  Although optimal transport (OT) problems admit closed form solutions in a
very few notable cases, e.g. in 1D or between Gaussians, these closed forms
have proved extremely fecund for practitioners to define tools inspired from
the OT geometry. On the other hand, the numerical resolution of OT problems
using entropic regularization has given rise to many applications, but because
there are no known closed-form solutions for entropic regularized OT problems,
these approaches are mostly algorithmic, not informed by elegant closed forms.
In this paper, we propose to fill the void at the intersection between these
two schools of thought in OT by proving that the entropy-regularized optimal
transport problem between two Gaussian measures admits a closed form. Contrary
to the unregularized case, for which the explicit form is given by the
Wasserstein-Bures distance, the closed form we obtain is differentiable
everywhere, even for Gaussians with degenerate covariance matrices. We obtain
this closed form solution by solving the fixed-point equation behind Sinkhorn's
algorithm, the default method for computing entropic regularized OT.
Remarkably, this approach extends to the generalized unbalanced case -- where
Gaussian measures are scaled by positive constants. This extension leads to a
closed form expression for unbalanced Gaussians as well, and highlights the
mass transportation / destruction trade-off seen in unbalanced optimal
transport. Moreover, in both settings, we show that the optimal transportation
plans are (scaled) Gaussians and provide analytical formulas of their
parameters. These formulas constitute the first non-trivial closed forms for
entropy-regularized optimal transport, thus providing a ground truth for the
analysis of entropic OT and Sinkhorn's algorithm.
