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Ruprecht-Karls-Universität Heidelberg Institute of Applied Mathematics Statistics of inverse problems Research Group
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Last edited on
Oct 18, 2021 by JJ
Master in Mathematics

Nathawut Phandoidaen

Minimax estimation of Wasserstein barycenters

Christof Schötz and Jan JOHANNES

In this Master’s thesis we study the space of probability measures with second finite moment under the 2–Wasserstein distance. From there, we introduce the Fréchet mean, namely the empirical 2–Wasserstein barycenter, as a sensible notion of a mean. Our statistical analysis begins with the observation of real random variables (Xij)1≤i≤n,1≤j≤pi where each experiment (Xij)1≤j≤pi is independently generated by an underlying non–observable random measure νi for 1 ≤ i ≤ n, pi ∈ N . At the same time, we assume that ν1,…,νn are independent realizations of a stochastic process themselves. The main focus of this thesis lies on finding an estimator for an appropriate structural mean measure and the study of its performance in a non–smoothed as well as a smoothed version. In a nonparametric context we provide lower and upper bounds for the minimax risk with respect to the 2–Wasserstein distance. Our theoretical results are applied to different examples, models and to a conclusive numerical experiment.

J. Bigot, R. Gouet, T. Klein, and A. López. Minimax convergence rate for estimating the Wasserstein barycenter of random measures on the real line. Technical report, arXiv:1606.03933, 2017.