Training Generative Networks with general Optimal Transport distances

We propose a new algorithm that uses an auxiliary Neural Network to calculate the transport distance between two data distributions and export an optimal transport map. In the sequel we use the aforementioned map to train Generative Networks. Unlike WGANs, where the Euclidean distance is implicitly used, this new method allows to use any transportation cost function that can be chosen to match the problem at hand. More specifically, it allows to use the squared distance as a transportation cost function, giving rise to the Wasserstein-2 metric for probability distributions, which has rich geometric properties that result in fast and stable gradients descends. It also allows to use image centered distances, like the Structure Similarity index, with notable differences in the results.