Silva. The observable structure of persistence modules. arXiv preprint arXiv:1405.5644, 2014.
  20. 20.
    F. Chazal, V. de Silva, M. Glisse, and S. Oudot. The structure and stability of persistence modules. arXiv preprint arXiv:1207.3674, 2012.
  21. 21.
    F. Chazal, L. Guibas, S. Oudot, and P. Skraba. Persistence-based clustering in riemannian manifolds. Journal of the ACM (JACM), 60(6):41, 2013.CrossRefMathSciNetGoogle Scholar
  22. 22.
    F. Chazal, L.J. Guibas, S.Y. Oudot, and P. Skraba. Analysis of scalar fields over point cloud data. In Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1021–1030. Society for Industrial and Applied Mathematics, 2009.Google Scholar
  23. 23.
    D. Cohen-Steiner, H. Edelsbrunner, and J. Harer. Stability of persistence diagrams. Discrete and Computational Geometry, 37(1):103–120, 2007.CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    D. Cohen-Steiner, H. Edelsbrunner, J. Harer, and Y. Mileyko. Lipschitz functions have \(L_p\)-stable persistence. Foundations of Computational Mathematics, 10(2):127–139, 2010.CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    D. Cohen-Steiner, H. Edelsbrunner, and D. Morozov. Vines and vineyards by updating persistence in linear time. In Proceedings of the twenty-second annual symposium on Computational geometry, pages 119–126. ACM, 2006.Google Scholar
  26. 26.
    W. Crawley-Boevey. Decomposition of pointwise finite-dimensional persistence modules. arXiv preprint arXiv:1210.0819, 2012.
  27. 27.
    J. Curry. Sheaves, cosheaves and applications. Ph.D. Dissertation, University of Pennsylvania, 2014.Google Scholar
  28. 28.
    M. d’Amico, P. Frosini, and C. Landi. Natural pseudo-distance and optimal matching between reduced size functions. Acta applicandae mathematicae, 109(2):527–554, 2010.CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Vin de Silva, Elizabeth Munch, and Amit Patel. Categorified reeb graphs. arXiv preprint arXiv:1501.04147, 2015.
  30. 30.
    H. Derksen and J. Weyman. Quiver representations. Notices of the AMS, 52(2):200–206, 2005.MATHMathSciNetGoogle Scholar
  31. 31.
    D.S. Dummit and R.M. Foote. Abstract algebra. Wiley, 1999.Google Scholar
  32. 32.
    H. Edelsbrunner and J. Harer. Computational topology: an introduction. American Mathematical Society, 2010.Google Scholar
  33. 33.
    D. Eisenbud. Commutative algebra with a view toward algebraic geometry. Springer, 1995.MATHGoogle Scholar
  34. 34.
    P. Frosini. Stable comparison of multidimensional persistent homology groups with torsion. Acta Applicandae Mathematicae, pages 1–12, 2010.Google Scholar
  35. 35.
    P. Frosini and M. Mulazzani. Size homotopy groups for computation of natural size distances. Bulletin of the Belgian Mathematical Society Simon Stevin, 6(3):455–464, 1999.MATHMathSciNetGoogle Scholar
  36. 36.
    Peter Gabriel. Unzerlegbare darstellungen i. Manuscripta Mathematica, 6(1):71–103, 1972.CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    A. Hatcher. Algebraic topology. Cambridge University Press, 2002.MATHGoogle Scholar
  38. 38.
    T. Ishkhanov. A topological method for shape comparison. In Computer Vision and Pattern Recognition Workshops, 2008. CVPRW’08. IEEE Computer Society Conference on, pages 1–4. IEEE, 2008.Google Scholar
  39. 39.
    S. Lang. Algebra, revised third edition. Graduate Texts in Mathematics, 2002.Google Scholar
  40. 40.
    M. Lesnick. The optimality of the interleaving distance on multidimensional persistence modules. Arxiv preprint arXiv:1106.5305v2, 2011.
  41. 41.
    M. Lesnick. Multidimensional interleavings and applications to topological inference. Ph.D. Dissertation, Stanford University, 2012.Google Scholar
  42. 42.
    C. Li, M. Ovsjanikov, and F. Chazal. Persistence-based structural recognition. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 1995–2002, 2013.Google Scholar
  43. 43.
    D. Morozov, K. Beketayev, and G. Weber. Interleaving distance between merge trees. Discrete and Computational Geometry, 49:22–45, 2013.CrossRefMATHMathSciNetGoogle Scholar
  44. 44.
    A. Verri, C. Uras, P. Frosini, and M. Ferri. On the use of size functions for shape analysis. Biological Cybernetics, 70(2):99–107, 1993.CrossRefMATHGoogle Scholar
  45. 45.
    L. Wasserman. All of statistics: a concise course in statistical inference. Springer Verlag, 2004.CrossRefGoogle Scholar
  46. 46.
    Cary Webb. Decomposition of graded modules. Proceedings of the American Mathematical Society, 94(4):565–571, 1985.CrossRefMATHMathSciNetGoogle Scholar
  47. 47.
    A. Zomorodian and G. Carlsson. Computing persistent homology. Discrete and Computational Geometry, 33(2):249–274, 2005.CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© SFoCM 2015

Authors and Affiliations

  1. 1.Institute for Mathematics and its ApplicationsMinneapolisUSA

Personalised recommendations