Related papers: The Monge-Kantorovich problem on Wasserstein space
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…
In this paper we characterize the so called uniformly rectifiable sets of David and Semmes in terms of the Wasserstein distance $W_2$ from optimal mass transport. To obtain this result, we first prove a localization theorem for the distance…
The theory of optimal transport of probability measures has wide-ranging applications across a number of different fields, including concentration of measure, machine learning, Markov chains, and economics. The generalisation of optimal…
By disintegration of transport plans it is introduced the notion of transport class. This allows to consider the Monge problem as a particular case of the Kantorovich transport problem, once a transport class is fixed. The transport problem…
We formulate an optimal transport problem for matrix-valued density functions. This is pertinent in the spectral analysis of multivariable time-series. The "mass" represents energy at various frequencies whereas, in addition to a usual…
Given $m < n$, we consider the problem of ``best'' approximating an $n\text{-d}$ probability measure $\rho$ via an $m\text{-d}$ measure $\nu$ such that $\mathrm{supp}\ \nu$ has bounded total ``complexity.'' When $\rho$ is concentrated near…
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…
The optimal (Monge-Kantorovich) transportation problem is discussed from several points of view. The Lagrangian formulation extends the action of the {\em Lagrangian} $L(v,x,t)$ from the set of orbits in $\R^n$ to a set of measure-valued…
We consider probability measures on $\mathbb{R}^{\infty}$ and study optimal transportation mappings for the case of infinite Kantorovich distance. Our examples include 1) quasi-product measures, 2) measures with certain symmetric…
In this work we prove the existence and uniqueness of the optimal transport map for $L^p$-Wasserstein distance with $p>1$, and particularly present an explicit expression of the optimal transport map for the case $p=2$. As an application,…
We investigate the properties of convex functions in the plane that satisfy a local inequality which generalizes the notion of sub-solution of Monge-Ampere equation for a Monge-Kantorovich problem with quadratic cost between non-absolutely…
This paper is devoted to the study of the Monge-Kantorovich theory of optimal mass transport and its applications, in the special case of one-dimensional and circular distributions. More precisely, we study the Monge-Kantorovich distances…
We introduce a one-parameter family of metrics on the space of Borel probability measures on Euclidean space with finite $p$th moment for $1\leq p <\infty$, called the $\textit{sliced Monge--Kantorovich metrics}$, which include the sliced…
We develop Brenier theorems on iterated Wasserstein spaces. For a separable Hilbert space $H$ and $N\geq 1$, we construct a full-support probability $\Lambda$ on $P_2^{N}(H)= P_2(... P_2(H)...)$ that is transport regular: for every $Q$ with…
We consider the following variant of the Monge-Kantorovich transportation problem. Let S be a finite set of point sites in d dimensions. A bounded set C in d-dimensional space is to be distributed among the sites p in S such that (i) each p…
In this paper, we investigate Monge-Kantorovich problems for which the absolute continuity of marginals is relaxed. For $X,Y\subseteq\mathbb{R}^{n+1}$ let $(X,\mathcal{B}_X,\mu)$ and $(Y,\mathcal{B}_Y,\nu)$ be two Borel probability spaces,…
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…
The optimal mass transportation was introduced by Monge some 200 years ago and is, today, the source of large number of results in analysis, geometry and convexity. Here I investigate a new, surprising link between optimal transformations…
This paper focuses on martingale optimal transport problems when the martingales are assumed to have bounded quadratic variation. First, we give a result that characterizes the existence of a probability measure satisfying some convex…
Using the dual formulation only, we show that the regularity of unbalanced optimal transport also called entropy-transport inherits from the regularity of standard optimal transport. We provide detailed examples of Riemannian manifolds and…