Uniqueness and approximate computation of optimal incomplete transportation plans

P. C. Álvarez-Esteban; E. del Barrio; J. A. Cuesta-Albertos; C. Matrán

Annales de l'I.H.P. Probabilités et statistiques (2011)

  • Volume: 47, Issue: 2, page 358-375
  • ISSN: 0246-0203

Abstract

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For α∈(0, 1) an α-trimming, P∗, of a probability P is a new probability obtained by re-weighting the probability of any Borel set, B, according to a positive weight function, f≤1/(1−α), in the way P∗(B)=∫Bf(x)P(dx). If P, Q are probability measures on euclidean space, we consider the problem of obtaining the best L2-Wasserstein approximation between: (a) a fixed probability and trimmed versions of the other; (b) trimmed versions of both probabilities. These best trimmed approximations naturally lead to a new formulation of the mass transportation problem, where a part of the mass need not be transported. We explore the connections between this problem and the similarity of probability measures. As a remarkable result we obtain the uniqueness of the optimal solutions. These optimal incomplete transportation plans are not easily computable, but we provide theoretical support for Monte-Carlo approximations. Finally, we give a CLT for empirical versions of the trimmed distances and discuss some statistical applications.

How to cite

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Álvarez-Esteban, P. C., et al. "Uniqueness and approximate computation of optimal incomplete transportation plans." Annales de l'I.H.P. Probabilités et statistiques 47.2 (2011): 358-375. <http://eudml.org/doc/242550>.

@article{Álvarez2011,
abstract = {For α∈(0, 1) an α-trimming, P∗, of a probability P is a new probability obtained by re-weighting the probability of any Borel set, B, according to a positive weight function, f≤1/(1−α), in the way P∗(B)=∫Bf(x)P(dx). If P, Q are probability measures on euclidean space, we consider the problem of obtaining the best L2-Wasserstein approximation between: (a) a fixed probability and trimmed versions of the other; (b) trimmed versions of both probabilities. These best trimmed approximations naturally lead to a new formulation of the mass transportation problem, where a part of the mass need not be transported. We explore the connections between this problem and the similarity of probability measures. As a remarkable result we obtain the uniqueness of the optimal solutions. These optimal incomplete transportation plans are not easily computable, but we provide theoretical support for Monte-Carlo approximations. Finally, we give a CLT for empirical versions of the trimmed distances and discuss some statistical applications.},
author = {Álvarez-Esteban, P. C., del Barrio, E., Cuesta-Albertos, J. A., Matrán, C.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {incomplete mass transportation problem; multivariate distributions; optimal transportation plan; similarity; trimming; uniqueness; trimmed probability; clt; CLT},
language = {eng},
number = {2},
pages = {358-375},
publisher = {Gauthier-Villars},
title = {Uniqueness and approximate computation of optimal incomplete transportation plans},
url = {http://eudml.org/doc/242550},
volume = {47},
year = {2011},
}

TY - JOUR
AU - Álvarez-Esteban, P. C.
AU - del Barrio, E.
AU - Cuesta-Albertos, J. A.
AU - Matrán, C.
TI - Uniqueness and approximate computation of optimal incomplete transportation plans
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2011
PB - Gauthier-Villars
VL - 47
IS - 2
SP - 358
EP - 375
AB - For α∈(0, 1) an α-trimming, P∗, of a probability P is a new probability obtained by re-weighting the probability of any Borel set, B, according to a positive weight function, f≤1/(1−α), in the way P∗(B)=∫Bf(x)P(dx). If P, Q are probability measures on euclidean space, we consider the problem of obtaining the best L2-Wasserstein approximation between: (a) a fixed probability and trimmed versions of the other; (b) trimmed versions of both probabilities. These best trimmed approximations naturally lead to a new formulation of the mass transportation problem, where a part of the mass need not be transported. We explore the connections between this problem and the similarity of probability measures. As a remarkable result we obtain the uniqueness of the optimal solutions. These optimal incomplete transportation plans are not easily computable, but we provide theoretical support for Monte-Carlo approximations. Finally, we give a CLT for empirical versions of the trimmed distances and discuss some statistical applications.
LA - eng
KW - incomplete mass transportation problem; multivariate distributions; optimal transportation plan; similarity; trimming; uniqueness; trimmed probability; clt; CLT
UR - http://eudml.org/doc/242550
ER -

References

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