Searcharxiv⌕ Search

arXiv subjects

Joseph Lehec

Publications and source records attributed to Joseph Lehec.

At least 19 recordsLinked to original sources
arXiv2025

On the stochastic proof of the Blaschke-Santal\'o inequality

In 2024, Courtade, Fathi and Mikulincer gave a proof of the symmetrized Talagrand inequality based on stochastic calculus, in the spirit of Borell's proof of the Pr\'ekopa-Leindler inequality. The symmetrized Talagrand inequality can be seen as a dual form of the functional Santal\'o inequality. The modest purpose of this note is to give a simplified version of the Courtade, Fathi and Mikulincer argument. Namely we first recall briefly Borell's original argument, and we then explain a simple twist in his proof that allows to recover the functional Santal\'o inequality directly, rather than in its dual form.

math.FA↗
arXiv2025

Thin-shell bounds via parallel coupling

We prove that for any log-concave random vector $X$ in $\mathbb{R}^n$ with mean zero and identity covariance, $$ \mathbb{E} (|X| - \sqrt{n})^2 \leq C $$ where $C > 0$ is a universal constant. Thus, most of the mass of the random vector $X$ is concentrated in a thin spherical shell, whose width is only $C / \sqrt{n}$ times its radius. This confirms the thin-shell conjecture in high dimensional convex geometry. Our method relies on the construction of a certain coupling between log-affine perturbations of the law of $X$ related to Eldan's stochastic localization and to the theory of non-linear filtering. Another ingredient is a recent breakthrough technique by Guan that was previously used in our proof of Bourgain's slicing conjecture, which is known to be implied by the thin-shell conjecture.

math.PR↗
arXiv2024

Affirmative Resolution of Bourgain's Slicing Problem using Guan's Bound

We provide the final step in the resolution of Bourgain's slicing problem in the affirmative. Thus we establish the following theorem: for any convex body $K \subseteq \mathbb{R}^n$ of volume one, there exists a hyperplane $H \subseteq \mathbb{R}^n$ such that $$ Vol_{n-1}(K \cap H) > c, $$ where $c > 0$ is a universal constant. Our proof combines Milman's theory of $M$-ellipsoids, stochastic localization with a recent bound by Guan, and stability estimates for the Shannon-Stam inequality by Eldan and Mikulincer.

math.MG↗
arXiv2024

Convergence in total variation for the kinetic Langevin algorithm

We prove non asymptotic total variation estimates for the kinetic Langevin algorithm in high dimension when the target measure satisfies a Poincar\'e inequality and has gradient Lipschitz potential. The main point is that the estimate improves significantly upon the corresponding bound for the non kinetic version of the algorithm, due to Dalalyan. In particular the dimension dependence drops from $O(n)$ to $O(\sqrt n)$.

math.PR↗
arXiv2022

The Langevin Monte Carlo algorithm in the non-smooth log-concave case

We prove non asymptotic polynomial bounds on the convergence of the Langevin Monte Carlo algorithm in the case where the potential is a convex function which is globally Lipschitz on its domain, typically the maximum of a finite number of affine functions on an arbitrary convex set. In particular the potential is not assumed to be gradient Lipschitz, in contrast with most existing works on the topic.

math.ST↗
arXiv2020

Stability of the logarithmic Sobolev inequality via the Föllmer Process

We study the stability and instability of the Gaussian logarithmic Sobolev inequality, in terms of covariance, Wasserstein distance and Fisher information, addressing several open questions in the literature. We first establish an improved logarithmic Sobolev inequality which is at the same time scale invariant and dimension free. As a corollary, we show that if the covariance of the measure is bounded by the identity, one may obtain a sharp and dimension-free stability bound in terms of the Fisher information matrix. We then investigate under what conditions stability estimates control the covariance, and when such control is impossible. For the class of measures whose covariance matrix is dominated by the identity, we obtain optimal dimension-free stability bounds which show that the deficit in the logarithmic Sobolev inequality is minimized by Gaussian measures, under a fixed covariance constraint. On the other hand, we construct examples showing that without the boundedness of the covariance, the inequality is not stable. Finally, we study stability in terms of the Wasserstein distance, and show that even for the class of measures with a bounded covariance matrix, it is hopeless to obtain a dimension-free stability result. The counterexamples provided motivate us to put forth a new notion of stability, in terms of proximity to mixtures of the Gaussian distribution. We prove new estimates (some dimension-free) based on this notion. These estimates are strictly stronger than some of the existing stability results in terms of the Wasserstein metric. Our proof techniques rely heavily on stochastic methods.

math.PR↗
arXiv2019

On Poincare and logarithmic Sobolev inequalities for a class of singular Gibbs measures

This note, mostly expository, is devoted to Poincar{é} and log-Sobolev inequalities for a class of Boltzmann-Gibbs measures with singular interaction. Such measures allow to model one-dimensional particles with confinement and singular pair interaction. The functional inequalities come from convexity. We prove and characterize optimality in the case of quadratic confinement via a factorization of the measure. This optimality phenomenon holds for all beta Hermite ensembles including the Gaussian unitary ensemble, a famous exactly solvable model of random matrix theory. We further explore exact solvability by reviewing the relation to Dyson-Ornstein-Uhlenbeck diffusion dynamics admitting the Hermite-Lassalle orthogonal polynomials as a complete set of eigenfunctions. We also discuss the consequence of the log-Sobolev inequality in terms of concentration of measure for Lipschitz functions such as maxima and linear statistics.

math.PR↗
arXiv2018

Poisson processes and a log-concave Bernstein theorem

We discuss interplays between log-concave functions and log-concave sequences. We prove a Bernstein-type theorem, which characterizes the Laplace transform of log-concave measures on the half-line in terms of log-concavity of the alternating Taylor coefficients. We establish concavity inequalities for sequences inspired by the Prékopa-Leindler and the Walkup theorems. One of our main tools is a stochastic variational formula for the Poisson average.

math.PR↗
arXiv2016

Transport-entropy inequalities and curvature in discrete-space Markov chains

We show that if the random walk on a graph has positive coarse Ricci curvature in the sense of Ollivier, then the stationary measure satisfies a W^1 transport-entropy inequality. Peres and Tetali have conjectured a stronger consequence, that a modified log-Sobolev inequality (MLSI) should hold, in analogy with the setting of Markov diffusions. We discuss how our entropy interpolation approach suggests a natural attack on the MLSI conjecture.

math.PR↗
arXiv2015

Borell's formula for a Riemannian manifold and applications

Borell's formula is a stochastic variational formula for the log-Laplace transform of a function of a Gaussian vector. We establish an extension of this to the Riemannian setting and give a couple of applications, including a new proof of a convolution inequality on the sphere due to Carlen, Lieb and Loss.

math.PR↗
arXiv2015

Sampling from a log-concave distribution with Projected Langevin Monte Carlo

We extend the Langevin Monte Carlo (LMC) algorithm to compactly supported measures via a projection step, akin to projected Stochastic Gradient Descent (SGD). We show that (projected) LMC allows to sample in polynomial time from a log-concave distribution with smooth potential. This gives a new Markov chain to sample from a log-concave distribution. Our main result shows in particular that when the target distribution is uniform, LMC mixes in $\tilde{O}(n^7)$ steps (where $n$ is the dimension). We also provide preliminary experimental evidence that LMC performs at least as well as hit-and-run, for which a better mixing time of $\tilde{O}(n^4)$ was proved by Lov{á}sz and Vempala.

math.PR↗
arXiv2015

Regularization in $L_1$ for the Ornstein--Uhlenbeck semigroup

Let $γ_n$ be the standard Gaussian measure on $\mathbb R^n$ and let $(Q_t)$ be the Ornstein--Ulhenbeck semigroup. Eldan and Lee recently established that for every non--negative function $f$ of integral $1$ and any time $t$ the following tail inequality holds true: \[ γ_n ( \{ Q_t f > r \} ) \leq C_t \, \frac{ (\log \log r)^4 }{r \sqrt{\log r}} , \quad \forall r>1 \] where $C_t$ is a constant depending on $t$ but not on the dimension. The purpose of the present paper is to simplify parts of their argument and to remove the $(\log \log r)^4$ factor.

math.PR↗
arXiv2013

Bounding the norm of a log-concave vector via thin-shell estimates

Chaining techniques show that if X is an isotropic log-concave random vector in R^n and Gamma is a standard Gaussian vector then E |X| < C n^{1/4} E |Gamma| for any norm |*|, where C is a universal constant. Using a completely different argument we establish a similar inequality relying on the thin-shell constant sigma_n = sup ((var|X|^){1/2} ; X isotropic and log-concave on R^n). In particular, we show that if the thin-shell conjecture sigma_n = O(1) holds, then n^{1/4} can be replaced by log (n) in the inequality. As a consequence, we obtain certain bounds for the mean-width, the dual mean-width and the isotropic constant of an isotropic convex body. In particular, we give an alternative proof of the fact that a positive answer to the thin-shell conjecture implies a positive answer to the slicing problem, up to a logarithmic factor.

math.FA↗
arXiv2013

Short probabilistic proof of the Brascamp-Lieb and Barthe theorems

We give a short proof of the Brascamp-Lieb theorem, which asserts that a certain general form of Young's convolution inequality is saturated by Gaussian functions. The argument is inspired by Borell's stochastic proof of the Prékopa-Leindler inequality and applies also to the reversed Brascamp-Lieb inequality, due to Barthe.

math.PR↗
arXiv2012

Cover times and generic chaining

A recent result of Ding, Lee and Peres expresses the cover time of the random walk on a graph in terms of generic chaining for the commute distance. Their proof is very involved and the purpose of this article is to present a simpler approach to this problem based on elementary hitting times estimates and chaining arguments. Unfortunately we fail to recover their full result, but not by much.

math.PR↗