  We propose a novel end-to-end non-minimax algorithm for training optimal
transport mappings for the quadratic cost (Wasserstein-2 distance). The
algorithm uses input convex neural networks and a cycle-consistency
regularization to approximate Wasserstein-2 distance. In contrast to popular
entropic and quadratic regularizers, cycle-consistency does not introduce bias
and scales well to high dimensions. From the theoretical side, we estimate the
properties of the generative mapping fitted by our algorithm. From the
practical side, we evaluate our algorithm on a wide range of tasks:
image-to-image color transfer, latent space optimal transport, image-to-image
style transfer, and domain adaptation.
