Statistical inference for Bures-Wasserstein barycenters (1901.00226v2)

Abstract: In this work we introduce the concept of Bures-Wasserstein barycenter $Q_$, that is essentially a Fr\'echet 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 $Q_$ 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.