  We show in this note that the Sobolev Discrepancy introduced in Mroueh et al
in the context of generative adversarial networks, is actually the weighted
negative Sobolev norm $||.||_{\dot{H}^{-1}(\nu_q)}$, that is known to linearize
the Wasserstein $W_2$ distance and plays a fundamental role in the dynamic
formulation of optimal transport of Benamou and Brenier. Given a Kernel with
finite dimensional feature map we show that the Sobolev discrepancy can be
approximated from finite samples. Assuming this discrepancy is finite, the
error depends on the approximation error in the function space induced by the
finite dimensional feature space kernel and on a statistical error due to the
finite sample approximation.
