  Generative adversarial networks (GANs) learn a target probability
distribution by optimizing a generator and a discriminator with minimax
objectives. This paper addresses the question of whether such optimization
actually provides the generator with gradients that make its distribution close
to the target distribution. We derive metrizable conditions, sufficient
conditions for the discriminator to serve as the distance between the
distributions by connecting the GAN formulation with the concept of sliced
optimal transport. Furthermore, by leveraging these theoretical results, we
propose a novel GAN training scheme, called slicing adversarial network (SAN).
With only simple modifications, a broad class of existing GANs can be converted
to SANs. Experiments on synthetic and image datasets support our theoretical
results and the SAN's effectiveness as compared to usual GANs. Furthermore, we
also apply SAN to StyleGAN-XL, which leads to state-of-the-art FID score
amongst GANs for class conditional generation on ImageNet 256$\times$256. Our
implementation is available on https://ytakida.github.io/san.
