Related papers: On the transport dimension of measures
Optimal Transport is a theory that allows to define geometrical notions of distance between probability distributions and to find correspondences, relationships, between sets of points. Many machine learning applications are derived from…
Wasserstein distances are metrics on probability distributions inspired by the problem of optimal mass transportation. Roughly speaking, they measure the minimal effort required to reconfigure the probability mass of one distribution in…
We consider random i.i.d. samples of absolutely continuous measures on bounded connected domains. We prove an upper bound on the $\infty$-transportation distance between the measure and the empirical measure of the sample. The bound is…
We suggest a new way of defining optimal transport of positive-semidefinite matrix-valued measures. It is inspired by a recent rendering of the incompressible Euler equations and related conservative systems as concave maximization…
For probability measures on a complete separable metric space, we present sufficient conditions for the existence of a solution to the Kantorovich transportation problem. We also obtain sufficient conditions (which sometimes also become…
Describing shapes by suitable measures in object segmentation, as proposed in [24], allows to combine the advantages of the representations as parametrized contours and indicator functions. The pseudo-Riemannian structure of optimal…
An upper bound for the Kantorovich transport distance between probability measures on multidimensional Euclidean spaces is given in terms of transport distances between one dimensional projections. This quantifies the Cram\'er-Wold…
Optimal transportation distances are valuable for comparing and analyzing probability distributions, but larger-scale computational techniques for the theoretically favorable quadratic case are limited to smooth domains or regularized…
Optimal Transport has received much attention in Machine Learning as it allows to compare probability distributions by exploiting the geometry of the underlying space. However, in its original formulation, solving this problem suffers from…
We consider the optimal mass transportation problem in $\RR^d$ with measurably parameterized marginals, for general cost functions and under conditions ensuring the existence of a unique optimal transport map. We prove a joint measurability…
Optimal transport is widely used in pure and applied mathematics to find probabilistic solutions to hard combinatorial matching problems. We extend the Wasserstein metric and other elements of optimal transport from the matching of sets to…
Since the introduction of the Sliced Wasserstein distance in the literature, its simplicity and efficiency have made it one of the most interesting surrogate for the Wasserstein distance in image processing and machine learning. However,…
We develop a theory of optimal transport relative to a distinguished subset, which acts as a reservoir of mass, allowing us to compare measures of different total variation. This relative transportation problem has an optimal solution and…
The paper proves transportation inequalities for probability measures on spheres for the Wasserstein metrics with respect to cost functions that are powers of the geodesic distance. Let $\mu$ be a probability measure on the sphere ${\bf…
Inspired by the recent theory of Entropy-Transport problems and by the $\mathbf{D}$-distance of Sturm on normalised metric measure spaces, we define a new class of complete and separable distances between metric measure spaces of possibly…
This is the lecture notes on the interplay between optimal transport and Riemannian geometry. On a Riemannian manifold, the convexity of entropy along optimal transport in the space of probability measures characterizes lower bounds of the…
In this paper, we prove a structure theorem for discrete optimal transportation plans. We show that, given any pair of discrete probability measures and a cost function, there exists an optimal transportation plan that can be expressed as…
Optimal transport and its related problems, including optimal partial transport, have proven to be valuable tools in machine learning for computing meaningful distances between probability or positive measures. This success has led to a…
This article connects the theory of extremal doubly stochastic measures to the geometry and topology of optimal transportation. We begin by reviewing an old question (# 111) of Birkhoff in probability and statistics [4], which is to give a…
This note exposes the differential topology and geometry underlying some of the basic phenomena of optimal transportation. It surveys basic questions concerning Monge maps and Kantorovich measures: existence and regularity of the former,…