A dimension free computational upper-bound for smooth optimal transport estimation

It is well-known that plug-in statistical estimation of optimal transport suffers from the curse of dimension. Despite recent efforts to improve the rate of estimation with the smoothness of the problem, the computational complexities of these recently proposed methods still degrade exponentially with the dimension. In this paper, thanks to a representation theorem, we derive a statistical estimator of smooth optimal transport which achieves in average a precision $\epsilon$ for a computational cost of $\tilde{\mathcal{O}}(\epsilon^{-2})$ when the smoothness increases, hence yielding a dimension free rate. Even though our result is theoretical in nature due to the large constants involved in our estimations, it settles the question of whether the smoothness of optimal solutions can be taken advantage of from a computational and statistical point of view.