Related papers: On the optimal map in the 2-dimensional random mat…
In this paper, we study the optimal transport problem induced by separable cost functions. In this framework, transportation can be expressed as the composition of two lower-dimensional movements. Through this reformulation, we prove that…
We consider robust variants of the standard optimal transport, named robust optimal transport, where marginal constraints are relaxed via Kullback-Leibler divergence. We show that Sinkhorn-based algorithms can approximate the optimal cost…
We study the optimal transport problem in the Euclidean space where the cost function is given by the value function associated with a Linear Quadratic minimization problem. Under appropriate assumptions, we generalize Brenier's Theorem…
We discuss the optimal matching solution for both the assignment problem and the matching problem in one dimension for a large class of convex cost functions. We consider the problem in a compact set with the topology both of the interval…
We propose two deep neural network-based methods for solving semi-martingale optimal transport problems. The first method is based on a relaxation/penalization of the terminal constraint, and is solved using deep neural networks. The second…
We introduce a constrained optimal transport problem where origins $x$ can only be transported to destinations $y\geq x$. Our statistical motivation is to describe the sharp upper bound for the variance of the treatment effect $Y-X$ given…
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…
Riemannian optimization uses local methods to solve optimization problems whose constraint set is a smooth manifold. A linear step along some descent direction usually leaves the constraints, and hence retraction maps are used to…
In this paper, we investigate optimal (partial) transport problems for which the target is a non-convex polygonal domain in \(\mathbb{R}^2\). For the complete optimal transport problem, we prove that the singular set is locally a smooth…
We give a new proof of the Caffarelli contraction theorem, which states that the Brenier optimal transport map sending the standard Gaussian measure onto a uniformly log-concave probability measure is Lipschitz. The proof combines a recent…
We study an optimization problem related to the approximation of given data by a linear combination of transformed modes. In the simplest case, the optimization problem reduces to a minimization problem well-studied in the context of proper…
In this paper we present a two-stage algorithm to solve the semi-discrete Optimal Transport Problem in the case where the support of the source measure is disconnected. We establish global linear convergence and local superlinear…
Obtaining solutions to Optimal Transportation (OT) problems is typically intractable when the marginal spaces are continuous. Recent research has focused on approximating continuous solutions with discretization methods based on i.i.d.…
We pose the Kantorovich optimal transport problem as a min-max problem with a Nash equilibrium that can be obtained dynamically via a two-player game, providing a framework for approximating optimal couplings. We prove convergence of the…
In this paper, we consider Strassen's version of optimal transport (OT) problem, which concerns minimizing the excess-cost probability (i.e., the probability that the cost is larger than a given value) over all couplings of two given…
Optimal transport (OT) has recently found widespread interest in machine learning. It allows to define novel distances between probability measures, which have shown promise in several applications. In this work, we discuss how to…
We study the estimation of optimal transport (OT) maps between an arbitrary source probability measure and a log-concave target probability measure. Our contributions are twofold. First, we propose a new evolution equation in the set of…
We study statistical inference for the optimal transport (OT) map (also known as the Brenier map) from a known absolutely continuous reference distribution onto an unknown finitely discrete target distribution. We derive limit distributions…
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.…
The bipartite matching problem is widely applied in the field of transportation; e.g., to find optimal matches between supply and demand over time and space. Recent efforts have been made on developing analytical formulas to estimate the…