  Generative Adversial Networks (GANs) have made a major impact in computer
vision and machine learning as generative models. Wasserstein GANs (WGANs)
brought Optimal Transport (OT) theory into GANs, by minimizing the
$1$-Wasserstein distance between model and data distributions as their
objective function. Since then, WGANs have gained considerable interest due to
their stability and theoretical framework. We contribute to the WGAN literature
by introducing the family of $(q,p)$-Wasserstein GANs, which allow the use of
more general $p$-Wasserstein metrics for $p\geq 1$ in the GAN learning
procedure. While the method is able to incorporate any cost function as the
ground metric, we focus on studying the $l^q$ metrics for $q\geq 1$. This is a
notable generalization as in the WGAN literature the OT distances are commonly
based on the $l^2$ ground metric. We demonstrate the effect of different
$p$-Wasserstein distances in two toy examples. Furthermore, we show that the
ground metric does make a difference, by comparing different $(q,p)$ pairs on
the MNIST and CIFAR-10 datasets. Our experiments demonstrate that changing the
ground metric and $p$ can notably improve on the common $(q,p) = (2,1)$ case.
