Wasserstein GAN With Quadratic Transport Cost

Wasserstein GANs are increasingly used in Computer Vision applications as they are easier to train. Previous WGAN variants mainly use the $l_1$ transport cost to compute the Wasserstein distance between the real and synthetic data distributions. The $l_1$ transport cost restricts the discriminator to be 1-Lipschitz. However, WGANs with $l_1$ transport cost were recently shown to not always converge. In this paper, we propose WGAN-QC, a WGAN with quadratic transport cost. Based on the quadratic transport cost, we propose an Optimal Transport Regularizer (OTR) to stabilize the training process of WGAN-QC. We prove that the objective of the discriminator during each generator update computes the exact quadratic Wasserstein distance between real and synthetic data distributions. We also prove that WGAN-QC converges to a local equilibrium point with finite discriminator updates per generator update. We show experimentally on a Dirac distribution that WGAN-QC converges, when many of the $l_1$ cost WGANs fail to [22]. Qualitative and quantitative results on the CelebA, CelebA-HQ, LSUN and the ImageNet dog datasets show that WGAN-QC is better than state-of-art GAN methods. WGAN-QC has much faster runtime than other WGAN variants.