On a generalization of cyclic monotonicity and distances among random vectors

Knott, M.; Smith, Cyril Stanley

doi:10.1016/0024-3795(94)90359-x

Cited by 42 publications

(43 citation statements)

References 5 publications

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“…In Euclidean space, this coincides with the cost studied by Gangbo and Swiech [17], who proved assertion 1 in Theorem 2.4 below in that setting, extending earlier partial results of Olkin and Rachev [30], Knott and Smith [22] and Ruschendorf and Uckelmann [35].…”

Section: Optimal Transport On Riemannian Manifoldssupporting

confidence: 86%

Wasserstein barycenters over Riemannian manifolds

Kim

Pass

2017

Advances in Mathematics

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We study barycenters in the space of probability measures on a Riemannian manifold, equipped with the Wasserstein metric. Under reasonable assumptions, we establish absolute continuity of the barycenter of general measures Ω ∈ P (P (M )) on Wasserstein space, extending on one hand, results in the Euclidean case (for barycenters between finitely many measures) of Agueh and Carlier [1] to the Riemannian setting, and on the other hand, results in the Riemannian case of Cordero-Erausquin, McCann, Schmuckenschläger [9] for barycenters between two measures to the multi-marginal setting. Our work also extends these results to the case where Ω is not finitely supported. As applications, we prove versions of Jensen's inequality on Wasserstein space and a generalized Brunn-Minkowski inequality for a random measurable set on a Riemannian manifold.

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Section: Optimal Transport On Riemannian Manifoldssupporting

confidence: 86%

Wasserstein barycenters over Riemannian manifolds

Kim

Pass

2017

Advances in Mathematics

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“…n , extending partial results for the same cost by Olkin and Rachev [19], Knott and Smith [14], and Rüschendorf and Uckelmann [22]. Gangbo andŚwiȩch's theorem was then reproved using a different argument by Rüschendorf and Uckelmann [23] and generalized by Heinich [11] to cost functions of the form c(…”

Section: Introductionmentioning

confidence: 87%

Uniqueness and Monge Solutions in the Multimarginal Optimal Transportation Problem

Pass¹

2011

SIAM J. Math. Anal.

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Abstract. We study a multimarginal optimal transportation problem. Under certain conditions on the cost function and the first marginal, we prove that the solution to the relaxed, Kantorovich version of the problem induces a solution to the Monge problem and that the solutions to both problems are unique. We also exhibit several examples of cost functions under which our conditions are satisfied, including one arising in a hedonic pricing model in mathematical economics.

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“…The input data are the empirical vectorial means x x k and covariance matrices † k of the k , and the unknown parameters of the barycenter are x y and † y . In terms of these, the map Y k .x/ D˛kx Cˇk (here˛k is a matrix andˇk a vector) has parameters [15] …”

Section: Poor-man Solutionsmentioning

confidence: 99%

“…which can be solved effectively using the following iterative scheme [15]: Propose an initial guess for † y , such as † 0 D…”

Section: Poor-man Solutionsmentioning

confidence: 99%

Explanation of Variability and Removal of Confounding Factors from Data through Optimal Transport

Tabak

Trigila

2017

Comm Pure Appl Math

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A methodology based on the theory of optimal transport is developed to attribute variability in data sets to known and unknown factors and to remove such attributable components of the variability from the data. Denoting by x the quantities of interest and by´the explanatory factors, the procedure transforms x into filtered variables y through a´-dependent map, so that the conditional probability distributions .xj´/ are pushed forward into a target distribution .y/, independent of´. Among all maps and target distributions that achieve this goal, the procedure selects the one that minimally distorts the original data: the barycenter of the .xj´/. Connections are found to unsupervised learning and to fundamental problems in statistics such as conditional density estimation and sampling. Particularly simple instances of the methodology are shown to be equivalent to k-means and principal component analysis. An application is shown to a time series of ground temperature hourly data across the United States.

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On a generalization of cyclic monotonicity and distances among random vectors

Cited by 42 publications

References 5 publications

Wasserstein barycenters over Riemannian manifolds

Wasserstein barycenters over Riemannian manifolds

Uniqueness and Monge Solutions in the Multimarginal Optimal Transportation Problem

Explanation of Variability and Removal of Confounding Factors from Data through Optimal Transport

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