  Optimal transport is a powerful framework for computing distances between
probability distributions. We unify the two main approaches to optimal
transport, namely Monge-Kantorovitch and Sinkhorn-Cuturi, into what we define
as Tsallis regularized optimal transport (\trot). \trot~interpolates a rich
family of distortions from Wasserstein to Kullback-Leibler, encompassing as
well Pearson, Neyman and Hellinger divergences, to name a few. We show that
metric properties known for Sinkhorn-Cuturi generalize to \trot, and provide
efficient algorithms for finding the optimal transportation plan with formal
convergence proofs. We also present the first application of optimal transport
to the problem of ecological inference, that is, the reconstruction of joint
distributions from their marginals, a problem of large interest in the social
sciences. \trot~provides a convenient framework for ecological inference by
allowing to compute the joint distribution --- that is, the optimal
transportation plan itself --- when side information is available, which is
\textit{e.g.} typically what census represents in political science.
Experiments on data from the 2012 US presidential elections display the
potential of \trot~in delivering a faithful reconstruction of the joint
distribution of ethnic groups and voter preferences.
