  It is well-known that plug-in statistical estimation of optimal transport
suffers from the curse of dimensionality. Despite recent efforts to improve the
rate of estimation with the smoothness of the problem, the computational
complexity of these recently proposed methods still degrades exponentially with
the dimension. In this paper, thanks to an infinite-dimensional sum-of-squares
representation, we derive a statistical estimator of smooth optimal transport
which achieves a precision $\varepsilon$ from $\tilde{O}(\varepsilon^{-2})$
independent and identically distributed samples from the distributions, for a
computational cost of $\tilde{O}(\varepsilon^{-4})$ when the smoothness
increases, hence yielding dimension-free statistical and computational rates,
with potentially exponentially dimension-dependent constants.
