Related papers: Grand-Canonical Optimal Transport
This note establishes that a generalization of $c$-cyclical monotonicity from the Monge-Kantorovich problem with two marginals gives rise to a sufficient condition for optimality also in the multi-marginal version of that problem. To obtain…
We study solutions to the multi-marginal Monge-Kantorovich problem which are concentrated on several graphs over the first marginal. We first present two general conditions on the cost function which ensure, respectively, that any solution…
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…
This article studies problems of optimal transport, by embedding them in a general functional analytic framework of convex optimization. This provides a unified treatment of a large class of related problems in probability theory and allows…
This paper is devoted to variational problems on the set of probability measures which involve optimal transport between unequal dimensional spaces. In particular, we study the minimization of a functional consisting of the sum of a term…
Optimal Transport is a foundational mathematical theory that connects optimization, partial differential equations, and probability. It offers a powerful framework for comparing probability distributions and has recently become an important…
Optimal mass transport is described by an approximation of transport cost via semi-discrete costs. The notions of optimal partition and optimal strong partition are given as well. We also suggest an algorithm for computation of Optimal…
Optimal transport has become part of the standard quantitative economics toolbox. It is the framework of choice to describe models of matching with transfers, but beyond that, it allows to: extend quantile regression; identify discrete…
Motivated by optimal re-balancing of a portfolio, we formalize an optimal transport problem in which the transported mass is scaled by a mass-change factor depending on the source and destination. This allows direct modeling of the creation…
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…
Semidiscrete optimal transport is a challenging generalization of the classical transportation problem in linear programming. The goal is to design a joint distribution for two random variables (one continuous, one discrete) with fixed…
A variant of the classical optimal transportation problem is: among all joint measures with fixed marginals and which are dominated by a given density, find the optimal one. Existence and uniqueness of solutions to this variant were…
We study the Lagrangian formulation of a class of the Monge-Kantorovich optimal transportation problem. It can be considered a stochastic optimal transportation problem for absolutely continuous stochastic processes. A cost function and…
Recent advances in large-scale optimal transport have greatly extended its application scenarios in machine learning. However, existing methods either not explicitly learn the transport map or do not support general cost function. In this…
We construct an explicit solution for the multimarginal transportation problem on the unit cube $[0,1]^3$ with the cost function $xyz$ and one-dimensional uniform projections. We show that the primal problem is concentrated on a set with…
This paper studies the geometry of the optimizer for the optimal transport problem with capacity constraints. We introduce the concept of c-capacity monotonicity, which is a generalization of c-cyclical monotonicity in optimal transport. We…
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…
We consider maps $T$ solving the optimal transport problem with a cost $c(x-y)$ modeled on the $p$-cost. For H\"older continuous marginals, we prove a $C^{1,\alpha}$-partial regularity result for $T $in the set $\{|T(x)-x|>0\}$.
We study a multi-marginal optimal transportation problem. Under certain conditions on the cost function and the first marginal, we prove that the solution to the relaxed, Kantorovich version of the problem induces a solution to the Monge…
Consider a transportation problem with sets of sources and sinks. There are profits and prices on the edges. The goal is to maximize the profit while meeting the following constraints; the total flow going out of a source must not exceed…