After presenting some background on optimal transport, we explain why the curse of dimensionality can be encountered in estimating optimal transport distances. we then explain computational techniques for solving optimal transport and in particular entropic regularization. We present Sinkhorn divergences and show how they have been proven to overcome the curse of dimension, but not for approximating optimal transport distances. Then, we show that smoothness of solutions of optimal transport can actually be leveraged to define statistical estimators which are amenable to computation. We use the dual formulation of optimal transport, whose minimizers enjoy a particular structure. These solutions can be written as a sum of squares in Reproducing Kernel Hilbert Spaces. As is standard in the SOS literature, it can be solved using an SDP formulation. By using a soft-penalty in Sobolev spaces on the optimal potential and a trace class positive self-adjoint operator, we can define an estimator of optimal transport which is both statistically and computationally efficient