  A new method is proposed for the solution of the data-driven optimal
transport barycenter problem and of the more general distributional barycenter
problem that the article introduces. The method improves on previous approaches
based on adversarial games, by slaving the discriminator to the generator,
minimizing the need for parameterizations and by allowing the adoption of
general cost functions. It is applied to numerical examples, which include
analyzing the MNIST data set with a new cost function that penalizes
non-isometric maps.
