Fréchet Means for Distributions of Persistence Diagrams. Turner, K.; Mileyko, Y.; Mukherjee, S.; and Harer, J. 52(1):44-70.
Fréchet Means for Distributions of Persistence Diagrams [link]Paper  doi  abstract   bibtex   
Given a distribution ρ\textbackslashrho on persistence diagrams and observations X1,…,Xn∼iidρX_\1\,\textbackslashldots ,X_\n\ \textbackslashmathop \\textbackslashsim \\textbackslashlimits \^\iid\ \textbackslashrho we introduce an algorithm in this paper that estimates a Fréchet mean from the set of diagrams X1,…,XnX_\1\,\textbackslashldots ,X_\n\. If the underlying measure ρ\textbackslashrho is a combination of Dirac masses ρ=1m∑mi=1δZi\textbackslashrho = \textbackslashfrac\1\\m\ \textbackslashsum _\i=1\\^\m\ \textbackslashdelta _\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 ρ\textbackslashrho . We illustrate the convergence of an empirical mean computed by the algorithm to a population mean by simulations from Gaussian random fields.
@article{turnerFrechetMeansDistributions2014,
  langid = {english},
  title = {Fréchet {{Means}} for {{Distributions}} of {{Persistence Diagrams}}},
  volume = {52},
  issn = {0179-5376, 1432-0444},
  url = {https://link.springer.com/article/10.1007/s00454-014-9604-7},
  doi = {10.1007/s00454-014-9604-7},
  abstract = {Given a distribution ρ\textbackslash{}rho on persistence diagrams and observations X1,…,Xn∼iidρX\_\{1\},\textbackslash{}ldots ,X\_\{n\} \textbackslash{}mathop \{\textbackslash{}sim \}\textbackslash{}limits \^\{iid\} \textbackslash{}rho we introduce an algorithm in this paper that estimates a Fréchet mean from the set of diagrams X1,…,XnX\_\{1\},\textbackslash{}ldots ,X\_\{n\}. If the underlying measure ρ\textbackslash{}rho is a combination of Dirac masses ρ=1m∑mi=1δZi\textbackslash{}rho = \textbackslash{}frac\{1\}\{m\} \textbackslash{}sum \_\{i=1\}\^\{m\} \textbackslash{}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 ρ\textbackslash{}rho . We illustrate the convergence of an empirical mean computed by the algorithm to a population mean by simulations from Gaussian random fields.},
  number = {1},
  journaltitle = {Discrete \& Computational Geometry},
  shortjournal = {Discrete Comput Geom},
  urldate = {2018-04-20},
  date = {2014-07-01},
  pages = {44-70},
  author = {Turner, Katharine and Mileyko, Yuriy and Mukherjee, Sayan and Harer, John},
  file = {/home/dimitri/Nextcloud/Zotero/storage/WFNRGRL6/Turner et al. - 2014 - Fréchet Means for Distributions of Persistence Dia.pdf;/home/dimitri/Nextcloud/Zotero/storage/TIA4XC3D/s00454-014-9604-7.html}
}
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