  Gaussian Process regression is a kernel method successfully adopted in many
real-life applications. Recently, there is a growing interest on extending this
method to non-Euclidean input spaces, like the one considered in this paper,
consisting of probability measures. Although a Positive Definite kernel can be
defined by using a suitable distance -- the Wasserstein distance -- the common
procedure for learning the Gaussian Process model can fail due to numerical
issues, arising earlier and more frequently than in the case of an Euclidean
input space and, as demonstrated in this paper, that cannot be avoided by
adding artificial noise (nugget effect) as usually done. This paper uncovers
the main reason of these issues, that is a non-stationarity relationship
between the Wasserstein-based squared exponential kernel and its
Euclidean-based counterpart. As a relevant result, the Gaussian Process model
is learned by assuming the input space as Euclidean and then an algebraic
transformation, based on the uncovered relation, is used to transform it into a
non-stationary and Wasserstein-based Gaussian Process model over probability
measures. This algebraic transformation is simpler than log-exp maps used in
the case of data belonging to Riemannian manifolds and recently extended to
consider the pseudo-Riemannian structure of an input space equipped with the
Wasserstein distance.
