L_p moments of random vectors via majorizing measures

For a random vector X in R^n, we obtain bounds on the size of a sample, for which the empirical p-th moments of linear functionals are close to the exact ones uniformly on an n-dimensional convex body K. We prove an estimate for a general random vector and apply it to several problems arising in geometric functional analysis. In particular, we find a short Lewis type decomposition for any finite dimensional subspace of L_p. We also prove that for an isotropic log-concave random vector, we only need about n^{p/2} \log n sample points so that the empirical p-th moments of the linear functionals are almost isometrically the same as the exact ones. We obtain a concentration estimate for the empirical moments. The main ingredient of the proof is the construction of an appropriate majorizing measure to bound a certain Gaussian process.Comment: 32 pages, to appear in Advances in Mathematic

A probabilistic approach to the geometry of the \ell_p^n-ball

This article investigates, by probabilistic methods, various geometric questions on B_p^n, the unit ball of \ell_p^n. We propose realizations in terms of independent random variables of several distributions on B_p^n, including the normalized volume measure. These representations allow us to unify and extend the known results of the sub-independence of coordinate slabs in B_p^n. As another application, we compute moments of linear functionals on B_p^n, which gives sharp constants in Khinchine's inequalities on B_p^n and determines the \psi_2-constant of all directions on B_p^n. We also study the extremal values of several Gaussian averages on sections of B_p^n (including mean width and \ell-norm), and derive several monotonicity results as p varies. Applications to balancing vectors in \ell_2 and to covering numbers of polyhedra complete the exposition.Comment: Published at http://dx.doi.org/10.1214/009117904000000874 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

A probabilistic approach to the geometry of the ℓᵨⁿ-ball

This article investigates, by probabilistic methods, various geometric questions on Bᵨⁿ, the unit ball of ℓᵨⁿ. We propose realizations in terms of independent random variables of several distributions on Bᵨⁿ, including the normalized volume measure. These representations allow us to unify and extend the known results of the sub-independence of coordinate slabs in Bᵨⁿ. As another application, we compute moments of linear functionals on Bᵨⁿ, which gives sharp constants in Khinchine’s inequalities on Bᵨⁿ and determines the ψ₂-constant of all directions on Bᵨⁿ. We also study the extremal values of several Gaussian averages on sections of Bᵨⁿ (including mean width and ℓ-norm), and derive several monotonicity results as p varies. Applications to balancing vectors in ℓ₂ and to covering numbers of polyhedra complete the exposition

Majorizing measures and proportional subsets of bounded orthonormal systems

In this article we prove that for any orthonormal system (\vphi_j)_{j=1}^n \subset L_2 that is bounded in $L_{\infty}$, and any $1 < k <n$, there exists a subset $I$ of cardinality greater than $n-k$ such that on \spa\{\vphi_i\}_{i \in I}, the $L_1$ norm and the $L_2$ norm are equivalent up to a factor $\mu (\log \mu)^{5/2}$, where $\mu = \sqrt{n/k} \sqrt{\log k}$. The proof is based on a new estimate of the supremum of an empirical process on the unit ball of a Banach space with a good modulus of convexity, via the use of majorizing measures

Invertibilité restreinte, distance au cube et covariance de matrices aléatoires

Dans cette thèse, on aborde trois thèmes : problème de sélection de colonnes dans une matrice, distance de Banach-Mazur au cube et estimation de la covariance de matrices aléatoires. Bien que les trois thèmes paraissent éloignés, les techniques utilisées se ressemblent tout au long de la thèse. Dans un premier lieu, nous généralisons le principe d'invertibilité restreinte de Bourgain-Tzafriri. Ce résultat permet d'extraire un "grand" bloc de colonnes linéairement indépendantes dans une matrice et d'estimer la plus petite valeur singulière de la matrice extraite. Nous proposons ensuite un algorithme déterministe pour extraire d'une matrice un bloc presque isométrique c est à dire une sous-matrice dont les valeurs singulières sont proches de 1. Ce résultat nous permet de retrouver le meilleur résultat connu sur la célèbre conjecture de Kadison-Singer. Des applications à la théorie locale des espaces de Banach ainsi qu'à l'analyse harmonique sont déduites. Nous donnons une estimation de la distance de Banach-Mazur d'un corps convexe de Rn au cube de dimension n. Nous proposons une démarche plus élémentaire, basée sur le principe d'invertibilité restreinte, pour améliorer et simplifier les résultats précédents concernant ce problème. Plusieurs travaux ont été consacrés pour approcher la matrice de covariance d'un vecteur aléatoire par la matrice de covariance empirique. Nous étendons ce problème à un cadre matriciel et on répond à la question. Notre résultat peut être interprété comme une quantification de la loi des grands nombres pour des matrices aléatoires symétriques semi-définies positives. L'estimation obtenue s'applique à une large classe de matrices aléatoiresIn this thesis, we address three themes : columns subset selection in a matrix, the Banach-Mazur distance to the cube and the estimation of the covariance of random matrices. Although the three themes seem distant, the techniques used are similar throughout the thesis. In the first place, we generalize the restricted invertibility principle of Bougain-Tzafriri. This result allows us to extract a "large" block of linearly independent columns inside a matrix and estimate the smallest singular value of the restricted matrix. We also propose a deterministic algorithm in order to extract an almost isometric block inside a matrix i.e a submatrix whose singular values are close to 1. This result allows us to recover the best known result on the Kadison-Singer conjecture. Applications to the local theory of Banach spaces as well as to harmonic analysis are deduced. We give an estimate of the Banach-Mazur distance between a symmetric convex body in Rn and the cube of dimension n. We propose an elementary approach, based on the restricted invertibility principle, in order to improve and simplify the previous results dealing with this problem. Several studies have been devoted to approximate the covariance matrix of a random vector by its sample covariance matrix. We extend this problem to a matrix setting and we answer the question. Our result can be interpreted as a quantified law of large numbers for positive semidefinite random matrices. The estimate we obtain, applies to a large class of random matricesPARIS-EST-Université (770839901) / SudocSudocFranceF

Kahane-Khinchine type inequalities for negative exponent

International audienceWe prove a concentration inequality for delta-concave measures over R-n. Using this result, we study the moments of order q of a norm with respect to a delta-concave measure over R-n. We obtain a lower bound for q is an element of ]-1, 0] and an upper bound for q is an element of ]0, + infinity[ in terms of the measure of the unit ball associated to the norm. This allows us to give Kahane-Khinchine type inequalities for negative exponent

Intérêt des CVIMAR dans le collage des brackets en céramique

Le collage en orthodontie se heurte aujourd'hui à ses effets délétères. Le mordançage acide, responsable d'une perte pure et simple d'émail, la persistance des resins tags après dépose, et les fractures récurrentes à l'interface émail-adhésif ne permettent pas de restituer la dent telle que le patient nous l'a confiée. L'utilisation des Ciments verres ionomères modifiés par adjonction de résine pourrait apporter une solution à ces inconvénients. Leur adhésion chimique intrinsèque au substrat amélaire, doublée de propriétés mécaniques améliorées, laissent espérer une adhésion équivalente pour une restitution ad integrum de l'émail. A l'heure d'une recrudescence des consultations adultes dans les services d'orthodontie, la demande en traitements invisibles explose. Les brackets céramiques à rétention mécanique cherchent à répondre à cette demande, autant qu'à éviter les travers de la rétention chimique par silanisation, responsable de trop fréquents dégâts amélaires. LA question posée par l'étude réalisée dans cette thèse est la suivante: les CVIMAR constituent-ils une alternative intéressante aux composites de collage traditionnels, dans le collage des brackets en céramique à rétention mécanique? Le CVIMAR peut-il supporter les contraintes imposées par les microclavetage des bases? Des questions subsidiaires, telles que le collage aux dents déciduales bovines, sont aussi abordées.RENNES1-BU Santé (352382103) / SudocSudocFranceF

Community detection in sparse networks via Grothendieck's inequality

We present a simple and flexible method to prove consistency of semidefinite optimization problems on random graphs. The method is based on Grothendieck's inequality. Unlike the previous uses of this inequality that lead to constant relative accuracy, we achieve any given relative accuracy by leveraging randomness. We illustrate the method with the problem of community detection in sparse networks, those with bounded average degrees. We demonstrate that even in this regime, various simple and natural semidefinite programs can be used to recover the community structure up to an arbitrarily small fraction of misclassified vertices. The method is general; it can be applied to a variety of stochastic models of networks and semidefinite programs

Subspaces and orthogonal decompositions generated by bounded orthogonal systems

We investigate properties of subspaces of L 2 spanned by subsets of a finite orthonormal system bounded in the L ∞ norm. We first prove that there exists an arbitrarily large subset of this orthonormal system on which the L 1 and the L 2 norms are clos

Majorizing measures and proprtional subsets of bounded orthonormal systems

In this article we prove that for any orthonormal system (φj)j=1n ⊂ L2. that is bounded in L∞, and any 1 < k < n, there exists a subset I of cardinality greater than n - k such that on span{φi} i∈I, the L1 norm and the L2 norm are equivalent up t