Large Deviations Principle for Bures-Wasserstein Barycenters

The Bures-Wasserstein space of Gaussian probability measures plays an important role in statistics and machine learning, where it has recently been appreciated that a canonical notion of averaging Gaussian measures arises as computing barycenters in the Bures-Wasserstein geometry. For empirical Bures-Wasserstein barycenters of independent, identically-distributed samples, much is known about their convergence to a population counterpart (strong law of large numbers, central limit theorem, rates of convergence, concentration inequalities, etc.).

In this talk, we add to this story by proving the large deviations principle for Bures-Wasserstein barycenters, which allows us to compute the exact exponential rate of decay of many different rare events of interest. We also give several statistical and probabilistic applications of these results.