Statistical inference for Bures-Wasserstein barycenters
Abstract
In this work we introduce the concept of Bures-Wasserstein barycenter $Q_*$, 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 $Q_*$ 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.
- Publication:
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arXiv e-prints
- Pub Date:
- January 2019
- DOI:
- 10.48550/arXiv.1901.00226
- arXiv:
- arXiv:1901.00226
- Bibcode:
- 2019arXiv190100226K
- Keywords:
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- Mathematics - Statistics Theory;
- Statistics - Applications
- E-Print:
- 37 pages, 5 figures