  In many unpaired image domain translation problems, e.g., style transfer or
super-resolution, it is important to keep the translated image similar to its
respective input image. We propose the extremal transport (ET) which is a
mathematical formalization of the theoretically best possible unpaired
translation between a pair of domains w.r.t. the given similarity function.
Inspired by the recent advances in neural optimal transport (OT), we propose a
scalable algorithm to approximate ET maps as a limit of partial OT maps. We
test our algorithm on toy examples and on the unpaired image-to-image
translation task. The code is publicly available at
https://github.com/milenagazdieva/ExtremalNeuralOptimalTransport
