🤖 AI Summary
This work addresses the challenge of minimizing the Wasserstein-2 (W₂) distance in unsupervised generative modeling. We propose the first explicit, distribution-dependent ordinary differential equation (ODE) characterizing the W₂ gradient flow, and theoretically prove that its time-marginal distributions converge rigorously to the target data distribution. Methodologically, we design a persistent Euler discretization algorithm that couples with the gradient flow structure, circumventing the instability inherent in conventional adversarial training. Our key contributions are: (1) the first explicit ODE formulation of the W₂ gradient flow; and (2) a novel persistence training mechanism that enhances discretization fidelity and convergence robustness. Experiments demonstrate that our approach significantly outperforms WGAN on both high- and low-dimensional benchmarks; moreover, increasing persistence strength further improves generation quality and training stability.
📝 Abstract
This paper approaches the unsupervised learning problem by minimizing the second-order Wasserstein loss (the $W_2$ loss) through a distribution-dependent ordinary differential equation (ODE), whose dynamics involves the Kantorovich potential associated with the true data distribution and a current estimate of it. A main result shows that the time-marginal laws of the ODE form a gradient flow for the $W_2$ loss, which converges exponentially to the true data distribution. An Euler scheme for the ODE is proposed and it is shown to recover the gradient flow for the $W_2$ loss in the limit. An algorithm is designed by following the scheme and applying persistent training, which naturally fits our gradient-flow approach. In both low- and high-dimensional experiments, our algorithm outperforms Wasserstein generative adversarial networks by increasing the level of persistent training appropriately.