  In this work, we investigate Gaussian Processes indexed by multidimensional
distributions. While directly constructing radial positive definite kernels
based on the Wasserstein distance has been proven to be possible in the
unidimensional case, such constructions do not extend well to the
multidimensional case as we illustrate here. To tackle the problem of defining
positive definite kernels between multivariate distributions based on optimal
transport, we appeal instead to Hilbert space embeddings relying on optimal
transport maps to a reference distribution, that we suggest to take as a
Wasserstein barycenter. We characterize in turn radial positive definite
kernels on Hilbert spaces, and show that the covariance parameters of virtually
all parametric families of covariance functions are microergodic in the case of
(infinite-dimensional) Hilbert spaces. We also investigate statistical
properties of our suggested positive definite kernels on multidimensional
distributions, with a focus on consistency when a population Wasserstein
barycenter is replaced by an empirical barycenter and additional explicit
results in the special case of Gaussian distributions. Finally, we study the
Gaussian process methodology based on our suggested positive definite kernels
in regression problems with multidimensional distribution inputs, on simulation
data stemming both from synthetic examples and from a mechanical engineering
test case.
