Related papers: 60 years of cyclic monotonicity: a survey
A general theory is provided delivering convergence of maximal cyclically monotone mappings containing the supports of coupling measures of sequences of pairs of possibly random probability measures on Euclidean space. The theory is based…
The Monge-Kantorovich mass transfer problem is equivalently formulated as a convex optimization problem for a potential function. In the light of this formulation an interative algorithm is developed for determining the solution. It is a…
The presentation covers prerequisite results from Topology and Measure Theory. This is then followed by an introduction into couplings and basic definitions for optimal transport. The Kantrorovich problem is then introduced and an existence…
We approach the problem of constructing a quantum analogue of the immensely fruitful classical transport cost theory of Monge from a new angle. Going back to the original motivations, by which the transport is a bilinear function of a mass…
We study the dual formulation of the Monge-Kantorovich optimal transportation problem, in particular under what circumstances it is permitted in an infinite dimensional setting to use cylindrical functions, i.e. functions of the form…
We propose an overview of optimal transport theory and its applications to econometric methodology. This review is specifically designed for practitioners, be they econometric theorists or applied econometricians. The review of applications…
The goal of the present work is three-fold. The first goal is to set foundational results on optimal transport in Lorentzian (pre-)length spaces, including cyclical monotonicity, stability of optimal couplings and Kantorovich duality…
The aim of this short note is to extend the recent variational proof of partial regularity for optimal transport maps to the case of continuous densities.
This note outlines a mean-field approach to dynamic optimal transport problems based on the recently proposed McKean-Pontryagin maximum principle. Key aspects of the proposed methodology include i) avoidance of sampling over stochastic…
Contraction theory is a mathematical framework for studying the convergence, robustness, and modularity properties of dynamical systems and algorithms. In this opinion paper, we provide five main opinions on the virtues of contraction…
We define the concept of a monotonic theory and show how to build efficient SMT (SAT Modulo Theory) solvers, including effective theory propagation and clause learning, for such theories. We present examples showing that monotonic theories…
Monotone systems, also known as order-preserving or cooperative systems, are prevalent in models of engineering applications such as transportation and biological networks. In this paper, we investigate the problem of finding a control…
We give new computable necessary conditions for a class of optimal transportation problems to have smooth solutions.
These lecture notes present the quantitative harmonic approximation result for quadratic optimal transport and general measures obtained by Goldman and Otto. The aim is to give a clear presentation of the proof of the main theorem with more…
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)…
We formulate and solve a class of finite-time transport and mixing problems in the set-oriented framework. The aim is to obtain optimal discrete-time perturbations in nonlinear dynamical systems to transport a specified initial measure on…
This paper investigates causal optimal transportation problems, in the framework of two Polish spaces, both endowed with filtrations. Specific concretizations yield primal problems equivalent to several classical problems of stochastic…
We introduce a cycle-expansion (fully deterministic) technique to compute the asymptotic behavior of arbitrary order transport moments. The theory is applied to different kinds of one-dimensional intermittent maps, and Lorentz gas with…
In optimal transport, quadratic regularization is a sparse alternative to entropic regularization: the solution measure tends to have small support. Computational experience suggests that the support decreases monotonically to the…
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…