Statistical inference for Bures–Wasserstein barycenters

By Alexey Kroshnin, Vladimir Spokoiny, Alexandra L. Suvorikova. The Annals of Applied Probability, 2019

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In this work we introduce the concept of Bures-Wasserstein barycenter $Q_*$, that is essentially a Frechet 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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/f5419a1503e8a8ba5c4190c00b520769945d2fe6