  We present a new multiscale algorithm for solving regularized Optimal
Transport problems on the GPU, with a linear memory footprint. Relying on
Sinkhorn divergences which are convex, smooth and positive definite loss
functions, this method enables the computation of transport plans between
millions of points in a matter of minutes. We show the effectiveness of this
approach on brain tractograms modeled either as bundles of fibers or as track
density maps. We use the resulting smooth assignments to perform label transfer
for atlas-based segmentation of fiber tractograms. The parameters -- blur and
reach -- of our method are meaningful, defining the minimum and maximum
distance at which two fibers are compared with each other. They can be set
according to anatomical knowledge. Furthermore, we also propose to estimate a
probabilistic atlas of a population of track density maps as a Wasserstein
barycenter. Our CUDA implementation is endowed with a user-friendly PyTorch
interface, freely available on the PyPi repository (pip install geomloss) and
at www.kernel-operations.io/geomloss.
