5 research outputs found
Entropy-Transport distances between unbalanced metric measure spaces
Inspired by the recent theory of Entropy-Transport problems and by the
-distance of Sturm on normalised metric measure spaces, we define a
new class of complete and separable distances between metric measure spaces of
possibly different total mass.
We provide several explicit examples of such distances, where a prominent
role is played by a geodesic metric based on the Hellinger-Kantorovich
distance. Moreover, we discuss some limiting cases of the theory, recovering
the "pure transport" -distance and introducing a new class of "pure
entropic" distances.
We also study in detail the topology induced by such Entropy-Transport
metrics, showing some compactness and stability results for metric measure
spaces satisfying Ricci curvature lower bounds in a synthetic sense.Comment: 36 pages. Comments are welcome
Sinkhorn Divergences for Unbalanced Optimal Transport
Optimal transport induces the Earth Mover's (Wasserstein) distance between
probability distributions, a geometric divergence that is relevant to a wide
range of problems. Over the last decade, two relaxations of optimal transport
have been studied in depth: unbalanced transport, which is robust to the
presence of outliers and can be used when distributions don't have the same
total mass; entropy-regularized transport, which is robust to sampling noise
and lends itself to fast computations using the Sinkhorn algorithm. This paper
combines both lines of work to put robust optimal transport on solid ground.
Our main contribution is a generalization of the Sinkhorn algorithm to
unbalanced transport: our method alternates between the standard Sinkhorn
updates and the pointwise application of a contractive function. This implies
that entropic transport solvers on grid images, point clouds and sampled
distributions can all be modified easily to support unbalanced transport, with
a proof of linear convergence that holds in all settings. We then show how to
use this method to define pseudo-distances on the full space of positive
measures that satisfy key geometric axioms: (unbalanced) Sinkhorn divergences
are differentiable, positive, definite, convex, statistically robust and avoid
any "entropic bias" towards a shrinkage of the measures' supports
Fast and Scalable Optimal Transport for Brain Tractograms
International audienceWe present a new multiscale algorithm for solving regular-ized 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