  We propose a new Integral Probability Metric (IPM) between distributions: the
Sobolev IPM. The Sobolev IPM compares the mean discrepancy of two distributions
for functions (critic) restricted to a Sobolev ball defined with respect to a
dominant measure $\mu$. We show that the Sobolev IPM compares two distributions
in high dimensions based on weighted conditional Cumulative Distribution
Functions (CDF) of each coordinate on a leave one out basis. The Dominant
measure $\mu$ plays a crucial role as it defines the support on which
conditional CDFs are compared. Sobolev IPM can be seen as an extension of the
one dimensional Von-Mises Cram\'er statistics to high dimensional
distributions. We show how Sobolev IPM can be used to train Generative
Adversarial Networks (GANs). We then exploit the intrinsic conditioning implied
by Sobolev IPM in text generation. Finally we show that a variant of Sobolev
GAN achieves competitive results in semi-supervised learning on CIFAR-10,
thanks to the smoothness enforced on the critic by Sobolev GAN which relates to
Laplacian regularization.
