Densité et dimension

Assouad, Patrick

doi:10.5802/aif.938

Assouad, Patrick

Annales de l'Institut Fourier, Tome 33 (1983) no. 3, pp. 233-282.

Résumé
Abstract

Une partie $𝒮$ de $2^{X}$ est appelée une classe de Vapnik-Cervonenkis si la croissance de la fonction $Δ^{𝒮} : r \to Sup {| A \cap | | A \subset X, | A | = r}$ est polynomiale; ces classes se trouvent être utiles en Statistique et en Calcul des Probabilités (voir par exemple Vapnik, Cervonenkis [V.N. Vapnik, A.YA. Cervonenkis, Theor. Prob. Appl., 16 (1971), 264-280], Dudley [R.M. Dudley, Ann. of Prob., 6 (1978), 899-929]).

Le présent travail est un essai de synthèse sur les classes de Vapnik-Cervonenkis. Mais il contient aussi beaucoup de résultats nouveaux, et notamment les deux résultats suivants :

- une partie $𝒮$ de $2^{X}$ est une classe de Vapnik-Cervonenkis si et seulement si le nombre d’atomes de la tribu engendrée par $r$ membres quelconques de $𝒮$ est majoré par un polynôme en $r$ ;

- si $𝒮$ est une partie de $2^{X}$ , chaque loi de probabilité $P$ sur la tribu engendrée par $𝒮$ définit un écart $d_{p} : S, S^{'} \to P (S Δ S^{'})$ sur la famille $𝒮$ , et on note dim $(𝒮, d_{p})$ la dimension d’entropie de l’espace $(𝒮, d_{p})$ ; la famille $𝒮$ est une classe de Vapnik-Cervonenkis si et seulement si la quantité Sup $\bar{\dim} (𝒮, d_{p})$ est finie.

On trouvera dans l’introduction les énoncés de plusieurs autres résultats nouveaux démontrés ici (dont certains sont indiqués sans démonstration dans ma note [P. Assouad, C.R.A.S., Paris, 292 (1981), 921-924]).

A class $𝒮$ of subsets of a set $X$ is called a Vapnik-Cervonenkis class if the growth of the function $Δ^{𝒮} : r \to Sup {| A \cap | | A \subset X, | A | = r}$ is polynomial ; these classes have proved to be useful in Statistics and Probability (see for example Vapnik, Cervonenkis [V.N. Vapnik, A.YA. Cervonenkis, Theor. Prob. Appl., 16 (1971), 264-280], Dudley [R.M. Dudley, Ann. of Prob., 6 (1978), 899-929]).

The present paper is a survey on Vapnik-Cervonenkis classes. Moreover we prove here many new results, among them the following:

- a subset $𝒮$ of $2^{X}$ is a Vapnik-Cervonenkis class if and only if the number of atoms of the $σ$ -algebra generated by any collection of $r$ members of $𝒮$ if no more than $C r^{s}$ (where $C$ and $s$ are two positive real numbers);

- if $𝒮$ is a subset of $2^{X}$ , every probability law $P$ on the $σ$ -algebra generated by $𝒮$ defines a semimetric $d_{p} : S, S^{'} \to P (S Δ S^{'})$ on the class $𝒮$ , and the entropy dimension of the space $(𝒮, d_{p})$ will be denoted $\bar{\dim} (𝒮, d_{p})$ ; the class $𝒮$ is a Vapnik-Cervonenkis class if and only if ${Sup}_{P} \bar{\dim} (𝒮, d_{p})$ is finite.

The present paper contains other new results, some of them being stated without proof in my note [P. Assouad, C.R.A.S., Paris, 292 (1981), 921-924]).

EuDML MR Zbl

DOI : 10.5802/aif.938

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     author = {Assouad, Patrick},
     title = {Densit\'e et dimension},
     journal = {Annales de l'Institut Fourier},
     pages = {233--282},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {33},
     number = {3},
     year = {1983},
     doi = {10.5802/aif.938},
     zbl = {0504.60006},
     mrnumber = {86j:05022},
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     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.938/}
}

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Assouad, Patrick. Densité et dimension. Annales de l'Institut Fourier, Tome 33 (1983) no. 3, pp. 233-282. doi : 10.5802/aif.938. https://aif.centre-mersenne.org/articles/10.5802/aif.938/

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