  Wasserstein barycenter, built on the theory of optimal transport, provides a
powerful framework to aggregate probability distributions, and it has
increasingly attracted great attention within the machine learning community.
However, it suffers from severe computational burden, especially for high
dimensional and continuous settings. To this end, we develop a novel continuous
approximation method for the Wasserstein barycenters problem given sample
access to the input distributions. The basic idea is to introduce a variational
distribution as the approximation of the true continuous barycenter, so as to
frame the barycenters computation problem as an optimization problem, where
parameters of the variational distribution adjust the proxy distribution to be
similar to the barycenter. Leveraging the variational distribution, we
construct a tractable dual formulation for the regularized Wasserstein
barycenter problem with c-cyclical monotonicity, which can be efficiently
solved by stochastic optimization. We provide theoretical analysis on
convergence and demonstrate the practical effectiveness of our method on real
applications of subset posterior aggregation and synthetic data.
