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- 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.