Related papers: Optimal Transport on Discrete Domains
We address optimal control problems on the space of measures for an objective containing a smooth functional and an optimal transport regularization. That is, the quadratic Monge-Kantorovich distance between a given prior measure and the…
We consider optimal transport problems where the cost for transporting a given probability measure $\mu_0$ to another one $\mu_1$ consists of two parts: the first one measures the transportation from $\mu_0$ to an intermediate (pivot)…
Optimal transport (OT) defines a powerful framework to compare probability distributions in a geometrically faithful way. However, the practical impact of OT is still limited because of its computational burden. We propose a new class of…
We introduce a new non-linear optimal transport formulation for a pair of probability measures on $\mathbb{R}^d$ sharing a common barycentre, in which admissible transference plans satisfy two martingale-type constraints. This bi-martingale…
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…
We propose a numerical algorithm for the computation of multi-marginal optimal transport (MMOT) problems involving general probability measures that are not necessarily discrete. By developing a relaxation scheme in which marginal…
This article gives an introduction to optimal transport, a mathematical theory that makes it possible to measure distances between functions (or distances between more general objects), to interpolate between objects or to enforce…
In the current book I suggest an off-road path to the subject of optimal transport. I tried to avoid prior knowledge of analysis, PDE theory and functional analysis, as much as possible. Thus I concentrate on discrete and semi-discrete…
We consider an optimal transport problem between laws of random probability measures: given a base cost function, we build the associated OT cost between probability measures that in turn we use to define the OT cost between probability…
Optimal transport is a geometrically intuitive, robust and flexible metric for sample comparison in data analysis and machine learning. Its formal Riemannian structure allows for a local linearization via a tangent space approximation. This…
Optimal mass transport, also known as the earth mover's problem, is an optimization problem with important applications in various disciplines, including economics, probability theory, fluid dynamics, cosmology and geophysics to cite a few.…
In this work, we construct a novel numerical method for solving the multi-marginal optimal transport problems with Coulomb cost. This type of optimal transport problems arises in quantum physics and plays an important role in understanding…
We consider Kantorovich optimal transportation problem in the case where the cost function and marginal distributions continuously depend on a parameter with values in a metric space. We prove the existence of approximate optimal Monge…
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…
We consider optimal transport problems where the cost is optimized over controlled dynamics and the end time is free. Unlike the classical setting, the search for optimal transport plans also requires the identification of optimal "stopping…
We introduce and analyze a statistical estimator for Monge transport maps: solutions to the quadratic optimal transport problem in Euclidean space. For absolutely continuous source measures, this map is uniquely defined as the gradient of a…
The aim of this article is to show that the Monge-Kantorovich problem is the limit of a sequence of entropy minimization problems when a fluctuation parameter tends down to zero. We prove the convergence of the entropic values to the…
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…
We introduce and study a multi-marginal optimal partial transport problem. Under a natural and sharp condition on the dominating marginals, we establish uniqueness of the optimal plan. Our strategy of proof establishes and exploits a…
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,…