A Class of Optimal Transport Regularized Formulations with Applications to Wasserstein GANs

Optimal transport costs (e.g. Wasserstein distances) are used for fitting high-dimensional distributions. For example, popular artificial intelligence algorithms such as Wasserstein Generative Adversarial Networks (WGANs) can be interpreted as fitting a black-box simulator of structured data with certain features (e.g. images) using the Wasserstein distance. We propose a regularization of optimal transport costs and study its computational and duality properties. We obtain running time improvements for fitting WGANs with no deterioration in testing performance, relative to current benchmarks. We also derive finite sample bounds for the empirical Wasserstein distance from our regularization.