Related papers: Optimal transportation on non-compact manifolds
Optimal transport (OT) has recently found widespread interest in machine learning. It allows to define novel distances between probability measures, which have shown promise in several applications. In this work, we discuss how to…
We consider the numerical solution of the optimal transport problem between densities that are supported on sets of unequal dimension. Recent work by McCann and Pass reformulates this problem into a non-local Monge-Amp\`ere type equation.…
We address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish space and dN is a geodesic Borel distance which makes (X,dN) a possibly branching geodesic space. We show that under some assumptions on the…
In this paper we consider the Benamou-Brenier formulation of optimal transport for nonlinear control affine systems on $\Rd$, removing the compactness assumption of the underlying manifold in previous work by the author. By using Bernard's…
We consider the optimal transportation problem on a globally hyperbolic spacetime for some cost function $c_2$, which corresponds to the optimal transportation problem on a complete Riemannian manifold where the cost function is the…
We propose a generalized curvature that is motivated by the optimal transport problem on $\mathbb{R}^d$ with cost induced by a Tonelli Lagrangian $L$. We show that non-negativity of the generalized curvature implies displacement convexity…
In this paper, we study Monge's problem on Riemannian manifolds $(M, g)$ with positive sectional curvature. Assuming that the source and target measures are absolutely continuous with respect to the Riemannian volume measure, we generalize…
We provide a solution to the problem of optimal transport by Brownian martingales in general dimensions whenever the transport cost satisfies certain subharmonic properties in the target variable, as well as a stochastic version of the…
We investigate the approximation of Monge--Kantorovich problems on general compact metric spaces, showing that optimal values, plans and maps can be effectively approximated via a fully discrete method. First we approximate optimal values…
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…
Given a transportation cost $c: M \times\bar M \to\mathbf{R}$, optimal maps minimize the total cost of moving masses from $M$ to $\bar M$. We find a pseudo-metric and a calibration form on $M\times\bar M$ such that the graph of an optimal…
A natural and important question in multi-marginal optimal transport is whether the \emph{Monge ansatz} is justified; does there exist a solution of Monge, or deterministic, form? We address this question for the quadratic cost when each…
We introduce a variant of Farber's topological complexity, defined for smooth compact orientable Riemannian manifolds, which takes into account only motion planners with the lowest possible "average length" of the output paths. We prove…
We investigate the small regularization limit of entropic optimal transport when the cost function is the Euclidean distance in dimensions $d > 1$, and the marginal measures are absolutely continuous with respect to the Lebesgue measure.…
The purpose of this paper is to show that in a finite dimensional metric space with Alexandrov's curvature bounded below, Monge's transport problem for the quadratic cost admits a unique solution.
We introduce a constrained optimal transport problem where origins $x$ can only be transported to destinations $y\geq x$. Our statistical motivation is to describe the sharp upper bound for the variance of the treatment effect $Y-X$ given…
A key inequality which underpins the regularity theory of optimal transport for costs satisfying the Ma--Trudinger--Wang condition is the Pogorelov second derivative bound. This translates to an apriori interior $C^1$ estimate for smooth…
The question of which costs admit unique optimizers in the Monge-Kantorovich problem of optimal transportation between arbitrary probability densities is investigated. For smooth costs and densities on compact manifolds, the only known…
We establish that solving an optimal transportation problem in which the source and target densities are defined on manifolds with different dimensions, is equivalent to solving a new nonlocal analog of the Monge-Amp\`ere equation,…
Let M and \bar M be n-dimensional manifolds equipped with suitable Borel probability measures \rho and \bar\rho. Ma, Trudinger & Wang gave sufficient conditions on a transportation cost c \in C^4(M \times \bar M) to guarantee smoothness of…