  Wasserstein GANs (WGANs), built upon the Kantorovich-Rubinstein (KR) duality
of Wasserstein distance, is one of the most theoretically sound GAN models.
However, in practice it does not always outperform other variants of GANs. This
is mostly due to the imperfect implementation of the Lipschitz condition
required by the KR duality. Extensive work has been done in the community with
different implementations of the Lipschitz constraint, which, however, is still
hard to satisfy the restriction perfectly in practice. In this paper, we argue
that the strong Lipschitz constraint might be unnecessary for optimization.
Instead, we take a step back and try to relax the Lipschitz constraint.
Theoretically, we first demonstrate a more general dual form of the Wasserstein
distance called the Sobolev duality, which relaxes the Lipschitz constraint but
still maintains the favorable gradient property of the Wasserstein distance.
Moreover, we show that the KR duality is actually a special case of the Sobolev
duality. Based on the relaxed duality, we further propose a generalized WGAN
training scheme named Sobolev Wasserstein GAN (SWGAN), and empirically
demonstrate the improvement of SWGAN over existing methods with extensive
experiments.
