# On the tightness of Gaussian concentration for convex functions

@article{Valettas2019OnTT,
title={On the tightness of Gaussian concentration for convex functions},
author={Petros Valettas},
journal={Journal d'Analyse Math{\'e}matique},
year={2019}
}
• P. Valettas
• Published 28 June 2017
• Mathematics
• Journal d'Analyse Mathématique
The concentration of measure phenomenon in Gauss' space states that every $L$-Lipschitz map $f$ on $\mathbb R^n$ satisfies $\gamma_{n} \left(\{ x : | f(x) - M_{f} | \geqslant t \} \right) \leqslant 2 e^{ - \frac{t^2}{ 2L^2} }, \quad t>0,$ where $\gamma_{n}$ is the standard Gaussian measure on $\mathbb R^{n}$ and $M_{f}$ is a median of $f$. In this work, we provide necessary and sufficient conditions for when this inequality can be reversed, up to universal constants, in the case when $f… Hypercontractivity, and Lower Deviation Estimates in Normed Spaces • Mathematics • 2019 We consider the problem of estimating probabilities of lower deviation$\mathbb P\{\|G\| \leqslant \delta \mathbb E\|G\|\}$in normed spaces with respect to the Gaussian measure. These estimates Dichotomies, structure, and concentration in normed spaces • Mathematics Advances in Mathematics • 2018 Abstract We use probabilistic, topological and combinatorial methods to establish the following deviation inequality: For any normed space X = ( R n , ‖ ⋅ ‖ ) there exists an invertible linear map T A Sharp Lower-tail Bound for Gaussian Maxima with Application to Bootstrap Methods in High Dimensions • Mathematics • 2018 Although there is an extensive literature on the maxima of Gaussian processes, there are relatively few non-asymptotic bounds on their lower-tail probabilities. The aim of this paper is to develop The equality cases of the Ehrhard–Borell inequality • Mathematics Advances in Mathematics • 2018 Abstract The Ehrhard–Borell inequality is a far-reaching refinement of the classical Brunn–Minkowski inequality that captures the sharp convexity and isoperimetric properties of Gaussian measures. Non asymptotic variance bounds and deviation inequalities by optimal transport • K. Tanguy • Mathematics Electronic Journal of Probability • 2019 The purpose of this note is to show how simple optimal transport arguments, on the real line, can be used in Superconcentration theory. As such, we derive non- asymptotic sharp variance bounds for Improved one-sided deviation inequalities under regularity assumptions for product measures • K. Tanguy • Mathematics ESAIM: Probability and Statistics • 2019 This note is concerned with lower tail estimates for product measures. Some improved deviation inequalities are obtained for functions satisfying some regularity and monotonicity assumptions. The Improved deviation inequalities under regularity assumptions for product measures This note is concerned with deviation inequalities for product measures. Some improved deviation inequalities are obtained for functions satisfying some regularity and monotonicity assumptions. The Remarks on Superconcentration and Gamma Calculus: Applications to Spin Glasses • K. Tanguy • Mathematics Progress in Probability • 2019 This note is concerned with the so-called superconcentration phenomenon. It shows that the Bakry-Emery’s Gamma calculus can provide relevant bound on the variance of function satisfying a inverse, ## References SHOWING 1-10 OF 64 REFERENCES A Remark on the Median and the Expectation of Convex Functions of Gaussian Vectors Ten years ago A. Ehrhard published an important paper, [1], in which he proved that if γn is a gaussian measure on R n, Φ is the normal distribution function, i.e \(\Phi (t)=\frac{1}{\sqrt{2\pi}}\int A Gaussian small deviation inequality for convex functions • Mathematics • 2016 Let$Z$be an$n$-dimensional Gaussian vector and let$f: \mathbb R^n \to \mathbb R$be a convex function. 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• Mathematics
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We prove a concentration inequality for the l n p norm on the l n p sphere for p,q > 0. This inequality, which generalizes results of Schechtman and Zinn (2000), is used to study the distance between