  The task of mapping two or more distributions to a shared representation has
many applications including fair representations, batch effect mitigation, and
unsupervised domain adaptation. However, most existing formulations only
consider the setting of two distributions, and moreover, do not have an
identifiable, unique shared latent representation. We use optimal transport
theory to consider a natural multiple distribution extension of the Monge
assignment problem we call the symmetric Monge map problem and show that it is
equivalent to the Wasserstein barycenter problem. Yet, the maps to the
barycenter are challenging to estimate. Prior methods often ignore
transportation cost, rely on adversarial methods, or only work for discrete
distributions. Therefore, our goal is to estimate invertible maps between two
or more distributions and their corresponding barycenter via a simple iterative
flow method. Our method decouples each iteration into two subproblems: 1)
estimate simple distributions and 2) estimate the invertible maps to the
barycenter via known closed-form OT results. Our empirical results give
evidence that this iterative algorithm approximates the maps to the barycenter.
