Description
Barycenters (or mean squared error estimators) play a distinguished role
in statistics and information theory. This concept is boring in
Euclidean spaces in the sense that it coincides with the weighted
average. However, non-Euclidean metrics and other distance-like
functions (such as relative entropies) are often more natural than the
flat metric from the viewpoint of applications.
First, we take the submanifold of centered Gaussian measures in the
space of square integrable random variables in R^d endowed with the
optimal transport (Wasserstein) distance to illustrate the challenges of
computing the barycenter in non-flat metrics. We will also discuss the
closely related Riemannian trace metric on positive operators, which is
defined by the Hessian of the Boltzmann entropy, from this viewpoint.
Then we turn to quantum information theory and consider generalized
quantum Hellinger divergences, that belong to the family of maximal
quantum f-divergences and behave like squared distances in some sense to
be clarified during the talk.
We derive a characterization of the barycenters for these divergences
and compare our results to those of Bhatia et al. [Lett. Math. Phys.
(2019), in press, arXiv:1901.01378v1]. We note that the characterization
given by Bhatia et al. is not correct in general, albeit it is true for
commuting operators.
Based on joint work with Jozsef Pitrik (arXiv:1903.10455)