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In this work we introduce the concept of Bures-Wasserstein barycenter $Q_*$, that is essentially a Frechet mean of some distribution $mathbb{P}$ supported on a subspace of positive semi-definite Hermitian operators $mathbb{H}_{+}(d)$. We allow a barycenter to be restricted to some affine subspace of $mathbb{H}_{+}(d)$ and 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 obtained results are connected to optimal transportation theory and can be applied to statistical inference in quantum mechanics.
We provide statistical theory for conditional and unconditional Wasserstein generative adversarial networks (WGANs) in the framework of dependent observations. We prove upper bounds for the excess Bayes risk of the WGAN estimators with respect to a m
In this thesis, we consider the Wasserstein barycenter problem of discrete probability measures from computational and statistical sides in two scenarios: (I) the measures are given and we need to compute their Wasserstein barycenter, and (ii) the me
In many applications, the dataset under investigation exhibits heterogeneous regimes that are more appropriately modeled using piece-wise linear models for each of the data segments separated by change-points. Although there have been much work on ch
Image interpolation, or image morphing, refers to a visual transition between two (or more) input images. For such a transition to look visually appealing, its desirable properties are (i) to be smooth; (ii) to apply the minimal required change in th
This paper gives a review of concentration inequalities which are widely employed in non-asymptotical analyses of mathematical statistics in a wide range of settings, from distribution-free to distribution-dependent, from sub-Gaussian to sub-exponent