Related papers: Sliced optimal transport: is it a suitable replace…
We introduce a general class of transport distances ${\rm WB}_{\Lambda}$ over the space of positive semi-definite matrix-valued Radon measures $\mathcal{M}(\Omega,\mathbb{S}_+^n)$, called the weighted Wasserstein-Bures distance. Such a…
In this paper, we give a new characterization of the cut locus of a point on a compact Riemannian manifold as the zero set of the optimal transport density solution of the Monge-Kantorovich equations, a PDE formulation of the optimal…
Wasserstein projections in the convex order were first considered in the framework of weak optimal transport, and found application in various problems such as concentration inequalities and martingale optimal transport. In dimension one,…
The conventional sliced Wasserstein is defined between two probability measures that have realizations as vectors. When comparing two probability measures over images, practitioners first need to vectorize images and then project them to…
Classical optimal transport problem seeks a transportation map that preserves the total mass betwenn two probability distributions, requiring their mass to be the same. This may be too restrictive in certain applications such as color or…
Let $\Omega\subset \mathbb{R}^n$ be non-empty, open and proper. Consider $Wb(\Omega)$, the space of finite Borel measures on $\Omega$ equipped with the partial transportation metric introduced by Figalli and Gigli that allows the creation…
The sliced Wasserstein distance $SW_2(\mu,\nu)$ compares high-dimensional probability measures by averaging one-dimensional optimal transport distances over linear projections. Although sliced Wasserstein distances are now standard…
Optimal Transport (OT) has attracted significant interest in the machine learning community, not only for its ability to define meaningful distances between probability distributions -- such as the Wasserstein distance -- but also for its…
This paper serves as a user's guide to sampling strategies for sliced optimal transport. We provide reminders and additional regularity results on the Sliced Wasserstein distance. We detail the construction methods, generation time…
Wasserstein 1 optimal transport maps provide a natural correspondence between points from two probability distributions, $\mu$ and $\nu$, which is useful in many applications. Available algorithms for computing these maps do not appear to…
We study the Wasserstein (or earthmover) metric on the space $P(X)$ of probability measures on a metric space $X$. We show that, if a finite metric space $X$ embeds stochastically with distortion $D$ in a family of finite metric trees, then…
Sliced Wasserstein (SW) distances offer an efficient method for comparing high-dimensional probability measures by projecting them onto multiple 1-dimensional probability distributions. However, identifying informative slicing directions…
We study the problem of optimal transport in tropical geometry and define the Wasserstein-$p$ distances in the continuous metric measure space setting of the tropical projective torus. We specify the tropical metric -- a combinatorial…
The duality theory of the Monge--Kantorovich transport problem is analyzed in a general setting. The spaces $X, Y$ are assumed to be polish and equipped with Borel probability measures $\mu$ and $\nu$. The transport cost function $c:X\times…
In this paper we study two basic facts of optimal transportation on Wiener space W. Our first aim is to answer to the Monge Problem on the Wiener space endowed with the Sobolev type norm (k,gamma) to the power of p (cases p = 1 and p > 1…
The Kantorovich metric provides a way of measuring the distance between two Borel probability measures on a metric space. This metric has a broad range of applications from bioinformatics to image processing, and is commonly linked to the…
Making sense of Wasserstein distances between discrete measures in high-dimensional settings remains a challenge. Recent work has advocated a two-step approach to improve robustness and facilitate the computation of optimal transport, using…
The aim of this paper is to prove isoperimetric inequalities on submanifolds of the Euclidean space using mass transportation methods. We obtain a sharp ?weighted isoperimetric inequality? and a nonsharp classical inequality similar to the…
We introduce a new non-linear optimal transport formulation for a pair of probability measures on $\mathbb{R}^d$ sharing a common barycentre, in which admissible transference plans satisfy two martingale-type constraints. This bi-martingale…
This paper studies the uniqueness of solutions to the dual optimal transport problem, both qualitatively and quantitatively (bounds on the diameter of the set of optimisers). On the qualitative side, we prove that when one marginal…