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Foundations of Computational Mathematics

, Volume 18, Issue 6, pp 1333–1396 | Cite as

Structure and Stability of the One-Dimensional Mapper

  • Mathieu Carrière
  • Steve Oudot
Article

Abstract

Given a continuous function \(f:X\rightarrow \mathbb {R}\) and a cover \(\mathcal {I}\) of its image by intervals, the Mapper is the nerve of a refinement of the pullback cover \(f^{-1}(\mathcal {I})\). Despite its success in applications, little is known about the structure and stability of this construction from a theoretical point of view. As a pixelized version of the Reeb graph of f, it is expected to capture a subset of its features (branches, holes), depending on how the interval cover is positioned with respect to the critical values of the function. Its stability should also depend on this positioning. We propose a theoretical framework that relates the structure of the Mapper to the one of the Reeb graphs, making it possible to predict which features will be present and which will be absent in the Mapper given the function and the cover, and for each feature, to quantify its degree of (in-)stability. Using this framework, we can derive guarantees on the structure of the Mapper, on its stability, and on its convergence to the Reeb graph as the granularity of the cover \(\mathcal {I}\) goes to zero.

Keywords

Topological data analysis Mapper Reeb graph Topological persistence 

Mathematics Subject Classification

55U10 68U05 

Notes

Acknowledgements

This work was supported by ERC grant Gudhi (ERC-2013-ADG-339025) and by ANR project TopData (ANR-13-BS01-0008). The authors would like to thank the anonymous referees for their constructive criticism and in particular for suggesting the connection to zigzag persistence, which had been overlooked in preliminary versions of this article. The second author acknowledges the support of ICERM and Brown University, as part of this work was carried out, while he was participating in the ICERM program Topology in Motion during the Fall of 2016.

Supplementary material

References

  1. 1.
    M. Alagappan, From 5 to 13: Redefining the Positions in Basketball. MIT Sloan Sports Analytics Conference, 2012.Google Scholar
  2. 2.
    A. Babu, Zigzag Coarsenings, Mapper Stability and Gene-network Analyses. PhD Thesis, 2013.Google Scholar
  3. 3.
    V. Barra and S. Biasotti, 3D Shape Retrieval and Classification using Multiple Kernel Learning on Extended Reeb graphs, The Visual Computer, 30 (2014), pp. 1247–1259.CrossRefGoogle Scholar
  4. 4.
    U. Bauer, X. Ge, and Y. Wang, Measuring Distance Between Reeb Graphs, in Proceedings of the 30th Symposium on Computational Geometry, 2014, pp. 464–473.Google Scholar
  5. 5.
    U. Bauer, E. Munch, and Y. Wang, Strong Equivalence of the Interleaving and Functional Distortion Metrics for Reeb Graphs, in Proceedings of the 31st Symposium on Computational Geometry, 2015.Google Scholar
  6. 6.
    S. Biasotti, D. Giorgi, M. Spagnuolo, and B. Falcidieno, Reeb Graphs for Shape Analysis and Applications, Theoretical Computer Science, 392 (2008), pp. 5–22.MathSciNetCrossRefGoogle Scholar
  7. 7.
    G. Carlsson, V. de Silva, and D. Morozov, Zigzag Persistent Homology and Real-valued Functions, in Proceedings of the 25th Symposium on Computational Geometry, 2009, pp. 247–256.Google Scholar
  8. 8.
    H. Carr and D. Duke, Joint Contour Nets, IEEE Transaction on Visualization and Computer Graphics, 20 (2014), pp. 1100–1113.CrossRefGoogle Scholar
  9. 9.
    M. Carrière, B. Michel, and S. Oudot, Statistical Analysis and Parameter Selection for Mapper, CoRR, abs/1706.00204 (2017).Google Scholar
  10. 10.
    M. Carrière and S. Oudot, Local Equivalence and Induced Metrics for Reeb Graphs, in Proceedings of the 33rd Symposium on Computational Geometry, 2017.Google Scholar
  11. 11.
    M. Carrière, S. Oudot, and M. Ovsjanikov, Local Signatures using Persistence Diagrams, HAL preprint, (2015).Google Scholar
  12. 12.
    A. Chattopadhyay, H. Carr, D. Duke, Z. Geng, and O. Saeki, Multivariate Topology Simplification, Computational Geometry, 58 (2016), pp. 1–24.MathSciNetCrossRefGoogle Scholar
  13. 13.
    F. Chazal, D. Cohen-Steiner, M. Glisse, L. Guibas, and S. Oudot, Proximity of Persistence Modules and their Diagrams, in Proceedings of the 25th Symposium on Computational Geometry, 2009, pp. 237–246.Google Scholar
  14. 14.
    F. Chazal, V. de Silva, M. Glisse, and S. Oudot, The Structure and Stability of Persistence Modules, Springer, 2016.zbMATHGoogle Scholar
  15. 15.
    F. Chazal, L. Guibas, S. Oudot, and P. Skraba, Analysis of scalar fields over point cloud data, in Proceedings of the 20th Symposium on Discrete Algorithm, 2009, pp. 1021–1030.CrossRefGoogle Scholar
  16. 16.
    F. Chazal and J. Sun, Gromov-Hausdorff Approximation of Filament Structure Using Reeb-type Graph, in Proceedings of the 30th Symposium on Computational Geometry, 2014.Google Scholar
  17. 17.
    D. Cohen-Steiner, H. Edelsbrunner, and J. Harer, Stability of Persistence Diagrams, Discrete and Computational Geometry, 37 (2007), pp. 103–120.MathSciNetCrossRefGoogle Scholar
  18. 18.
    D. Cohen-Steiner, H. Edelsbrunner, and J. Harer, Extending persistence using Poincaré and Lefschetz duality, Foundation of Computational Mathematics, 9 (2009), pp. 79–103.CrossRefGoogle Scholar
  19. 19.
    E. Colin de Verdière, G. Ginot, and X. Goaoc, Multinerves and Helly numbers of acyclic families, in Proceedings of the 28th Symposium on Computational Geometry, 2012, pp. 209–218.Google Scholar
  20. 20.
    V. de Silva, E. Munch, and A. Patel, Categorified Reeb Graphs, Discrete and Computational Geometry, 55 (2016), pp. 854–906.MathSciNetCrossRefGoogle Scholar
  21. 21.
    T. Dey, F. Fan, and Y. Wang, Graph Induced Complex on Point Data, in Proceedings of the 29th Symposium on Computational Geometry, 2013, pp. 107–116.Google Scholar
  22. 22.
    T. Dey, F. Mémoli, and Y. Wang, Multiscale Mapper: Topological Summarization via Codomain Covers, in Proceedings of the 27th Symposium on Discrete Algorithms, 2016, pp. 997–1013.Google Scholar
  23. 23.
    T. Dey and Y. Wang, Reeb Graphs: Approximation and Persistence, Discrete and Computational Geometry, 49 (2013), pp. 46–73.MathSciNetCrossRefGoogle Scholar
  24. 24.
    H. Edelsbrunner, J. Harer, and A. Patel, Reeb Spaces of Piecewise Linear Mappings, in Proceedings of the 24th Symposium on Computational Geometry, 2008, pp. 242–250.Google Scholar
  25. 25.
    H. Edelsbrunner, D. Letscher, and A. Zomorodian, Topological Persistence and Simplification, Discrete and Computational Geometry, 28 (2002), pp. 511–533.MathSciNetCrossRefGoogle Scholar
  26. 26.
    W. Harvey, Y. Wang, and R. Wenger, A randomized O(m log m) time algorithm for computing Reeb graphs of arbitrary simplicial complexes, in Proceedings of the 26th Symposium on Computational Geometry, 2010, pp. 267–276.Google Scholar
  27. 27.
    V. Kurlin, A one-dimensional homologically persistent skeleton of an unstructured point cloud in any metric space, in Proceedings of the 13th Symposium on Geometry Processing, 2015.CrossRefGoogle Scholar
  28. 28.
    D. Morozov, Homological Illusions of Persistence and Stability, Ph.D. dissertation, Department of Computer Science, Duke University, 2008.Google Scholar
  29. 29.
    E. Munch and B. Wang, Convergence between Categorical Representations of Reeb Space and Mapper, in Proceedings of the 32nd Symposium on Computational Geometry, vol. 51, 2016, pp. 53:1–53:16.Google Scholar
  30. 30.
    M. Nicolau, A. Levine, and G. Carlsson, Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival, Proceedings of the National Academy of Science, 108 (2011), pp. 7265–7270.CrossRefGoogle Scholar
  31. 31.
    G. Reeb, Sur les points singuliers d’une forme de Pfaff complètement intégrable ou d’une fonction numérique, Compte Rendu de l’Académie des Science de Paris, 222 (1946), pp. 847–849.zbMATHGoogle Scholar
  32. 32.
    G. Singh, F. Mémoli, and G. Carlsson, Topological Methods for the Analysis of High Dimensional Data Sets and 3D Object Recognition, in Symposium on Point Based Graphics, 2007.Google Scholar
  33. 33.
    R. B. Stovner, On the Mapper Algorithm. Master Thesis, 2012.Google Scholar
  34. 34.
    W. Sutherland, Introduction to Metric and Topological Spaces, Oxford University Press, 2009.Google Scholar

Copyright information

© SFoCM 2017

Authors and Affiliations

  1. 1.Inria Saclay, 1 rue Honoré d’Estienne d’Orves, Bâtiment Alan TuringEcole PolytechniquePalaiseauFrance

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