Statistical inference for Bures-Wasserstein barycenters

Abstract

In this work we introduce the concept of Bures-Wasserstein barycenter $Q_{\ast }$, that is essentially a Fréchet 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 $Q_{\ast }$ 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.

37 pages, 5 figures