Learning to generate Wasserstein barycenters
Authors: Julien Lacombe, Julie Digne, Nicolas Courty, Nicolas Bonneel
Keywords: Wasserstein barycenters, Optimal Transport
Abstract: Optimal transport is a notoriously difficult problem to solve numerically, with current approaches often remaining intractable for very large scale applications such as those encountered in machine learning. Wasserstein barycenters -- the problem of finding measures in-between given input measures in the optimal transport sense -- is even more computationally demanding. 
By training a deep convolutional neural network, we improve by a factor of 60 the computational speed of Wasserstein barycenters over the fastest state-of-the-art approach on the GPU, resulting in milliseconds computational times on $512\times512$ regular grids.
We show that our network, trained on Wasserstein barycenters of pairs of measures, generalizes well to the problem of finding Wasserstein barycenters of more than two measures. We validate our approach on synthetic shapes generated via Constructive Solid Geometry as well as on the ``Quick, Draw'' sketches dataset.