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


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.


Persistence diagram Fréchet mean Topological data analysis Alexandrov space Persistent homology 



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.


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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

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