Related papers: Conditional Optimal Transport on Function Spaces
In this paper we extend recent developments in computational optimal transport to the setting of Riemannian manifolds. In particular, we show how to learn optimal transport maps from samples that relate probability distributions defined on…
In this note, we propose polynomial-time algorithms solving the Monge and Kantorovich formulations of the $\infty$-optimal transport problem in the discrete and finite setting. It is the first time, to the best of our knowledge, that…
Discrete optimal transportation problems arise in various contexts in engineering, the sciences and the social sciences. Often the underlying cost criterion is unknown, or only partly known, and the observed optimal solutions are corrupted…
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…
We present a neural framework for learning conditional optimal transport (OT) maps between probability distributions. Our approach introduces a conditioning mechanism capable of processing both categorical and continuous conditioning…
We propose a novel holistic approach for safe autonomous exploration and map building based on constrained Bayesian optimisation. This method finds optimal continuous paths instead of discrete sensing locations that inherently satisfy…
We study a single-period optimal transport problem on $\mathbb{R}^2$ with a covariance-type cost function $c(x,y) = (x_1-y_1)(x_2-y_2)$ and a backward martingale constraint. We show that a transport plan $\gamma$ is optimal if and only if…
The inverse optimal transport problem is to find the underlying cost function from the knowledge of optimal transport plans. While this amounts to solving a linear inverse problem, in this work we will be concerned with the nonlinear…
We identify a condition for regularity of optimal transport maps that requires only three derivatives of the cost function, for measures given by densities that are only bounded above and below. This new condition is equivalent to the weak…
We consider an optimal transport problem with backward martingale constraint. The objective function is given by the scalar product of a pseudo-Euclidean space $S$. We show that the supremums over maps and plans coincide, provided that the…
We establish numerical methods for solving the martingale optimal transport problem (MOT) - a version of the classical optimal transport with an additional martingale constraint on transport's dynamics. We prove that the MOT value can be…
Optimal transport (OT) naturally arises in a wide range of machine learning applications but may often become the computational bottleneck. Recently, one line of works propose to solve OT approximately by searching the \emph{transport plan}…
We discuss the Monge problem of mass transportation in the framework of stochastic thermodynamics and revisit the problem of the Landauer limit for finite-time thermodynamics, a problem that got the interest of Krzysztof Gawedzki in the…
Many causal and structural parameters in economics can be identified and estimated by computing the value of an optimization program over all distributions consistent with the model and the data. Existing tools apply when the data is…
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…
Functional lifting methods provide a tool for approximating solutions of difficult non-convex problems by embedding them into a larger space. In this work, we investigate a mathematically rigorous formulation based on embedding into the…
We provide a method to solve optimization problem when objective function is a complex stochastic simulator of an urban transportation system. To reach this goal, a Bayesian optimization framework is introduced. We show how the choice of…
We investigate the transportation problem under a Monge cost structure and derive compact formulas for optimal dual solutions based on the northwest-corner rule. As an application illustrating how these formulas yield structural insight…
We provide a new proof of the known partial regularity result for the optimal transportation map (Brenier map) between two sets. Contrary to the existing regularity theory for the Monge-Amp{\`e}re equation, which is based on the maximum…
The Monge-Amp\`{e}re equation arises in the theory of optimal transport. When more complicated cost functions are involved in the optimal transportation problem, which are motivated e.g. from economics, the corresponding equation for the…