  Variational problems that involve Wasserstein distances and more generally
optimal transport (OT) theory are playing an increasingly important role in
data sciences. Such problems can be used to form an examplar measure out of
various probability measures, as in the Wasserstein barycenter problem, or to
carry out parametric inference and density fitting, where the loss is measured
in terms of an optimal transport cost to the measure of observations. Despite
being conceptually simple, such problems are computationally challenging
because they involve minimizing over quantities (Wasserstein distances) that
are themselves hard to compute. Entropic regularization has recently emerged as
an efficient tool to approximate the solution of such variational Wasserstein
problems. In this paper, we give a thorough duality tour of these
regularization techniques. In particular, we show how important concepts from
classical OT such as c-transforms and semi-discrete approaches translate into
similar ideas in a regularized setting. These dual formulations lead to smooth
variational problems, which can be solved using smooth, differentiable and
convex optimization problems that are simpler to implement and numerically more
stable that their un-regularized counterparts. We illustrate the versatility of
this approach by applying it to the computation of Wasserstein barycenters and
gradient flows of spatial regularization functionals.
