Statistical inference for Bures–Wasserstein barycenters

Abstract

In this work we introduce the concept of Bures–Wasserstein barycenter ${\mathit{Q}}_{\ast }$, that is essentially a Fréchet mean of some distribution $\mathbb{P}$ supported on a subspace of positive semi-definite d-dimensional Hermitian operators ${\mathbb{H}}_{+}(\mathit{d})$. We allow a barycenter to be constrained to some affine subspace of ${\mathbb{H}}_{+}(\mathit{d})$, and we provide conditions ensuring its existence and uniqueness. We also investigate convergence and concentration properties of an empirical counterpart of ${\mathit{Q}}_{\ast }$ 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.