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Kantorovich duality for general transport costs and applications
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In this article, a general notion of transport cost is introduced, which encompasses many costs used in the literature, including the classical one and weak transport costs introduced by Talagrand and Marton in the 90's.Abstract:
We introduce a general notion of transport cost that encompasses many costs used in the literature (including the classical one and weak transport costs introduced by Talagrand and Marton in the 90's), and prove a Kantorovich type duality theorem. As a by-product we obtain various applications in different directions: we give a short proof of a result by Strassen on the existence of a martingale with given marginals, we characterize the associated transport-entropy inequalities together with the log-Sobolev inequality restricted to convex/concave functions. Some explicit examples of discrete measures satisfying weak transport-entropy inequalities are also given.read more
Citations
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Causal transport in discrete time and applications
TL;DR: A dynamic programming principle is established that links the causal transport problem to the transport problem for general costs recently considered by Gozlan et al. and gives conditions under which the celebrated Knothe-Rosenblatt rearrangement can be viewed as a causal analogue to the Brenier's map.
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A new class of costs for optimal transport planning
TL;DR: In this article, the authors studied a class of optimal transport planning problems where the reference cost involves a non-linear function G(x, p) representing the transport cost between the Dirac mesure x and a target probability p, and they derived a Kantorovich-Rubinstein version of the dual problem allowing to show existence in some regular cases.
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Sampling of probability measures in the convex order by Wasserstein projection
TL;DR: The motivation is the design of sampling techniques preserving the convex order in order to approximate Martingale Optimal Transport problems by using linear programming solvers and convergence of the Wasserstein projection based sampling methods as the sample sizes tend to infinity.
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On a mixture of brenier and strassen theorems
Nathael Gozlan,Nicolas Juillet +1 more
TL;DR: In this paper, the authors give a characterization of optimal transport plans for a variant of the usual quadratic transport cost introduced in [33], where the optimal plans are composition of a deterministic transport given by the gradient of a continuously differentiable convex function followed by a martingale coupling.
References
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Book
The Probabilistic Method
TL;DR: A particular set of problems - all dealing with “good” colorings of an underlying set of points relative to a given family of sets - is explored.
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Optimal Transport: Old and New
TL;DR: In this paper, the authors provide a detailed description of the basic properties of optimal transport, including cyclical monotonicity and Kantorovich duality, and three examples of coupling techniques.
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Topics in Optimal Transportation
TL;DR: In this paper, the metric side of optimal transportation is considered from a differential point of view on optimal transportation, and the Kantorovich duality of the optimal transportation problem is investigated.
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Functions of Bounded Variation and Free Discontinuity Problems
TL;DR: The Mumford-Shah functional minimiser of free continuity problems as mentioned in this paper is a special function of the Mumfordshah functional and has been shown to be a function of free discontinuity set.
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Concentration Inequalities: A Nonasymptotic Theory of Independence
TL;DR: Deep connections with isoperimetric problems are revealed whilst special attention is paid to applications to the supremum of empirical processes.