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Lipschitz Functions Have L p -Stable Persistence

Abstract

We prove two stability results for Lipschitz functions on triangulable, compact metric spaces and consider applications of both to problems in systems biology. Given two functions, the first result is formulated in terms of the Wasserstein distance between their persistence diagrams and the second in terms of their total persistence.

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References

  1. 1.

    P.N. Benfey, B. Scheres, Root development, J. Curr. Biol. 10, R813–815 (2000).

  2. 2.

    S.M. Brady, D.A. Orlando, J.-Y. Lee, J.Y. Wang, J. Koch, J.R. Dinneny, D. Mace, U. Ohler, P.N. Benfey, A high-resolution root spatiotemporal map reveals dominant expression patterns, Science 318, 801–806 (2007).

  3. 3.

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

  4. 4.

    D. Cohen-Steiner, H. Edelsbrunner, J. Harer, Extending persistence using Poincaré and Lefschetz duality, Found. Comput. Math. 9, 79–103 (2009) and 133–134.

  5. 5.

    M.-L. Dequéant, E. Glynn, K. Gaudenz, M. Wahl, J. Chen, A. Mushegian, O. Pourquié, A complex oscillating network of signaling genes underlies the mouse segmentation clock, Science 314, 1595–1598 (2006).

  6. 6.

    M.-L. Dequéant, S. Ahnert, H. Edelsbrunner, T.M.A. Fink, E.F. Glynn, G. Hattem, A. Kudlicki, Y. Mileyko, J. Morton, A.R. Mushegian, L. Pachter, M. Rowicka, A. Shiu, B. Sturmfels, O. Pourquié, Comparison of pattern detection methods in microarray time series of the segmentation clock, PLoS ONE 3, e2856 (2008). doi:10.1371/journal.pone.0002856.

  7. 7.

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

  8. 8.

    H. Edelsbrunner, D. Morozov, V. Pascucci, Persistence-sensitive simplification of functions on 2-manifolds, in Proc. 22nd Ann. Sympos. Comput. Geom. (2006), pp. 127–134.

  9. 9.

    P. Frosini, C. Landi, Size theory as a topological tool for computer vision, Pattern Recognit. Image Anal. 9, 596–603 (1999).

  10. 10.

    L.V. Kantorovich, On the translocation of masses, C.R. (Dokl.) Acad. Sci. URSS 37, 199–226 (1942).

  11. 11.

    G. Monge, Mémoire sur la théorie des déblais et des remblais, in Histoire de l’Académie Royale des Sciences de Paris (1781), pp. 666–704.

  12. 12.

    J.R. Munkres, Elements of Algebraic Topology (Addison-Wesley, Redwood City, 1984).

  13. 13.

    O. Pourquié, The segmentation clock: converting embryonic time into spatial pattern, Science 301, 328–330 (2003).

  14. 14.

    L.N. Wasserstein, Markov processes over denumerable products of spaces describing large systems of automata, Probl. Inf. Transm. 5, 47–52 (1969).

  15. 15.

    A. Zomorodian, G. Carlsson, Computing persistent homology, Discrete Comput. Geom. 33, 249–274 (2005).

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Author information

Correspondence to Herbert Edelsbrunner.

Additional information

This research is partially supported by the Defense Advanced Research Projects Agency (DARPA) under grants HR0011-05-1-0007 and HR0011-05-1-0057 and by CNRS under grant PICS-3416.

Communicated By Peter Olver.

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Cohen-Steiner, D., Edelsbrunner, H., Harer, J. et al. Lipschitz Functions Have L p -Stable Persistence. Found Comput Math 10, 127–139 (2010). https://doi.org/10.1007/s10208-010-9060-6

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Keywords

  • Continuous functions
  • Metric spaces
  • Persistent homology
  • Wasserstein distance
  • Total persistence
  • Stability
  • Gene expression
  • Comparison
  • Classification

Mathematics Subject Classification (2000)

  • 55N99
  • 68W30
  • 92-08