Olivier Guédon

A probabilistic approach to the geometry of the ℓᵨⁿ-ball

This article investigates, by probabilistic methods, various geometric questions on B n p , the unit ball of ln p . We propose realizations in terms of independent random variables of several distributions on B n p , 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 n p . As another application, we compute moments of linear functionals on B n p , which gives sharp constants in Khinchines inequalities on B n p and determines the 2-constant of all directions on B n p . We also study the extremal values of several Gaussian averages on sections of B n p (including mean width and l-norm), and derive several monotonicity results as p varies. Applications to balancing vectors in l 2 and to covering numbers of polyhedra complete the exposition.

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.

The Extreme Points of Subsets of s -Concave Probabilities and a Geometric Localization Theorem

Abstract We prove that the extreme points of the set of s-concave probability measures satisfying a linear constraint are some Dirac measures and some s-affine probabilities supported by a segment. From this we deduce that the constrained maximization of a convex functional on the s-concave probability measures is reduced to this small set of extreme points. This gives a new approach to a localization theorem due to Kannan, Lovász and Simonovits which happens to be very useful in geometry to obtain inequalities for integrals like concentration and isoperimetric inequalities. Roughly speaking, the study of such inequalities is reduced to these extreme points.

Condition number of a square matrix with i.i.d. columns drawn from a convex body

We study the smallest singular value of a square random matrix with i.i.d. columns drawn from an isotropic log-concave distribution. An important example is obtained by sampling vectors uniformly dis- tributed in an isotropic convex body. We deduce that the condition number of such matrices is of the order of the size of the matrix and give an estimate on its tail behavior.

On the interval of fluctuation of the singular values of random matrices

Let

Supremum of a Process in Terms of Trees

A

Small Ball Estimates for Quasi-Norms

be a matrix whose columns

On the Expectation of Operator Norms of Random Matrices

X_1,\dots, X_N

Inverse Littlewood---Offord Problems for Quasi-norms

are independent random vectors in

A stability result for mean width of Lp-centroid bodies

\mathbb{R}^n