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# 20th European Young Statisticians Meeting

14-18 August 2017
Uppsala University
Europe/Stockholm timezone

## Fréchet means and Procrustes analysis in Wasserstein space

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

Speaker

### Speaker

Dr Yoav Zemel (Ecole polytechnique fédérale de Lausanne)

### Description

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)

### Co-author

Prof. Victor Panaretos (Ecole polytechnique fédérale de Lausanne)

### Presentation Materials

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