Large Deviations Principle for Bures-Wasserstein Barycenters
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914951092764672 |
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| author | Jaffe, Adam Quinn Santoro, Leonardo V. |
| author_facet | Jaffe, Adam Quinn Santoro, Leonardo V. |
| contents | We prove the large deviations principle for empirical Bures-Wasserstein barycenters of independent, identically-distributed samples of covariance matrices and covariance operators. As an application, we explore some consequences of our results for the phenomenon of dimension-free concentration of measure for Bures-Wasserstein barycenters. Our theory reveals a novel notion of exponential tilting in the Bures-Wasserstein space, which, in analogy with Crámer's theorem in the Euclidean case, solves the relative entropy projection problem under a constraint on the barycenter. Notably, this method of proof is easy to adapt to other geometric settings of interest; with the same method, we obtain large deviations principles for empirical barycenters in Riemannian manifolds and the univariate Wasserstein space, and we obtain large deviations upper bounds for empirical barycenters in the general multivariate Wasserstein space. In fact, our results are the first known large deviations principles for Fréchet means in any non-linear metric space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_11384 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Large Deviations Principle for Bures-Wasserstein Barycenters Jaffe, Adam Quinn Santoro, Leonardo V. Probability Optimization and Control Statistics Theory 49Q22, 60F10, 62R30 We prove the large deviations principle for empirical Bures-Wasserstein barycenters of independent, identically-distributed samples of covariance matrices and covariance operators. As an application, we explore some consequences of our results for the phenomenon of dimension-free concentration of measure for Bures-Wasserstein barycenters. Our theory reveals a novel notion of exponential tilting in the Bures-Wasserstein space, which, in analogy with Crámer's theorem in the Euclidean case, solves the relative entropy projection problem under a constraint on the barycenter. Notably, this method of proof is easy to adapt to other geometric settings of interest; with the same method, we obtain large deviations principles for empirical barycenters in Riemannian manifolds and the univariate Wasserstein space, and we obtain large deviations upper bounds for empirical barycenters in the general multivariate Wasserstein space. In fact, our results are the first known large deviations principles for Fréchet means in any non-linear metric space. |
| title | Large Deviations Principle for Bures-Wasserstein Barycenters |
| topic | Probability Optimization and Control Statistics Theory 49Q22, 60F10, 62R30 |
| url | https://arxiv.org/abs/2409.11384 |