Journal ArticleDOI
On a Kantorovich Problem with a Density Constraint
TLDR
In this paper, the existence and uniqueness of a solution of the Kantorovich optimal transport problem with a density constraint on measures on an infinite-dimensional space is proved, and the admissible transport plan is nonnegative and majorized by a given constraint function.Abstract:
The Kantorovich optimal transport problemwith a density constraint onmeasures on an infinite-dimensional space is considered. In this setting, the admissible transport plan is nonnegative and majorized by a given constraint function. The existence and the uniqueness of a solution of this problem are proved.read more
Citations
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Journal ArticleDOI
Kantorovich problems and conditional measures depending on a parameter
TL;DR: In this article, the authors studied the Borel measurability of conditional measures and the Kantorovich optimal transportation with respect to a parametric family of measures and mappings.
Posted Content
The multistochastic Monge-Kantorovich problem.
TL;DR: The multistsochastic Monge-Kantorovich problem on the product of measures with fixed projections is a generalization of the multimarginal MKK problem.
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Kantorovich problems and conditional measures depending on a parameter
TL;DR: In this article, the authors studied the Borel measurability of measures on a parameter in the case of parametric families of measures and mappings and provided sufficient conditions for the existence of conditional probabilities measurably depending on a parametric family of measures.
Journal ArticleDOI
Задача Канторовича оптимальной транспортировки мер: новые направления исследований
TL;DR: In this article , the authors proposed a method to improve the quality of the data collected by the system by using the information from the users' own data points of interest (e.g., the data points from the user's phone, the phone number, etc.).
References
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Book
Gradient Flows: In Metric Spaces and in the Space of Probability Measures
TL;DR: In this article, Gradient flows and curves of Maximal slopes of the Wasserstein distance along geodesics are used to measure the optimal transportation problem in the space of probability measures.
Journal ArticleDOI
On the Translocation of Masses
TL;DR: In this paper, Kantorovich and Akad defined a translocation of masses as a function Ψ(e, e′) defined for pairs of (B)-sets e, e − ∈ R such that: (1) it is nonnegative and absolutely additive with respect to each of its arguments, (2) Φ (e, R) = Φ(e), Ψ (R, e−∆), Ω(R, E − ∆), e− ∆ = Π(e−∀ −∆ −
Journal ArticleDOI
The geometry of optimal transportation
TL;DR: In this paper, the existence and uniqueness of optimal maps are discussed. But the uniqueness of the optimal map is not discussed. And the role of the map in finding the optimal solution is left open.
BookDOI
Mass transportation problems
TL;DR: In this article, a modification of the Monge-Kantorovich Problem with relaxed or additional constraints is presented. But this modification is restricted to the case where the Kantorovich-type metrics are applied to various Probabilistic-Type Limit Theorems.
Book ChapterDOI
A User’s Guide to Optimal Transport
Luigi Ambrosio,Nicola Gigli +1 more
TL;DR: In this paper, the authors provide a quick and reasonably account of the classical theory of optimal mass transportation and its more recent developments, including the metric theory of gradient flows, geometric and functional inequalities related to optimal transportation, the first and second order differential calculus in the Wasserstein space and the synthetic theory of metric measure spaces with Ricci curvature bounded from below.