One of the challenges in the study of generative adversarial networks (GANs) is the difficulty of its performance control. Lipschitz constraint is essential in guaranteeing training stability for GANs. Although heuristic methods such as weight clipping, gradient penalty and spectral normalization have been proposed to enforce Lipschitz constraint, it is still difficult to achieve a solution that is both practically effective and theoretically provably satisfying a Lipschitz constraint. In this paper, we introduce the boundedness and continuity (BC) conditions to enforce the Lipschitz constraint on the discriminator functions of GANs. We prove theoretically that GANs with discriminators meeting the BC conditions satisfy the Lipschitz constraint. We present a practically very effective implementation of a GAN based on a convolutional neural network (CNN) by forcing the CNN to satisfy the BC conditions (BC–GAN). We show that as compared to recent techniques including gradient penalty and spectral normalization, BC–GANs have not only better performances but also lower computational complexity.

生成对抗网络（GAN）研究的挑战之一是其性能控制的难度。Lipschitz约束对于保证GAN的训练稳定性至关重要。尽管已提出了权重裁剪，梯度罚分和频谱归一化之类的启发式方法来增强Lipschitz约束，但仍然难以实现既实用又有效且理论上可满足Lipschitz约束的解决方案。在本文中，我们介绍了有界和连续性（BC）条件，以在Lipsitzitz约束上强制执行GAN的区分函数。我们从理论上证明，具有满足BC条件的鉴别器的GAN满足Lipschitz约束。我们通过强制CNN满足BC条件（BC–GAN），提出了基于卷积神经网络（CNN）的GAN的一种非常有效的实现。我们证明，与包括梯度罚分和频谱归一化的最新技术相比，BC–GAN不仅具有更好的性能，而且具有较低的计算复杂度。