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)
