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A measure theoretical approach is presented to study the Monge-Kantorovich optimal mass transport problem. This approach together with Kantorovich duality provide an effective tool to answer a long standing question about the support of…
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…
This paper presents a widely applicable approach to solving (multi-marginal, martingale) optimal transport and related problems via neural networks. The core idea is to penalize the optimization problem in its dual formulation and reduce it…
This paper investigates causal optimal transportation problems, in the framework of two Polish spaces, both endowed with filtrations. Specific concretizations yield primal problems equivalent to several classical problems of stochastic…
We prove existence of an optimal transport map in the Monge-Kantorovich problem associated to a cost $c(x,y)$ which is not finite everywhere, but coincides with $|x-y|^2$ if the displacement $y-x$ belongs to a given convex set $C$ and it is…
Monotonicity and convex analysis arise naturally in the framework of multi-marginal optimal transport theory. However, a comprehensive multi-marginal monotonicity and convex analysis theory is still missing. To this end we study extensions…
We consider a multimarginal optimal transport, which includes as a particular case the Wasserstein barycenter problem. In this problem one has to find an optimal coupling between $m$ probability measures, which amounts to finding a tensor…
We introduce and investigate properties of a variant of the semi-discrete optimal transport problem. In this problem, one is given an absolutely continuous source measure and cost function, along with a finite set which will be the support…
Weak optimal transport generalizes the classical theory of optimal transportation to nonlinear cost functions and covers a range of problems that lie beyond the traditional theory - including entropic transport, martingale transport, and…
In this paper, we address the numerical solution to the multimarginal optimal transport (MMOT) with pairwise costs. MMOT, as a natural extension from the classical two-marginal optimal transport, has many important applications including…
In this series of lectures we introduce the Monge-Kantorovich problem of optimally transporting one distribution of mass onto another, where optimality is measured against a cost function c(x,y). Connections to geometry, inequalities, and…
For a family of probability spaces $\{(X_k,\mathcal{B}_{X_k},\mu_k)\}_{k=1}^N$ and a cost function $c: X_1\times\cdots\times X_N\to \mathbb{R}$ we consider the Monge-Kantorovich problem \begin{align*}\tag{MK}\label{MONKANT}…
Optimal transport is the problem of designing a joint distribution for two random variables with fixed marginals. In virtually the entire literature on this topic, the objective is to minimize expected cost. This paper is the first to study…
A probabilistic method for solving the Monge-Kantorovich mass transport problem on $R^d$ is introduced. A system of empirical measures of independent particles is built in such a way that it obeys a doubly indexed large deviation principle…
We consider the multidimensional Monge-Kantrovich transport problem in an abstract setting. Our main results state that if a cost function and marginal measures are invariant by a family of transformations, then a solution of the Kantrovich…
The classical problem of optimal transportation can be formulated as a linear optimization problem on a convex domain: among all joint measures with fixed marginals find the optimal one, where optimality is measured against a cost function.…
By investigating model-independent bounds for exotic options in financial mathematics, a martingale version of the Monge-Kantorovich mass transport problem was introduced in \cite{BeiglbockHenry…
We present a primal-dual dynamical formulation of the multi-marginal optimal transport problem for (semi-)convex cost functions. Even in the two-marginal setting, this formulation applies to cost functions not covered by the classical…
We investigate the structure of optimal transport plans, dual optimizers, and geodesic paths for the quadratic barycentric transport problem.
In this work we study a modification of the Monge-Kantorovich problem taking into account path dependence and interaction effects between particles. We prove existence of solutions under mild conditions on the data, and after imposing…