Uniqueness and approximate computation of optimal incomplete transportation plans

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

Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 2, pp. 358-375.

Pour α∈(0, 1), une α-coupe P∗ d'une probabilité P selon une fonction positive f majorée par 1/(1-α) est la probabilité obtenue pour tout ensemble de Borel B par P∗(B)=∫Bf(x)P(dx). Si P, Q sont deux probabilités sur l'espace euclidien, on considère le problème de minimiser la distance de Wasserstein L2 entre (a) une probabilité et ses versions coupées (b) les versions coupées de deux probabilités. Ce problème mène naturellement à une nouvelle formulation du problème de transport de masse, où une partie de la masse ne doit pas être transportée. Nous explorons les liaisons entre ce problème et la similitude des mesures de probabilité. Un de nos résultats remarquables est l'unicité du transport de masse. Ces plans de transport optimal incomplets ne sont pas facilement calculables mais nous fournissons un appui théorique pour des approximations de Monte-Carlo. Enfin, nous donnons un TCL pour les versions empiriques des distances coupées et discutons certaines applications statistiques.

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.

Classification : 49Q20, 60A10, 60B10, 28A50
Mots clés : incomplete mass transportation problem, multivariate distributions, optimal transportation plan, similarity, trimming, uniqueness, trimmed probability, clt

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