  Barycentric averaging is a principled way of summarizing populations of
measures. Existing algorithms for estimating barycenters typically parametrize
them as weighted sums of Diracs and optimize their weights and/or locations.
However, these approaches do not scale to high-dimensional settings due to the
curse of dimensionality. In this paper, we propose a scalable and general
algorithm for estimating barycenters of measures in high dimensions. The key
idea is to turn the optimization over measures into an optimization over
generative models, introducing inductive biases that allow the method to scale
while still accurately estimating barycenters. We prove local convergence under
mild assumptions on the discrepancy showing that the approach is well-posed. We
demonstrate that our method is fast, achieves good performance on
low-dimensional problems, and scales to high-dimensional settings. In
particular, our approach is the first to be used to estimate barycenters in
thousands of dimensions.
