Document Open Access Logo

The Density of Expected Persistence Diagrams and its Kernel Based Estimation

Authors Frédéric Chazal, Vincent Divol

Thumbnail PDF


  • Filesize: 0.78 MB
  • 15 pages

Document Identifiers

Author Details

Frédéric Chazal
Vincent Divol

Cite AsGet BibTex

Frédéric Chazal and Vincent Divol. The Density of Expected Persistence Diagrams and its Kernel Based Estimation. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 26:1-26:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


Persistence diagrams play a fundamental role in Topological Data Analysis where they are used as topological descriptors of filtrations built on top of data. They consist in discrete multisets of points in the plane R^2 that can equivalently be seen as discrete measures in R^2. When the data come as a random point cloud, these discrete measures become random measures whose expectation is studied in this paper. First, we show that for a wide class of filtrations, including the Cech and Rips-Vietoris filtrations, the expected persistence diagram, that is a deterministic measure on R^2, has a density with respect to the Lebesgue measure. Second, building on the previous result we show that the persistence surface recently introduced in [Adams et al., 2017] can be seen as a kernel estimator of this density. We propose a cross-validation scheme for selecting an optimal bandwidth, which is proven to be a consistent procedure to estimate the density.
  • topological data analysis
  • persistence diagrams
  • subanalytic geometry


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Henry Adams, Tegan Emerson, Michael Kirby, Rachel Neville, Chris Peterson, Patrick Shipman, Sofya Chepushtanova, Eric Hanson, Francis Motta, and Lori Ziegelmeier. Persistence images: a stable vector representation of persistent homology. Journal of Machine Learning Research, 18(8):1-35, 2017. Google Scholar
  2. Christophe Biscio and Jesper Møller. The accumulated persistence function, a new useful functional summary statistic for topological data analysis, with a view to brain artery trees and spatial point process applications. arXiv preprint arXiv:1611.00630, 2016. Google Scholar
  3. Omer Bobrowski, Matthew Kahle, Primoz Skraba, et al. Maximally persistent cycles in random geometric complexes. The Annals of Applied Probability, 27(4):2032-2060, 2017. Google Scholar
  4. Peter Bubenik. Statistical topological data analysis using persistence landscapes. The Journal of Machine Learning Research, 16(1):77-102, 2015. Google Scholar
  5. Mickaël Buchet, Frédéric Chazal, Steve Y Oudot, and Donald R Sheehy. Efficient and robust persistent homology for measures. Computational Geometry, 58:70-96, 2016. Google Scholar
  6. F. Chazal, D. Cohen-Steiner, L. J. Guibas, F. Memoli, and S. Y. Oudot. Gromov-hausdorff stable signatures for shapes using persistence. Computer Graphics Forum (proc. SGP 2009), pages 1393-1403, 2009. Google Scholar
  7. F. Chazal, V. de Silva, and S. Oudot. Persistence stability for geometric complexes. Geometriae Dedicata, 173(1):193-214, 2014. Google Scholar
  8. Frédéric Chazal, Vin de Silva, Marc Glisse, and Steve Oudot. The structure and stability of persistence modules. SpringerBriefs in Mathematics. Springer, 2016. Google Scholar
  9. Frédéric Chazal and Vincent Divol. The density of expected persistence diagrams and its kernel based estimation. Extended version of a paper to appear in the proceedings of the Symposium of Computational Geometry 2018, 2018. URL:
  10. Frédéric Chazal, Brittany Terese Fasy, Fabrizio Lecci, Alessandro Rinaldo, and Larry Wasserman. Stochastic convergence of persistence landscapes and silhouettes. In Proceedings of the thirtieth annual symposium on Computational geometry, page 474. ACM, 2014. Google Scholar
  11. Frédéric Chazal, Marc Glisse, Catherine Labruère, and Bertrand Michel. Convergence rates for persistence diagram estimation in topological data analysis. Journal of Machine Learning Research, 16:3603-3635, 2015. URL:
  12. Yen-Chi Chen, Daren Wang, Alessandro Rinaldo, and Larry Wasserman. Statistical analysis of persistence intensity functions. arXiv preprint arXiv:1510.02502, 2015. Google Scholar
  13. David Cohen-Steiner, Herbert Edelsbrunner, and John Harer. Stability of persistence diagrams. Discrete &Computational Geometry, 37(1):103-120, 2007. Google Scholar
  14. HSM Coxeter. The circumradius of the general simplex. The Mathematical Gazette, pages 229-231, 1930. Google Scholar
  15. Trinh Khanh Duy, Yasuaki Hiraoka, and Tomoyuki Shirai. Limit theorems for persistence diagrams. arXiv preprint arXiv:1612.08371, 2016. Google Scholar
  16. B. T. Fasy, F. Lecci, A. Rinaldo, L. Wasserman, S. Balakrishnan, A. Singh, et al. Confidence sets for persistence diagrams. The Annals of Statistics, 42(6):2301-2339, 2014. Google Scholar
  17. D. Morozov H. Edelsbrunner. Persistent homology. In Handbook of Discrete and Computational Geometry (3rd Ed - To appear). CRC Press (to appear), 2017. Google Scholar
  18. Matthew Kahle, Elizabeth Meckes, et al. Limit the theorems for betti numbers of random simplicial complexes. Homology, Homotopy and Applications, 15(1):343-374, 2013. Google Scholar
  19. Genki Kusano, Kenji Fukumizu, and Yasuaki Hiraoka. Kernel method for persistence diagrams via kernel embedding and weight factor. arXiv preprint arXiv:1706.03472, 2017. Google Scholar
  20. Genki Kusano, Yasuaki Hiraoka, and Kenji Fukumizu. Persistence weighted gaussian kernel for topological data analysis. In International Conference on Machine Learning, pages 2004-2013, 2016. Google Scholar
  21. Claire Lacour, Pascal Massart, and Vincent Rivoirard. Estimator selection: a new method with applications to kernel density estimation. arXiv preprint arXiv:1607.05091, 2016. Google Scholar
  22. Michel Ledoux and Michel Talagrand. Probability in Banach Spaces: isoperimetry and processes. Springer Science &Business Media, 2013. Google Scholar
  23. F. Morgan. Geometric Measure Theory: A Beginner’s Guide. Elsevier Science, 2016. Google Scholar
  24. Jan Reininghaus, Stefan Huber, Ulrich Bauer, and Roland Kwitt. A stable multi-scale kernel for topological machine learning. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 4741-4748, 2015. Google Scholar
  25. M. Shiota. Geometry of Subanalytic and Semialgebraic Sets. Progress in mathematics. Springer, 1997. Google Scholar
  26. Charles J Stone. An asymptotically optimal window selection rule for kernel density estimates. The Annals of Statistics, pages 1285-1297, 1984. Google Scholar
  27. Katharine Turner, Yuriy Mileyko, Sayan Mukherjee, and John Harer. Fréchet means for distributions of persistence diagrams. Discrete &Computational Geometry, 52(1):44-70, 2014. Google Scholar
  28. Yuhei Umeda. Time series classification via topological data analysis. Transactions of the Japanese Society for Artificial Intelligence, 32(3):D-G72_1, 2017. Google Scholar
  29. D Yogeshwaran, Robert J Adler, et al. On the topology of random complexes built over stationary point processes. The Annals of Applied Probability, 25(6):3338-3380, 2015. Google Scholar
  30. D. Yogeshwaran, Eliran Subag, and Robert J. Adler. Random geometric complexes in the thermodynamic regime. Probability Theory and Related Fields, 167(1):107-142, Feb 2017. URL:
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail