Related papers: Optimal Transportation of Vector-Valued Measures
In this paper, for $\mu$ and $\nu$ two probability measures on $\mathbb{R}^d$ with finite moments of order $\rho\ge 1$, we define the respective projections for the $W_\rho$-Wasserstein distance of $\mu$ and $\nu$ on the sets of probability…
We study a variant of the dynamical optimal transport problem in which the energy to be minimised is modulated by the covariance matrix of the distribution. Such transport metrics arise naturally in mean-field limits of certain ensemble…
This short note gives a proof of the triangle inequality based on the Kantorovich duality formula for the Wasserstein distances of exponent $p\in[1,+\infty)$ in the case of a general Polish space. In particular it avoids the "glueing of…
Optimal transport has found widespread applications in signal processing and machine learning. Among its many equivalent formulations, optimal transport seeks to reconstruct a random variable/vector with a prescribed distribution at the…
We shall present a measure theoretical approach for which together with the Kantorovich duality provide an efficient tool to study the optimal transport problem. Specifically, we study the support of optimal plans where the cost function…
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…
We generalize a well-known result of L. Caffarelli on Lipschitz estimates for optimal transportation $T$ between uniformly log-concave probability measures. Let $T : \R^d \to \R^d$ be an optimal transportation pushing forward $\mu =…
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 study the optimal transport between two probability measures on the real line, where the transport plans are laws of one-step martingales. A quasi-sure formulation of the dual problem is introduced and shown to yield a complete duality…
In this paper, we establish a Kantorovich duality for weak optimal total variation transport problems. As consequences, we recover a version of duality formula for partial optimal transports established by Caffarelli and McCann; and we also…
We propose and study a new approach to the topologization of spaces of (possibly not all) future-directed causal curves in a stably causal spacetime. It relies on parametrizing the curves "in accordance" with a chosen time function. Thus…
We characterise equality cases in matrix H\"older's inequality and develop a divergence formulation of optimal transport of vector measures. As an application, we reprove the representation formula for measures in the polar cone to monotone…
This is the first part of a general description in terms of mass transport for time-evolving interacting particles systems, at a mesoscopic level. Beyond kinetic theory, our framework naturally applies in biology, computer vision, and…
Stability of the value function and the set of minimizers w.r.t. the given data is a desirable feature of optimal transport problems. For the classical Kantorovich transport problem, stability is satisfied under mild assumptions and in…
We discuss the relation between the Wasserstein distance of order 1 between probability distributions on a metric space, arising in the study of Monge-Kantorovich transport problem, and the spectral distance of noncommutative geometry.…
Optimal transport (OT) provides powerful tools for comparing probability measures in various types. The Wasserstein distance which arises naturally from the idea of OT is widely used in many machine learning applications. Unfortunately,…
These notes constitute a sort of Crash Course in Optimal Transport Theory. The different features of the problem of Monge-Kantorovitch are treated, starting from convex duality issues. The main properties of space of probability measures…
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…
Optimal transport provides an inherently geometric and highly structured framework for studying spaces of probability measures, supplying a rich theoretical toolkit for contemporary statistics, machine learning, and generative modelling. In…
We provide a unifying interpretation of various optimal transport problems as a minimisation of a linear functional over the set of all Choquet representations of a given pair of probability measures ordered with respect to a certain convex…