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14-18 August 2017
Uppsala University
Europe/Stockholm timezone

Fréchet means and Procrustes analysis in Wasserstein space

15 Aug 2017, 15:30
Ångströmslaboratoriet (Uppsala University)


Uppsala University



Dr Yoav Zemel (Ecole polytechnique fédérale de Lausanne)


We consider three interlinked problems in stochastic geometry: (1) constructing optimal multicouplings of random vectors; (2) determining the Fréchet mean of probability measures in Wasserstein space; and (3) registering collections of randomly deformed spatial point processes. We demonstrate how these problems are canonically interpreted through the prism of the theory of optimal transportation of measure on $\mathbb R^d$. We provide explicit solutions in the one dimensional case, consistently solve the registration problem and establish convergence rates and a (tangent space) central limit theorem for Cox processes. When $d>1$, the solutions are no longer explicit and we propose a steepest descent algorithm for deducing the Fréchet mean in problem (2). Supplemented by uniform convergence results for the optimal maps, this furnishes a solution to the multicoupling problem (1). The latter is then utilised, as in the case $d=1$, in order to construct consistent estimators for the registration problem (3). While the consistency results parallel their one-dimensional counterparts, their derivation requires more sophisticated techniques from convex analysis. This is joint work with Victor M. Panaretos

Primary author

Dr Yoav Zemel (Ecole polytechnique fédérale de Lausanne)


Prof. Victor Panaretos (Ecole polytechnique fédérale de Lausanne)

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