A topological study of functional data and Fréchet functions of metric measure spaces

Abstract

We study the persistent homology of both functional data on compact topological spaces and structural data presented as compact metric measure spaces. One of our goals is to define persistent homology so as to capture primarily properties of the shape of a signal, eliminating otherwise highly persistent homology classes that may exist simply because of the nature of the domain on which the signal is defined. We investigate the stability of these invariants using metrics that downplay regions where signals are weak. The distance between two signals is small if they exhibit high similarity in regions where they are strong, regardless of the nature of their full domains, in particular allowing different homotopy types. Consistency and estimation of persistent homology of metric measure spaces from data are studied within this framework. We also apply the methodology to the construction of multi-scale topological descriptors for data on compact Riemannian manifolds via metric relaxations derived from the heat kernel.

This is a preview of subscription content, access via your institution.

Fig. 1

References

  1. Arnaudon, M., Barbaresco, F.: Medians and means in Riemannian geometry: existence, uniqueness and computation. In: Nielsen, F., Bhatia, R. (eds.) Matrix Information Geometry, pp. 169–197. Springer, Berlin (2013)

    MATH  Chapter  Google Scholar 

  2. Bérard, P., Besson, G., Gallot, S.: Embedding Riemannian manifolds by their heat kernel. Geom. Funct. Anal. (GAFA) 4(4), 373–398 (1994)

    MathSciNet  MATH  Article  Google Scholar 

  3. Bhattacharya, R., Patrangenaru, V.: Large sample theory of intrinsic and extrinsic sample means on manifolds. Ann. Stat. 31(1), 1–29 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  4. Blumberg, A.J., Gal, I., Mandell, M.A., Pancia, M.: Robust statistics, hypothesis testing, and confidence intervals for persistent homology on metric measure spaces. Found. Comput. Math. 14(4), 745–789 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  5. Carlsson, G.: Topology and data. Bull. Am. Math. Soc. 46(2), 255–308 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  6. Cerri, A., Di Fabio, B., Ferri, M., Frosini, P., Landi, C.: Betti numbers in multidimensional persistent homology are stable functions. Math. Methods Appl. Sci. 36, 1543–1557 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  7. Chazal, F., Cohen-Steiner, D., Glisse, M., Guibas, L., Oudot, S.: Proximity of persistence modules and their diagrams. In: Proceedings of the 25th Annual ACM Symposium on Computational Geometry, pp. 237–246 (2009)

  8. Chazal, F., Cohen-Steiner, D., Mérigot, Q.: Geometric inference for probability measures. Found. Comput. Math. 11, 733–751 (2011). https://doi.org/10.1007/s10208-011-9098-0

    MathSciNet  MATH  Article  Google Scholar 

  9. Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. Discrete Comput. Geom. 37(1), 103–120 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  10. Coifman, R.R., Lafon, S.: Diffusion maps. Appl. Comput. Harmon. Anal. 21(1), 5–30 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  11. de Silva, V., Chazal, F., Glisse, M., Oudot, S.: The Structure and Stability of Persistence Modules. Springer Briefs in Mathematics. Springer, Berlin (2016)

    MATH  Google Scholar 

  12. Díaz Martínez, D., Lee, C., Kim, P., Mio, W.: Probing the geometry of data with diffusion Fréchet functions. Appl. Comput. Harmon. Anal. (2018a). https://doi.org/10.1016/j.acha.2018.01.003

    MATH  Article  Google Scholar 

  13. Díaz Martínez, D., Mémoli, F., Mio, W.: The shape of data and probability measures. Appl. Comput. Harmon. Anal. (2018b). https://doi.org/10.1016/j.acha.2018.03.003

    MATH  Article  Google Scholar 

  14. Dudley, R.M.: Real Analysis and Probability, vol. 74. Cambridge University Press, Cambridge (2002)

    MATH  Book  Google Scholar 

  15. Edelsbrunner, H., Harer, J.: Persistent homology: a survey. Contemp. Math. 453, 257–282 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  16. Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. Discrete Comput. Geom. 28, 511–533 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  17. Frosini, P.: Measuring shapes by size functions. In: Intelligent Robots and Computer Vision X: Algorithms and Techniques, pp. 122–133. International Society for Optics and Photonics (1992)

  18. Frosini, P., Landi, C., Mémoli, F.: The persistent homotopy type distance. Homol. Homotopy Appl. (2017). https://doi.org/10.4310/HHA.2019.v21.n2.a13

    MATH  Article  Google Scholar 

  19. Grigor’yan, A.: Heat Kernel and Analysis on Manifolds. Studies in Advanced Mathematics, vol. 47. American Mathematical Society, Providence (2009)

    MATH  Google Scholar 

  20. Gromov, M.: Metric Structures for Riemannian and Non-Riemannian Spaces. Springer, Berlin (2007)

    MATH  Google Scholar 

  21. Grove, K., Karcher, H.: How to conjugate \({C}^1\)-close group actions. Math. Z. 132, 11–20 (1973)

    MathSciNet  MATH  Article  Google Scholar 

  22. Kasue, A., Kumura, H.: Spectral convergence of Riemannian manifolds. Tohoku Math. J. Second Ser. 46(2), 147–179 (1994)

    MathSciNet  MATH  Article  Google Scholar 

  23. Lesnick, M.: The theory of the interleaving distance on multidimensional persistence modules. Found. Comput. Math. 15(3), 613–650 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  24. Mémoli, F.: Gromov–Wasserstein distances and the metric approach to object matching. Found. Comput. Math. 11(4), 417–487 (2011a)

    MathSciNet  MATH  Article  Google Scholar 

  25. Mémoli, F.: A spectral notion of Gromov–Wasserstein distance and related methods. Appl. Comput. Harmon. Anal. 30(3), 363–401 (2011b)

    MathSciNet  MATH  Article  Google Scholar 

  26. Robins, V.: Towards computing homology from finite approximations. In: Proceedings of the 14th Summer Conference on General Topology and Its Applications (1999)

  27. Weed, J., Bach, F.: Sharp asymptotic and finite-sample rates of convergence of empirical measures in Wasserstein distance. arXiv:1707.00087 (2017)

Download references

Acknowledgements

This research was supported in part by NSF Grants DMS-1418007, DMS-1722995 and DMS-1723003.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Haibin Hang.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Hang, H., Mémoli, F. & Mio, W. A topological study of functional data and Fréchet functions of metric measure spaces. J Appl. and Comput. Topology 3, 359–380 (2019). https://doi.org/10.1007/s41468-019-00037-8

Download citation

Keywords

  • Persistent homology
  • Functional data
  • Metric measure spaces

Mathematics Subject Classification

  • 55N35
  • 62-07
  • 60B05