Towards Generalized Implementation of Wasserstein Distance in GANs

TL;DR

This paper introduces a relaxed dual form of Wasserstein distance called Sobolev duality, enabling a generalized WGAN training scheme (SWGAN) that relaxes Lipschitz constraints and improves performance.

Contribution

It presents the Sobolev duality as a more general framework, relaxing Lipschitz constraints in WGANs and proposing SWGAN, which empirically outperforms existing methods.

Findings

SWGAN demonstrates improved training stability.

Relaxed duality maintains Wasserstein's gradient properties.

Empirical results show better sample quality.

Abstract

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.