  Optimal transport aims to estimate a transportation plan that minimizes a
displacement cost. This is realized by optimizing the scalar product between
the sought plan and the given cost, over the space of doubly stochastic
matrices. When the entropy regularization is added to the problem, the
transportation plan can be efficiently computed with the Sinkhorn algorithm.
Thanks to this breakthrough, optimal transport has been progressively extended
to machine learning and statistical inference by introducing additional
application-specific terms in the problem formulation. It is however
challenging to design efficient optimization algorithms for optimal transport
based extensions. To overcome this limitation, we devise a general
forward-backward splitting algorithm based on Bregman distances for solving a
wide range of optimization problems involving a differentiable function with
Lipschitz-continuous gradient and a doubly stochastic constraint. We illustrate
the efficiency of our approach in the context of continuous domain adaptation.
Experiments show that the proposed method leads to a significant improvement in
terms of speed and performance with respect to the state of the art for domain
adaptation on a continually rotating distribution coming from the standard two
moon dataset.
