• DOI:10.1007/s00454-014-9604-7
• Corpus ID: 14293062

Fréchet Means for Distributions of Persistence Diagrams

@article{Turner2014FrchetMF,
  title={Fr{\'e}chet Means for Distributions of Persistence Diagrams},
  author={Katharine Turner and Yuriy Mileyko and Sayan Mukherjee and John Harer},
  journal={Discrete \& Computational Geometry},
  year={2014},
  volume={52},
  pages={44-70}
}

• Katharine Turner, Yuriy Mileyko, +1 author J. Harer
• Published 13 June 2012
• Mathematics, Computer Science
• Discrete & Computational Geometry

Given a distribution $$\rho $$ρ on persistence diagrams and observations $$X_{1},\ldots ,X_{n} \mathop {\sim }\limits ^{iid} \rho $$X1,…,Xn∼iidρ we introduce an algorithm in this paper that estimates a Fréchet mean from the set of diagrams $$X_{1},\ldots ,X_{n}$$X1,…,Xn. If the underlying measure $$\rho $$ρ is a combination of Dirac masses $$\rho = \frac{1}{m} \sum _{i=1}^{m} \delta _{Z_{i}}$$ρ=1m∑i=1mδZi 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 $$\rho $$ρ. We illustrate the convergence of an empirical mean computed by the algorithm to a population mean by simulations from Gaussian random fields.