  The objective in statistical Optimal Transport (OT) is to consistently
estimate the optimal transport plan/map solely using samples from the given
source and target marginal distributions. This work takes the novel approach of
posing statistical OT as that of learning the transport plan's kernel mean
embedding from sample based estimates of marginal embeddings. The proposed
estimator controls overfitting by employing maximum mean discrepancy based
regularization, which is complementary to $\phi$-divergence (entropy) based
regularization popularly employed in existing estimators. A key result is that,
under very mild conditions, $\epsilon$-optimal recovery of the transport plan
as well as the Barycentric-projection based transport map is possible with a
sample complexity that is completely dimension-free. Moreover, the implicit
smoothing in the kernel mean embeddings enables out-of-sample estimation. An
appropriate representer theorem is proved leading to a kernelized convex
formulation for the estimator, which can then be potentially used to perform OT
even in non-standard domains. Empirical results illustrate the efficacy of the
proposed approach.
