  Domain Translation is the problem of finding a meaningful correspondence
between two domains. Since in a majority of settings paired supervision is not
available, much work focuses on Unsupervised Domain Translation (UDT) where
data samples from each domain are unpaired. Following the seminal work of
CycleGAN for UDT, many variants and extensions of this model have been
proposed. However, there is still little theoretical understanding behind their
success. We observe that these methods yield solutions which are approximately
minimal w.r.t. a given transportation cost, leading us to reformulate the
problem in the Optimal Transport (OT) framework. This viewpoint gives us a new
perspective on Unsupervised Domain Translation and allows us to prove the
existence and uniqueness of the retrieved mapping, given a large family of
transport costs. We then propose a novel framework to efficiently compute
optimal mappings in a dynamical setting. We show that it generalizes previous
methods and enables a more explicit control over the computed optimal mapping.
It also provides smooth interpolations between the two domains. Experiments on
toy and real world datasets illustrate the behavior of our method.
