  This paper is focused on the study of entropic regularization in optimal
transport as a smoothing method for Wasserstein estimators, through the prism
of the classical tradeoff between approximation and estimation errors in
statistics. Wasserstein estimators are defined as solutions of variational
problems whose objective function involves the use of an optimal transport cost
between probability measures. Such estimators can be regularized by replacing
the optimal transport cost by its regularized version using an entropy penalty
on the transport plan. The use of such a regularization has a potentially
significant smoothing effect on the resulting estimators. In this work, we
investigate its potential benefits on the approximation and estimation
properties of regularized Wasserstein estimators. Our main contribution is to
discuss how entropic regularization may reach, at a lower computational cost,
statistical performances that are comparable to those of un-regularized
Wasserstein estimators in statistical learning problems involving
distributional data analysis. To this end, we present new theoretical results
on the convergence of regularized Wasserstein estimators. We also study their
numerical performances using simulated and real data in the supervised learning
problem of proportions estimation in mixture models using optimal transport.
