arXiv subjects

Olivier Guédon

Publications and source records attributed to Olivier Guédon.

17 recordsLinked to original sources

arXiv2026

On partition functions of Gaussian random variables

For $β\in \mathbb{R}$, we define the partition function $$ Z_β(X) = \sum_{i=1}^N \exp(βX_i) $$ of a centered Gaussian random vector $X=(X_1, \ldots, X_N)$ with $\mathbb{E}[X_i^2]= 1$. For $q \in \mathbb{R}$, we prove a complete phase transition for the generalized $L_q$-means of $Z_β(X)$ at $q=1$. The centered Gaussian vector with covariance matrix $Δ_N$ obtained from a regular simplex configuration of unit vectors maximizes these means for $q<1$, and minimizes them for $q>1$. The two regimes are governed by different principles. For $q > 1$ {and $β\ne0$}, the moment functional is globally strongly convex on the entire set of correlation matrices, with an explicit modulus of convexity and a quantitative centroid-shape stability estimate. For $q<1$, the strategy is different. We prove a universal comparison for log-concave profiles of reverse Brascamp--Lieb type. Specializing this result to the Gumbel profile yields a Laplace-transform comparison between the partition functions.

math.FA↗

arXiv2026

An extension of a theorem of Chevet

We adopt a modern approach to extend a comparison inequality of Chevet for Gaussian processes to a broader class of functions. Furthermore, we consider centered Gaussian random vectors associated with a family of $n+1$ vectors on the unit sphere $S^{n-1}$ and investigate configurations of the vertices for which these new functionals could be maximum for the regular simplex.

math.FA↗

arXiv2024

Talagrand's mathematical journey to the Abel Prize 2024

Michel Talagrand (Centre National de la Recherche Scientifique, France) has been awarded the prestigious Abel Prize for 2024 for his work in probability theory, functional analysis, and statistical physics. In this note, we introduce the Abel Prize Laureate and his main contributions.

math.HO↗

arXiv2022

Asymptotics of the Inertia Moments and the Variance Conjecture in Schatten Balls

We study the first and second orders of the asymptotic expansion, as the dimension goes to infinity, of the moments of the Hilbert-Schmidt norm of a uniformly distributed matrix in the p-Schatten unit ball. We consider the case of matrices with real, complex or quaternionic entries, self-adjoint or not. When p > 3, this asymptotic expansion allows us to establish a generalized version of the variance conjecture for the family of p-Schatten unit balls of self-adjoint matrices.

math.FA↗

arXiv2019

On the geometry of polytopes generated by heavy-tailed random vectors

We study the geometry of centrally-symmetric random polytopes, generated by $N$ independent copies of a random vector $X$ taking values in $\mathbb{R}^n$. We show that under minimal assumptions on $X$, for $N \gtrsim n$ and with high probability, the polytope contains a deterministic set that is naturally associated with the random vector---namely, the polar of a certain floating body. This solves the long-standing question on whether such a random polytope contains a canonical body. Moreover, by identifying the floating bodies associated with various random vectors we recover the estimates that have been obtained previously, and thanks to the minimal assumptions on $X$ we derive estimates in cases that had been out of reach, involving random polytopes generated by heavy-tailed random vectors (e.g., when $X$ is $q$-stable or when $X$ has an unconditional structure). Finally, the structural results are used for the study of a fundamental question in compressive sensing---noise blind sparse recovery.

math.PR↗

arXiv2019

Random polytopes obtained by matrices with heavy tailed entries

Let $Γ$ be an $N\times n$ random matrix with independent entries and such that in each row entries are i.i.d. Assume also that the entries are symmetric, have unit variances, and satisfy a small ball probabilistic estimate uniformly. We investigate properties of the corresponding random polytope $Γ^* B_1^N$ in $\mathbb{R}$ (the absolute convex hull of rows of $Γ$). In particular, we show that $$ ΓB_1^N \supset b^{-1} \left( B_{\infty}^n \cap \sqrt{\ln (N/n)}\, B_2^n \right). $$ where $b$ depends only on parameters in small ball inequality. This extends results of \cite{LPRT} and recent results of \cite{KKR}. This inclusion is equivalent to so-called $\ell_1$-quotient property and plays an important role in compressive sensing (see \cite{KKR} and references therein).

math.FA↗

arXiv2016

Dimension free bounds for the Hardy--Littlewood maximal operator associated to convex sets

This survey is based on a series of lectures given by the authors at the working seminar "Convexité et Probabilités" at UPMC Jussieu, Paris, during the spring 2013. It is devoted to maximal inequalities associated to symmetric convex sets in high dimensional linear spaces, a topic mainly developed between 1982 and 1990 but recently renewed by further advances. The series focused on proving for these maximal functions inequalities in $L^p(\mathbb{R}^n)$ with bounds independent of the dimension $n$, for all $p \in (1, +\infty]$ in the best cases. This program was initiated in 1982 by Elias Stein, who obtained the first theorem of this kind for the family of Euclidean balls in arbitrary dimension. We present several results along this line, proved by Bourgain, Carbery and Müller during the period 1986--1990, and a new one due to Bourgain (2014) for the family of cubes in arbitrary dimension. We complete the cube case with negative results for the weak type $(1, 1)$ constant, due to Aldaz, Aubrun and Iakovlev--Strömberg between 2009 and 2013.

math.FA↗

arXiv2016

On the expectation of operator norms of random matrices

We prove estimates for the expected value of operator norms of Gaussian random matrices with independent and mean-zero entries, acting as operators from $\ell^m_{p^*}$ to $\ell_q^n$, $1\leq p^* \leq 2 \leq q \leq \infty$.

math.PR↗

arXiv2015

Inverse Littlewood-Offord problems for Quasi-Norms

Given a star-shaped domain $K\subseteq \mathbb R^d$, $n$ vectors $v_1,\dots,v_n \in \mathbb R^d$, a number $R>0$, and i.i.d. random variables $η_1,\dots,η_n$, we study the geometric and arithmetic structure of the set of vectors $V = \{v_1,\dots,v_n\}$ under the assumption that the small ball probability \[\sup_{x\in \mathbb R^d}~\mathbb P\Bigg(\sum_{j=1}^nη_jv_j\in x+RK\Bigg)\] does not decay too fast as $n\to \infty$. This generalises the case where $K$ is the Euclidean ball, which was previously studied by Nguyen-Vu and Tao-Vu.

math.PR↗

arXiv2015

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

Let $A$ be a matrix whose columns $X_1,\dots, X_N$ are independent random vectors in $\mathbb{R}^n$. Assume that the tails of the 1-dimensional marginals decay as $\mathbb{P}(|\langle X_i, a\rangle|\geq t)\leq t^{-p}$ uniformly in $a\in S^{n-1}$ and $i\leq N$. Then for $p>4$ we prove that with high probability $A/{\sqrt{n}}$ has the Restricted Isometry Property (RIP) provided that Euclidean norms $|X_i|$ are concentrated around $\sqrt{n}$. We also show that the covariance matrix is well approximated by the empirical covariance matrix and establish corresponding quantitative estimates on the rate of convergence in terms of the ratio $n/N$. Moreover, we obtain sharp bounds for both problems when the decay is of the type $ \exp({-t^α})$ with $α\in (0,2]$, extending the known case $α\in[1, 2]$.

math.PR↗

arXiv2015

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.

math.ST↗

arXiv2015

Small ball estimates for quasi-norms

This note contains two types of small ball estimates for random vectors in finite dimensional spaces equipped with a quasi-norm. In the first part, we obtain bounds for the small ball probability of random vectors under some smoothness assumptions on their density function. In the second part, we obtain Littlewood-Offord type estimates for quasi-norms. This generalizes a result which was previously obtained by Friedland and Sodin and by Rudelson and Vershynin.

math.PR↗

arXiv2014

Thin-shell concentration for convex measures

We prove that for $s<0$, $s$-concave measures on ${\mathbb R}^n$ satisfy a thin shell concentration similar to the log-concave one. It leads to a Berry-Esseen type estimate for their one dimensional marginal distributions. We also establish sharp reverse Hölder inequalities for $s$-concave measures.

math.FA↗

arXiv2013

Concentration phenomena in high dimensional geometry

The purpose of this note is to present several aspects of concentration phenomena in high dimensional geometry. At the heart of the study is a geometric analysis point of view coming from the theory of high dimensional convex bodies. The topic has a broad audience going from algorithmic convex geometry to random matrices. We have tried to emphasize different problems relating these areas of research. Another connected area is the study of probability in Banach spaces where some concentration phenomena are related with good comparisons between the weak and the strong moments of a random vector.

math.FA↗

arXiv2012

Moment estimates for convex measures

Let $p\geq 1$, $\eps >0$, $r\geq (1+\eps) p$, and $X$ be a $(-1/r)$-concave random vector in $\R^n$ with Euclidean norm $|X|$. We prove that $(\E |X|^{p})^{1/{p}}\leq c (C(\eps) \E|X|+σ_{p}(X))$, where $σ_{p}(X)=\sup_{|z|\leq 1}(\E| |^{p})^{1/p}$, $C(\eps)$ depends only on $\eps$ and $c$ is a universal constant. Moreover, if in addition $X$ is centered then $(\E |X|^{-p})^{-1/{p}}\geq c(\eps) (\E|X| - C σ_{p}(X))$.

math.PR↗

arXiv2011

Sparsity and non-Euclidean embeddings

We present a relation between sparsity and non-Euclidean isomorphic embeddings. We introduce a general restricted isomorphism property and show how it enables to construct embeddings of $\ell_p^n$, $p > 0$, into various type of Banach or quasi-Banach spaces. In particular, for $0 0$.

math.FA↗

arXiv2011

Interpolating Thin-Shell and Sharp Large-Deviation Estimates For Isotropic Log-Concave Measures

Given an isotropic random vector $X$ with log-concave density in Euclidean space $\Real^n$, we study the concentration properties of $|X|$ on all scales, both above and below its expectation. We show in particular that: \[ ¶(\abs{|X| -\sqrt{n}} \geq t \sqrt{n}) \leq C \exp(-c n^{1/2} \min(t^3,t)) \;\;\; \forall t \geq 0 ~, \] for some universal constants $c,C>0$. This improves the best known deviation results on the thin-shell and mesoscopic scales due to Fleury and Klartag, respectively, and recovers the sharp large-deviation estimate of Paouris. Another new feature of our estimate is that it improves when $X$ is $ψ_α$ ($α\in (1,2]$), in precise agreement with Paouris' estimates. The upper bound on the thin-shell width $\sqrt{\Var(|X|)}$ we obtain is of the order of $n^{1/3}$, and improves down to $n^{1/4}$ when $X$ is $ψ_2$. Our estimates thus continuously interpolate between a new best known thin-shell estimate and the sharp large-deviation estimate of Paouris. As a consequence, a new best known bound on the Cheeger isoperimetric constant appearing in a conjecture of Kannan--Lovász--Simonovits is deduced.

math.FA↗