Generative Modeling by Minimizing the Wasserstein-2 Loss

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.