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One of the challenges in the study of Generative Adversarial Networks (GANs)is the difficulty of its performance control. Lipschitz constraint is essentialin guaranteeing training stability for GANs. Although heuristic methods such asweight clipping, gradient penalty and spectral normalization have been proposedto enforce Lipschitz constraint, it is still difficult to achieve a solutionthat is both practically effective and theoretically provably satisfying aLipschitz constraint. In this paper, we introduce the boundedness andcontinuity ($BC$) conditions to enforce the Lipschitz constraint on thediscriminator functions of GANs. We prove theoretically that GANs withdiscriminators meeting the BC conditions satisfy the Lipschitz constraint. Wepresent a practically very effective implementation of a GAN based on aconvolutional neural network (CNN) by forcing the CNN to satisfy the $BC$conditions (BC-GAN). We show that as compared to recent techniques includinggradient penalty and spectral normalization, BC-GANs not only have betterperformances but also lower computational complexity.

Lipschitz Constrained GANs via Boundedness and Continuity