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- ItemGaussian processes with multidimensional distribution inputs via optimal transport and Hilbertian embedding(Ithaca, NY : Cornell University Library, 2020) Bachoc, François; Suvorikova, Alexandra; Ginsbourger, David; Loubes, Jean-Michel; Spokoiny, VladimirIn this work, we propose a way to construct Gaussian processes indexed by multidimensional distributions. More precisely, we tackle the problem of defining positive definite kernels between multivariate distributions via notions of optimal transport and appealing to Hilbert space embeddings. Besides presenting a characterization of radial positive definite and strictly positive definite kernels on general Hilbert spaces, we investigate the statistical properties of our theoretical and empirical kernels, focusing in particular on consistency as well as the special case of Gaussian distributions. A wide set of applications is presented, both using simulations and implementation with real data.
- ItemStatistical inference for Bures--Wasserstein barycenters(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Kroshnin, Alexey; Spokoiny, Vladimir; Suvorikova, AlexandraIn this work we introduce the concept of Bures--Wasserstein barycenter $Q_*$, that is essentially a Fréchet mean of some distribution $P$ supported on a subspace of positive semi-definite $d$-dimensional Hermitian operators $H_+(d)$. We allow a barycenter to be constrained to some affine subspace of $H_+(d)$, and we provide conditions ensuring its existence and uniqueness. We also investigate convergence and concentration properties of an empirical counterpart of $Q_*$ in both Frobenius norm and Bures--Wasserstein distance, and explain, how the obtained results are connected to optimal transportation theory and can be applied to statistical inference in quantum mechanics.