Distributional barycenter problem through data-driven flows

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 distributional barycenter problem provides a conceptual and computational toolbox for central problems in pattern recognition, such as the simulation of conditional distributions, the construction of a representative for a family of distributions indexed by a covariate and a new class of data-based generative models. The method proposed improves on previous approaches based on adversarial games, by slaving the discriminator to the generator and minimizing the need for parameterizations. It applies not only to a discrete family of distributions, but to more general distributions conditioned to factors z of any cardinality and type. The methodology is applied to numerical examples, including an analysis of the MNIST data set with a new cost function that penalizes non-isometric maps.