WIAS Preprint No. 2788, (2020)

Statistical inference for Bures--Wasserstein barycenters


  • Kroshnin, Alexey
  • Spokoiny, Vladimir
    ORCID: 0000-0002-2040-3427
  • Suvorikova, Alexandra
    ORCID: 0000-0001-9115-7449

2010 Mathematics Subject Classification

  • 60F05


  • Bures-Wasserstein barycenters, central limit theorem, optimal transport




In 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.

Appeared in

Download Documents