🤖 AI Summary
This work addresses the training instability and slow convergence of Wasserstein Generative Adversarial Networks (WGANs). We propose a distribution-dependent ordinary differential equation (ODE) framework grounded in the gradient flow of the Wasserstein-1 distance, and derive a novel generative model—W1-FE—via forward Euler discretization. Our contribution is threefold: (i) we reformulate WGAN theory from an ODE dynamical systems perspective for the first time; (ii) we rigorously prove that persistent training improves performance *only* under this specific gradient flow structure, thereby refuting heuristic, dynamics-agnostic augmentation strategies; and (iii) empirical results demonstrate that W1-FE consistently outperforms standard WGAN across low- to high-dimensional tasks, achieving faster convergence and superior sample quality. These findings validate both the effectiveness and necessity of ODE-guided persistent training.
📝 Abstract
This paper proposes a new theoretical lens to view Wasserstein generative adversarial networks (WGANs). To minimize the Wasserstein-1 distance between the true data distribution and our estimate of it, we derive a distribution-dependent ordinary differential equation (ODE) which represents the gradient flow of the Wasserstein-1 loss, and show that a forward Euler discretization of the ODE converges. This inspires a new class of generative models that naturally integrates persistent training (which we call W1-FE). When persistent training is turned off, we prove that W1-FE reduces to WGAN. When we intensify persistent training, W1-FE is shown to outperform WGAN in training experiments from low to high dimensions, in terms of both convergence speed and training results. Intriguingly, one can reap the benefits only when persistent training is carefully integrated through our ODE perspective. As demonstrated numerically, a naive inclusion of persistent training in WGAN (without relying on our ODE framework) can significantly worsen training results.