Related papers: Quantitative Stability in Discrete Optimal Transpo…
This paper studies the uniqueness of solutions to the dual optimal transport problem, both qualitatively and quantitatively (bounds on the diameter of the set of optimisers). On the qualitative side, we prove that when one marginal…
We establish novel quantitative stability results for optimal transport problems with respect to perturbations in the target measure. We provide explicit bounds on the stability of optimal transport potentials and maps, which are relevant…
We study the stability of entropically regularized optimal transport with respect to the marginals. Lipschitz continuity of the value and H\"older continuity of the optimal coupling in $p$-Wasserstein distance are obtained under general…
Quadratically regularized optimal transport (QOT) is a sparse alternative to entropic optimal transport. We develop a quantitative stability theory for QOT under perturbations of the marginals, the transport cost function, and the…
We study the quantitative stability of the mapping that to a measure associates its pushforward measure by a fixed (non-smooth) optimal transport map. We exhibit a tight H\"older-behavior for this operation under minimal assumptions. Our…
We prove quantitative bounds on the stability of optimal transport maps and Kantorovich potentials from a fixed source measure $\rho$ under variations of the target measure $\mu$, when the cost function is the squared Riemannian distance on…
The Monge-Kantorovich problem for the infinite Wasserstein distance presents several peculiarities. Among them the lack of convexity and then of a direct duality. We study in dimension 1 the dual problem introduced by Barron, Bocea and…
We study stability and sample complexity properties of divergence regularized optimal transport (DOT). First, we obtain quantitative stability results for optimizers of DOT measured in Wasserstein distance, which are applicable to a wide…
We use a Korevaar-style maximum principle approach to show the following: Fixing a $C^{2}$ bound on the log densities of a set of smooth measures, there is a quantifiably-sized Wasserstein neighborhood over which all pairs of such measures…
We analyze optimal transport problems with additional entropic cost evaluated along curves in the Wasserstein space which join two probability measures $m_0,m_1$. The effect of the additional entropy functional results into an elliptic…
In this paper, we prove a structure theorem for discrete optimal transportation plans. We show that, given any pair of discrete probability measures and a cost function, there exists an optimal transportation plan that can be expressed as…
This work studies the quantitative stability of the quadratic optimal transport map between a fixed probability density $\rho$ and a probability measure $\mu$ on R^d , which we denote T$\mu$. Assuming that the source density $\rho$ is…
We introduce folded optimal transport, as a method to extend a cost or distance defined on the extreme boundary of a convex to the whole convex, related to convex extension. This construction broadens the framework of standard optimal…
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.…
We show quantitative stability results for the geometric "cells" arising in semi-discrete optimal transport problems. Our results show two types of stability, the first is stability of the associated Laguerre cells in measure, without any…
The Cauchy problem for a multidimensional linear transport equation with discontinuous coefficient is investigated. Provided the coefficient satisfies a one-sided Lipschitz condition, existence, uniqueness and weak stability of solutions…
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…
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 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…
Semi-discrete optimal transport problems, which evaluate the Wasserstein distance between a discrete and a generic (possibly non-discrete) probability measure, are believed to be computationally hard. Even though such problems are…