Estimation of Optimal Transport in Generative Models

As Wasserstein Generative Adversial Networks (WGANs) were introduced, optimal transport emerged as a toolkit to be used to devise loss functions for learning probabilistic models in machine learning. Especially the 1-Wasserstein distance is a popular choice as such a loss function, expressed in its dual Kantorovich form, where one is required to variationally maximize the sum of expectations over data of two constrained target functions. Different heurestics are applied to enforce the constraints of being 1-Lipschitz, such as the weight clipping or gradient penalty methods. In this talk, we wish to address how well these methods are estimating the 1-Wasserstein distance, and into which directions we could look to perhaps do better.