Discrete & Computational Geometry

, Volume 52, Issue 1, pp 44–70 | Cite as

Fréchet Means for Distributions of Persistence Diagrams

  • Katharine Turner
  • Yuriy Mileyko
  • Sayan Mukherjee
  • John Harer
Article

Abstract

Given a distribution \(\rho \) on persistence diagrams and observations \(X_{1},\ldots ,X_{n} \mathop {\sim }\limits ^{iid} \rho \) we introduce an algorithm in this paper that estimates a Fréchet mean from the set of diagrams \(X_{1},\ldots ,X_{n}\). If the underlying measure \(\rho \) is a combination of Dirac masses \(\rho = \frac{1}{m} \sum _{i=1}^{m} \delta _{Z_{i}}\) then we prove the algorithm converges to a local minimum and a law of large numbers result for a Fréchet mean computed by the algorithm given observations drawn iid from \(\rho \). We illustrate the convergence of an empirical mean computed by the algorithm to a population mean by simulations from Gaussian random fields.

Keywords

Persistence diagram Fréchet mean Topological data analysis Alexandrov space Persistent homology 

Notes

Acknowledgments

SM and KT would like to acknowledge Shmuel Weinberger for discussions and insight. SM and KT would like to acknowledge E. Subag with help in obtaining persistence diagrams computed from random Gaussian fields and explaining the generative model. JH and YM are pleased to acknowledge the support from grants DTRA: HDTRA1-08-BRCWMD, DARPA: D12AP00001On, AFOSR: FA9550-10-1-0436, and NIH (Systems Biology): 5P50-GM081883. SM is pleased to acknowledge support from grants NIH (Systems Biology): 5P50-GM081883, AFOSR: FA9550-10-1-0436, and NSF CCF-1049290.

References

  1. 1.
    Adler, R.J., Bobrowski, O., Borman, M.S., Subag, E., Weinberger, S.: Persistent homology for random fields and complexes. In: Berger, J.O., Tony Cai, T., Johnstone, I.M. (eds.) Borrowing Strength: Theory Powering Applications—A Festschrift for Lawrence D. Brown, vol. 6. Institute of Mathematical Statistics, Beachwood (2010)Google Scholar
  2. 2.
    Bendich, P., Mukherjee, S., Wang B.: Local homology transfer and stratification learning. In: ACM-SIAM Symposium on Discrete Algorithms (2012)Google Scholar
  3. 3.
    Birdson, M.R., Haefliger, A.: Metric Spaces of Non-positive Curvature. Springer-Verlag, Berlin (1999)CrossRefGoogle Scholar
  4. 4.
    Bubenik, P., Carlsson, G., Kim, P.T., Luo, Z.-M.: Statistical topology via Morse theory, persistence, and nonparametric estimation. In: Viana, M.A.G., Wynn, H.P. (eds.) Algebraic Methods in Statistics and Probability II. Contemporary Mathematics, vol. 516, pp. 75–92. American Mathematical Society, Providence (2010)CrossRefGoogle Scholar
  5. 5.
    Burago, Y., Gromov, M., Perel’man, G.: A.D. Alexandrov spaces with curvature bounded below. Russ. Math. Surv. 47(2), 1–58 (1992)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Chazal, F., Cohen-Steiner, D., Lieutier, A.: A sampling theory for compact sets in Euclidean space. Discrete Comput. Geom. 41, 461–479 (2009)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Cohen-Steiner, D., Edelsbrunner, H., Harer, J., Mileyko, Y.: Lipschitz functions have \({L}_p\)-stable persistence. Found. Comput. Math. 10, 127–139 (2010). doi:  10.1007/s10208-010-9060-6 CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Edelsbrunner, H., Harer, J.: Computational Topology: An Introduction. American Mathematical Society, Providence (2010)Google Scholar
  9. 9.
    Gromov, M.: Hyperbolic groups. In: Gersten, S.M. (ed.) Essays in Group Theory. Mathematical Sciences Research Institute Publications, vol. 8, pp. 75–263. Springer, New York (1987)Google Scholar
  10. 10.
    Kahle, M.: Topology of random clique complexes. Discrete Math. 309(6), 1658–1671 (2009)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Kahle, M.: Random geometric complexes (2011). http://arxiv.org/abs/0910.1649
  12. 12.
    Kahle, M., Meckes, E.: Limit theorems for Betti numbers of random simplicial complexes (2010). http://arxiv.org/abs/1009.4130v3
  13. 13.
    Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. Ann. Math. 169, 903–991 (2009)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Lunagómez, S., Mukherjee, S., Wolpert, R.L.: Geometric representations of hypergraphs for prior specification and posterior sampling (2009). http://arxiv.org/abs/0912.3648
  15. 15.
    Lytchak, A.: Open map theorem for metric spaces. St. Petersbg. Math. J. 17(3), 477–491 (2006)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Mileyko, Y., Mukherjee, S., Harer, J.: Probability measures on the space of persistence diagrams. Inverse Probab. 27(12), 124007 (2012)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Molchanov, I.: Theory of Random Sets. Springer, London (2005)MATHGoogle Scholar
  18. 18.
    Munkres, J.: Algorithms for the assignment and transportation problems. J. Soc. Ind. Appl. Math. 5(1), 32–38 (1957)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Niyogi, P., Smale, S., Weinberger, S.: Finding the homology of submanifolds with high confidence from random samples. Discrete Comput. Geom. 39, 419–441 (2008)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Niyogi, P., Smale, S., Weinberger, S.: A topological view of unsupervised a topological view of unsupervised learning from noisy data. Manuscript (2008)Google Scholar
  21. 21.
    Ohta, S.: Barycenters in Alexandrov spaces with curvature bounded below. Adv. Geom. 12, 571–587 (2012)MATHMathSciNetGoogle Scholar
  22. 22.
    Penrose, M.D.: Random Geometric Graphs. Oxford University Press, New York (2003)CrossRefMATHGoogle Scholar
  23. 23.
    Penrose, M.D., Yukich, J.E.: Central limit theorems for some graphs in computational geometry. Ann. Appl. Probab. 11(4), 1005–1041 (2001)MATHMathSciNetGoogle Scholar
  24. 24.
    Petrunin, A.: Semiconcave functions in Alexandrov’s geometry. Surv. Differ. Geom. 11, 137–201 (2007)CrossRefMathSciNetGoogle Scholar
  25. 25.
    Sturm, K.-T.: Probability measures on metric spaces of nonpositive curvature. In: Auscher, P., Coulhon, T., Grigoryan, A. (eds.) Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces, vol. 338. American Mathematical Society, Providence (2002)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Katharine Turner
    • 1
  • Yuriy Mileyko
    • 2
  • Sayan Mukherjee
    • 3
  • John Harer
    • 4
  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA
  2. 2.Department of MathematicsUniversity of Hawaii at ManoaHonoluluUSA
  3. 3.Departments of Statistical Science, Computer Science, and Mathematics, Institute for Genome Sciences & PolicyDuke UniversityDurhamUSA
  4. 4.Departments of Mathematics, Computer Science and Electrical and Computer EngineeringCenter for Systems Biology, Duke UniversityDurhamUSA

Personalised recommendations