Related papers: Localisation for constrained transports I: theory

Optimal maps, solutions to the optimal transportation problems, are completely determined by the corresponding c-convex potential functions. In this paper, we give simple sufficient conditions for a smooth function to be c-convex when the…

F. Wehrung has asked: Given a family $\mathcal{C}$ of subsets of a set $\Omega$, under what conditions will there exist a total ordering on $\Omega$ under which every member of $\mathcal{C}$ is convex? <p> Note that if $A$ and $B$ are…

This paper proves some results concerning the polar factorisation of an integrable vector-valued function u into the composition of the gradient of a convex function with a measure-preserving mapping. Not every integrable function has a…

Optimal transport (OT) provides effective tools for comparing and mapping probability measures. We propose to leverage the flexibility of neural networks to learn an approximate optimal transport map. More precisely, we present a new and…

We study the nonlocal vectorial transport equation $\partial_ty+ (\mathbb{P} y \cdot \nabla) y=0$ on bounded domains of $\mathbb{R}^d$ where $\mathbb{P}$ denotes the Leray projector. This equation was introduced to obtain the unique optimal…

We investigate the estimation of an optimal transport map between probability measures on an infinite-dimensional space and reveal its minimax optimal rate. Optimal transport theory defines distances within a space of probability measures,…

In this paper we establish a general framework in which the verification of support theorems for generalized convex functions acting between an algebraic structure and an ordered algebraic structure is still possible. As for the domain…

We study optimal transport between two high-dimensional distributions $\mu,\nu$ in $R^n$ from an algorithmic perspective: given $x \sim \mu$, find a close $y \sim \nu$ in $poly(n)$ time, where $n$ is the dimension of $x,y$. Thus, running…

This paper proposes a unified framework for designing robustness in optimization under uncertainty using gauge sets, convex sets that generalize distance and capture how distributions may deviate from a nominal reference. Representing…

We consider optimal transportation of measures on metric and topological spaces in the case where the cost function and marginal distributions depend on a parameter with values in a metric space. The Hausdorff distance between the sets of…

Consider the problem of optimally matching two measures on the circle, or equivalently two periodic measures on the real line, and suppose the cost of matching two points satisfies the Monge condition. We introduce a notion of locally…

Optimal transport from the volume measure to a convex combination of Dirac measures yields a tessellation of a Riemannian manifold into pieces of arbitrary relative size. This tessellation is studied for the cost functions…

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…

Let R be a sufficiently saturated o-minimal expansion of a real closed field, let O be the convex hull of the rationals in R, and let st: O^n \to \mathbb{R}^n be the standard part map. For X \subseteq R^n define st(X):=st(X \cap O^n). We…

We study the problem of estimating a function $T$ given independent samples from a distribution $P$ and from the pushforward distribution $T_\sharp P$. This setting is motivated by applications in the sciences, where $T$ represents the…

A local convergence rate is established for a Gauss orthogonal collocation method applied to optimal control problems with control constraints. If the Hamiltonian possesses a strong convexity property, then the theory yields convergence for…

The study of convex functions - in particular, of their optimization (really minimization) is one of the most important fields of applied mathematics. Convexity seems to be one of those incredibly well-chosen hypotheses which is just…

We consider the problem of optimal transportation with general cost between a empirical measure and a general target probability on R d , with d $\ge$ 1. We extend results in [19] and prove asymptotic stability of both optimal transport…

The aim of this paper is to recover displacement interpolations of probability measures, in the sense of the Optimal Transport theory, by semiclassical measures associated with solutions of Schr\"odinger equations defined on the flat torus.…

Given a function $F$ transforming a probability measure $\mu$ into another one $F(\mu)$, we study the existence and regularity of a transport representation of it. That is, we ask whether we can represent the image $F(\mu)$ of the input…