Statistical inference for Bures-Wasserstein barycenters

01/02/2019

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by   Alexey Kroshnin, et al.

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In this work we introduce the concept of Bures-Wasserstein barycenter Q_*, that is essentially a Fréchet mean of some distribution supported on a subspace of positive semi-definite Hermitian operators _̋+(d). We allow a barycenter to be restricted to some affine subspace of _̋+(d) and provide conditions ensuring its existence and uniqueness. We also investigate convergence and concentration properties of an empirical counterpart of Q_* in both Frobenious 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.