Statistical inference for Bures--Wasserstein barycenters
- Kroshnin, Alexey
- Spokoiny, Vladimir
- Suvorikova, Alexandra
2010 Mathematics Subject Classification
- 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.
- Ann. Appl. Probab., 31 (2021), pp. 1264--1298, DOI 10.1214/20-AAP1618 .