  Wasserstein barycenters provide a geometrically meaningful way to aggregate
probability distributions, built on the theory of optimal transport. They are
difficult to compute in practice, however, leading previous work to restrict
their supports to finite sets of points. Leveraging a new dual formulation for
the regularized Wasserstein barycenter problem, we introduce a stochastic
algorithm that constructs a continuous approximation of the barycenter. We
establish strong duality and use the corresponding primal-dual relationship to
parametrize the barycenter implicitly using the dual potentials of regularized
transport problems. The resulting problem can be solved with stochastic
gradient descent, which yields an efficient online algorithm to approximate the
barycenter of continuous distributions given sample access. We demonstrate the
effectiveness of our approach and compare against previous work on synthetic
examples and real-world applications.
