The Monge-Kantorovich problem: achievements, connections, and perspectives

Богачев, В. И.; Колесников, Александр Викторович

doi:10.1070/rm2012v067n05abeh004808

Cited by 152 publications

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“…We refer to [3], [24], where the reader can find comprehensive information about the solvability, uniqueness, and regularity issues. (2) Given a probability measure ν satisfying xdν = 0 find a solution to the elliptic Kähler-Einstein equation…”

Section: Notations Definitions and Previously Known Resultsmentioning

confidence: 99%

“…The results of Corollaries 6.4, 6.5 are dimension-free and have natural analogues for infinite-dimensional measures. For instance, some natural estimates of this type holds for the potential ϕ of the optimal transporation T (x) = x + ∇ϕ(x) pushing forward g · γ onto a (infinite dimensional) Gaussian measure γ, where ∇ϕ is understood as a gradient along the Cameron-Martin space of γ (see [16], [3], [4] and the references therein).…”

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confidence: 99%

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Remarks on curvature in the transportation metric

Alexander

2017

Anal Math

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Abstract. According to a classical result of E. Calabi any hyperbolic affine hypersphere endowed with its natural Hessian metric has a non-positive Ricci tensor. The affine hyperspheres can be described as the level sets of solutions to the "hyperbolic" toric Kähler-Einstein equation e Φ = det D 2 Φ on proper convex cones. We prove a generalization of this theorem by showing that for every Φ solving this equation on a proper convex domain Ω the corresponding metric measure space (D 2 Φ, e Φ dx) has a non-positive Bakry-Émery tensor. Modifying the Calabi computations we obtain this result by applying the tensorial maximum principle to the weighted Laplacian of the Bakry-Émery tensor. Our computations are carried out in a generalized framework adapted to the optimal transportation problem for arbitrary target and source measures. For the optimal transportation of the log-concave probability measures we prove a third-order uniform dimension-free apriori estimate in the spirit of the second-order Caffarelli contraction theorem, which has numerous applications in probability theory.

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Section: Notations Definitions and Previously Known Resultsmentioning

confidence: 99%

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confidence: 99%

Remarks on curvature in the transportation metric

Alexander

2017

Anal Math

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“…However, most density or distribution functions do not possess such a property or are not in the space of L p , Sobolev, Besov and so on. Therefore a statistical distance, called Wasserstein metric [1], is introduced. Intuitively, if each density function is viewed as a unit amount of "earth", the distance between two density functions is the minimum cost of the energy of turning one pile into the other.…”

Section: §1 Introductionmentioning

confidence: 99%

Distribution function estimates by Wasserstein metric and Bernstein approximation for C −1 functions

Zheng

2015

Appl. Math. J. Chin. Univ.

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The aim of the paper is to estimate the density functions or distribution functions measured by Wasserstein metric, a typical kind of statistical distances, which is usually required in the statistical learning. Based on the classical Bernstein approximation, a scheme is presented.To get the error estimates of the scheme, the problem turns to estimating the L1 norm of the Bernstein approximation for monotone C −1 functions, which was rarely discussed in the classical approximation theory. Finally, we get a probability estimate by the statistical distance. §1 IntroductionIn the statistical learning such as pattern recognition and classification, one of the key problems is to find the conditional probability distribution or conditional expectation, which is based on the distributions estimated from sampling data. There are a lot of schemes to estimate the density functions and the corresponding distribution functions(e.g. [3,5,6]). However, most of them are based on the traditional mathematical norms such as L p (usually mean square error), Sobolev and Besov, which require the continuity of the approximated function (the density or distribution functions). However, most density or distribution functions do not possess such a property or are not in the space of L p , Sobolev, Besov and so on. Therefore a statistical distance, called Wasserstein metric [1], is introduced. Intuitively, if each density function is viewed as a unit amount of "earth", the distance between two density functions is the minimum cost of the energy of turning one pile into the other. Because of this analogy, the metric is known in computer science as the earth mover's distance. The following two examples are presented to show why we prefer Wasserstein metric in our study.

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“…This problem is known in measure theory as the Monge transport problem [22,23] and a lot of results have been obtained regarding the existence of the optimal map and its properties [24]. One of the most interesting cases is the quadratic one, in which the cost is given by the convex function w(x,y) = x − y 2 , and we have to minimize the functional…”

Section: A the Monge-kantorovič Problem And Monge-ampère Equationmentioning

confidence: 99%

A New Approach to the Matching Problem

Mertens¹

2014

Physics

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We propose a simple yet very predictive form, based on a Poisson's equation, for the functional dependence of the cost from the density of points in the Euclidean bipartite matching problem. This leads, for quadratic costs, to the analytic prediction of the large N limit of the average cost in dimension d = 1,2 and of the subleading correction in higher dimension. A nontrivial scaling exponent,, which differs from the monopartite's one, is found for the subleading correction. We argue that the same scaling holds true for a generic cost exponent in dimension d > 2.

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The Monge-Kantorovich problem: achievements, connections, and perspectives

Cited by 152 publications

References 148 publications

Remarks on curvature in the transportation metric

Remarks on curvature in the transportation metric

Distribution function estimates by Wasserstein metric and Bernstein approximation for C −1 functions

A New Approach to the Matching Problem

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