Dario Cordero-Erausquin
We review some simple techniques based on monotone mass transport that allow to obtain transport-type inequalities for any log-concave probability measure, and for more general measures as well. We discuss quantitative forms of these inequalities, with application to the Brascamp-Lieb variance inequality.
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We prove, using optimal transport tools, weighted Poincar'e inequalities for log-concave random vectors satisfying some centering conditions. We recover by this way similar results by Klartag and Barthe-Cordero-Erausquin for log-concave random vectors with symmetries. In addition, we prove that the variance conjecture is true for increments of log-concave martingales.
We prove a sharp inequality conjectured by Bobkov on the measure of dilations of Borel sets in $\mathbb{R}^n$ by a $s$-concave probability. Our result gives a common generalization of an inequality of Nazarov, Sodin and Volberg and a concentration inequality of Guédon. Applying our inequality to the level sets of functions satisfying a Remez type inequality, we deduce, as it is classical, that these functions enjoy dimension free distribution inequalities and Kahane-Khintchine type inequalities with positive and negative exponent, with respect to an arbitrary $s$-concave probability.
We establish concentration inequalities in the class of ultra log-concave distributions. In particular, we show that ultra log-concave distributions satisfy Poisson concentration bounds. As an application, we derive concentration bounds for the intrinsic volumes of a convex body, which generalizes and improves a result of Lotz, McCoy, Nourdin, Peccati, and Tropp (2019).
Geometric and Functional Inequalities for Log-concave Probability Sequences
该研究针对对数凹概率序列，证明了整数集上的伸缩不等式及对应的泛函偏差不等式，揭示对数仿射序列为相关集合极点，借此简化已有卷积不等式证明并得到离散Prékopa-Leindler不等式。
Extending results of Harg{é} and Hu for the Gaussian measure, we prove inequalities for the covariance Cov$_μ(f, g)$ where $μ$ is a general product probability measure on $\mathbb{R}^d$ and $f,g: \mathbb{R}^d \to \mathbb{R}$ satisfy some convexity or log-concavity assumptions, with possibly some symmetries.
Sharp Rosenthal-type inequalities for mixtures and log-concave variables
本文为固定分布混合的独立随机变量和矩建立了带尖锐常数的 Rosenthal 型不等式，并在对数凹情形下确定了极值元。
We investigate the role of convexity in Rényi entropy power inequalities. After proving that a general Rényi entropy power inequality in the style of Bobkov-Chistyakov (2015) fails when the Rényi parameter $r\in(0,1)$, we show that random vectors with $s$-concave densities do satisfy such a Rényi entropy power inequality. Along the way, we establish the convergence in the Central Limit Theorem for Rényi entropies of order $r\in(0,1)$ for log-concave densities and for compactly supported, spherically symmetric and unimodal densities, complementing a celebrated result of Barron (1986). Additiona…
In this paper, we use the semi-group method and an adaptation of the $L^2-$method of Hörmander to establish some $Φ-$entropy inequalities and asymmetric covariance estimates for the strictly convex measures in $\mathbb R^n$. These inequalities extends the ones for the strictly log-concave measures to more general setting of convex measures. The $Φ-$entropy inequalities are turned out to be sharp in the special case of Cauchy measures. Finally, we show that the similar inequalities for log-concave measures can be obtained from our results in the limiting case.
We introduce a transport-majorization argument that establishes a majorization in the convex order between two densities, based on control of the gradient of a transportation map between them. As applications, we give elementary derivations of some delicate Fourier analytic inequalities, which in turn yield geometric "slicing-inequalities" in both continuous and discrete settings. As a further consequence of our investigation we prove that any strongly log-concave probability density majorizes the Gaussian density and thus the Gaussian density maximizes the Rényi and Tsallis entropies of all o…
Majorization, Entropy and Concentration Inequalities for Discrete $α$-Concave Random Variables
针对离散 $\alpha$-凹随机变量（$-1/3<\alpha<0$），提出最大方差不等式及凸优超原理，推广对数凹情形，并导出逆向熵幂不等式、Lévy 集中函数界及多项式型集中不等式。
We study functional inequalities (Poincaré, Cheeger, log-Sobolev) for probability measures obtained as perturbations. Several explicit results for general measures as well as log-concave distributions are given.The initial goal of this work was to obtain explicit bounds on the constants in view of statistical applications for instance. These results are then applied to the Langevin Monte-Carlo method used in statistics in order to compute Bayesian estimators.
We show that the modified log-Sobolev constant for a natural Markov chain which converges to an $r$-homogeneous strongly log-concave distribution is at least $1/r$. Applications include a sharp mixing time bound for the bases-exchange walk for matroids, and a concentration bound for Lipschitz functions over these distributions.
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.
Anti-concentration inequalities for log-concave variables on the real line
本文针对实直线上的对数凹随机变量，在离散与连续情形下证明了精确的反集中不等式，采用初等优化方法恢复并推广了已有结果。
Concentration inequalities for log-concave sequences
本文通过凸序中的主次关系，为满足精确对数凹度的随机变量推导出集中不等式、矩界和熵界，并改进了现有文献中的结果。
Entropy-variance inequalities for discrete log-concave random variables via degree of freedom
本文利用离散自由度概念证明整数值对数凹随机变量的精确最小熵-方差不等式，证明几何分布能最小化最小熵，并由此推导出改进的离散 Rényi 熵幂不等式。
Concave and other generalizations of stochastic Gronwall inequalities
本文将一类随机 Gronwall 不等式推广到 Bihari–LaSalle 型的凹性及其他非线性情形，用于证明 Lévy 过程驱动的路径依赖 SDE 在非 Lipschitz 条件下全局解的存在唯一性。
Brascamp--Lieb-type, weighted Poincaré-type and related analytic inequalities are studied for multidimensional Cauchy distributions and more general $κ$-concave probability measures (in the hierarchy of convex measures). In analogy with the limiting (infinite-dimensional log-concave) Gaussian model, the weighted inequalities fully describe the measure concentration and large deviation properties of this family of measures. Cheeger-type isoperimetric inequalities are investigated similarly, giving rise to a common weight in the class of concave probability measures under consideration.
We develop in this paper an amelioration of the method given by S. Bobkov and M. Ledoux in GAFA (2000). We prove by Prekopa-Leindler Theorem an optimal modified logarithmic Sobolev inequality adapted for all log-concave measure on $\dR^n$. This inequality implies results proved by Bobkov and Ledoux, the Euclidean Logarithmic Sobolev inequality generalized in the last years and it also implies some convex logarithmic Sobolev inequalities for large entropy.
We study the minimax identity for a non-decreasing upper-semicontinuous utility function satisfying mild growth assumption. In contrast to the classical setting, we do not impose the assumption that the utility function is concave. By considering the concave envelope of the utility function we obtain equalities and inequalities between the robust utility functionals of an initial utility function and its concavification. Furthermore, we prove similar equalities and inequalities in the case of implementing an upper bound on the final endowment of the initial model.
We prove an isoperimetric inequality for the uniform measure on a uniformly convex body and for a class of uniformly log-concave measures (that we introduce). These inequalities imply (up to universal constants) the log-Sobolev inequalities proved by Bobkov--Ledoux as well as the isoperimetric inequalities due to Bakry-Ledoux and Bobkov--Zegarlinski. We also recover a concentration inequality for uniformly convex bodies, similar to that proved by Gromov--Milman.
We show that the uniform distribution minimizes entropy among all one-dimensional symmetric log-concave distributions with fixed variance, as well as various generalizations of this fact to Rényi entropies of orders less than 1 and with moment constraints involving $p$-th absolute moments with $p\leq 2$. As consequences, we give new capacity bounds for additive noise channels with symmetric log-concave noises, as well as for timing channels involving positive signal and noise where the noise has a decreasing log-concave density. In particular, we show that the capacity of an additive noise cha…
The Poisson transport map
本文构造了从泊松点过程到超对数凹测度的收缩传输映射，克服了离散情形下传递函数不等式的障碍，并得到修正对数索博列夫不等式的最佳常数。
On stochastic forms of functional isoperimetric inequalities
本文研究了p-凹函数类中的功能等周不等式的随机形式，推导出Sobolev型不等式，并验证了Zhang的仿射Sobolev不等式在随机模型中的期望成立，同时恢复了几何和确定性结果。
Two-sided bounds are explored for concentration functions and Rényi entropies in the class of discrete log-concave probability distributions. They are used to derive certain variants of the entropy power inequalities.
An important theme in recent work in asymptotic geometric analysis is that many classical implications between different types of geometric or functional inequalities can be reversed in the presence of convexity assumptions. In this note, we explore the extent to which different notions of distance between probability measures are comparable for log-concave distributions. Our results imply that weak convergence of isotropic log-concave distributions is equivalent to convergence in total variation, and is further equivalent to convergence in relative entropy when the limit measure is Gaussian.
We derive two concentration inequalities for linear functions of log-concave distributions: an enhanced version of the classical Brascamp--Lieb concentration inequality, and an inequality quantifying log-concavity of marginals in a manner suitable for obtaining variance and tail probability bounds. These inequalities are applied to the statistical mechanics problem of estimating the fluctuations of random surfaces of the $\nablaφ$ type. The classical Brascamp--Lieb inequality bounds the fluctuations whenever the interaction potential is uniformly convex. We extend these bounds to the case of c…
Let $K$ be an isotropic convex body in $\R^n$. Given $\eps>0$, how many independent points $X_i$ uniformly distributed on $K$ are needed for the empirical covariance matrix to approximate the identity up to $\eps$ with overwhelming probability? Our paper answers this question posed by Kannan, Lovasz and Simonovits. More precisely, let $X\in\R^n$ be a centered random vector with a log-concave distribution and with the identity as covariance matrix. An example of such a vector $X$ is a random point in an isotropic convex body. We show that for any $\eps>0$, there exists $C(\eps)>0$, such that if…
We prove that the exponent of the entropy of one dimensional projections of a log-concave random vector defines a 1/5-seminorm. We make two conjectures concerning reverse entropy power inequalities in the log-concave setting and discuss some examples.
In this paper we intend to give a comprehensive approach of functional inequalities for diffusion processes under some "curvature" assumptions. Our notion of curvature coincides with the usual $Γ_2$ curvature of Bakry and Emery in the case of a (reversible) drifted Brownian motion, but differs for more general diffusion processes. Our approach using simple coupling arguments together with classical stochastic tools, allows us to obtain new results, to recover and to extend already known results, giving in many situations explicit (though non optimal) bounds. In particular, we show new results …
We find sufficient conditions for a probability measure $μ$ to satisfy an inequality of the type $$ \int_{\R^d} f^2 F\Bigl(\frac{f^2}{\int_{\R^d} f^2 d μ} \Bigr) d μ\le C \int_{\R^d} f^2 c^{*}\Bigl(\frac{|\nabla f|}{|f|} \Bigr) d μ+ B \int_{\R^d} f^2 d μ, $$ where $F$ is concave and $c$ (a cost function) is convex. We show that under broad assumptions on $c$ and $F$ the above inequality holds if for some $δ>0$ and $ε>0$ one has $$ \int_{0}^ε Φ\Bigl(δc\Bigl[\frac{t F(\frac{1}{t})}{{\mathcal I}_μ(t)} \Bigr] \Bigr) dt < \infty, $$ where ${\mathcal I}_μ$ is the isoperimetric function of $μ$ and $Φ…
We study the Restricted Isometry Property of a random matrix $Γ$ with independent isotropic log-concave rows. To this end, we introduce a parameter $Γ_{k,m}$ that controls uniformly the operator norm of sub-matrices with $k$ rows and $m$ columns. This parameter is estimated by means of new tail estimates of order statistics and deviation inequalities for norms of projections of an isotropic log-concave vector.
The goal of this paper is to push forward the study of those properties of log-concave measures that help to estimate their Poincar{é} constant. First we revisit E. Milman's result [40] on the link between weak (Poincar{é} or concentration) inequalities and Cheeger's inequality in the logconcave cases, in particular extending localization ideas and a result of Latala, as well as providing a simpler proof of the nice Poincar{é} (dimensional) bound in the inconditional case. Then we prove alternative transference principle by concentration or using various distances (total variation, Wasserstein…
We prove that for a probability measure on $\mathbb{R}^n$, the Poincaré inequality for convex functions is equivalent to the weak transportation inequality with a quadratic-linear cost. This generalizes recent results by Gozlan et al. and Feldheim et al., concerning probability measures on the real line. The proof relies on modified logarithmic Sobolev inequalities of Bobkov-Ledoux type for convex and concave functions, which are of independent interest. We also present refined concentration inequalities for general (not necessarily Lipschitz) convex functions, complementing recent results by …
Transport-entropy forms of direct and Converseblaschke-Santal{ó} inequalities
本文研究Blaschke-Santaló不等式及其Mahler猜想对应物的输运-熵表述，得到s-凹函数的直接与逆不等式，并证明Mahler猜想等价于球面上加强的对数Sobolev型不等式。
In this paper, we give necessary and sufficient conditions for Talagrand's like transportation cost inequalities on the real line. This brings a new wide class of examples of probability measures enjoying a dimension-free concentration of measure property. Another byproduct is the characterization of modified Log-Sobolev inequalities for Log-concave probability measures on R.
Minimum entropy of a log-concave variable for fixed variance
本文证明固定方差下对数凹变量的最小熵由指数分布达到，并据此推导加性噪声信道容量上界及改进反向熵幂不等式常数。
Growth estimates on optimal transport maps via concentration inequalities
本文通过集中不等式和映射单调性，为Brenier映射的多项式上界提供替代证明并扩展结果，涵盖从高斯到对数凹测度的二次增长情形。
We introduce a notion called entropic independence that is an entropic analog of spectral notions of high-dimensional expansion. Informally, entropic independence of a background distribution $μ$ on $k$-sized subsets of a ground set of elements says that for any (possibly randomly chosen) set $S$, the relative entropy of a single element of $S$ drawn uniformly at random carries at most $O(1/k)$ fraction of the relative entropy of $S$. Entropic independence is the analog of the notion of spectral independence, if one replaces variance by entropy. We use entropic independence to derive tight mix…
We prove logarithmic Sobolev inequalities and concentration results for convex functions and a class of product random vectors. The results are used to derive tail and moment inequalities for chaos variables (in spirit of Talagrand and Arcones, Gine). We also show that the same proof may be used for chaoses generated by log-concave random variables, recovering results by Lochowski and present an application to exponential integrability of Rademacher chaos.
The Brunn-Minkowski and Prékopa-Leindler inequalities admit a variety of proofs that are inspired by convexity. Nevertheless, the former holds for compact sets and the latter for integrable functions so it seems that convexity has no special signficance. On the other hand, it was recently shown that the Brunn-Minkowski inequality, specialized to convex sets, follows from a local stochastic dominance for naturally associated random polytopes. We show that for the subclass of $\log$-concave functions and associated stochastic approximations, a similar stochastic dominance underlies the Prékopa-L…
The goal of the article is to improve constants in the infimum convolution inequalities (IC for short) which were introduced by R. Latała and J.O. Wojtaszczyk. We show that the exponential distribution satisfies IC with constant $2$ but not with constant $1$, which implies that linear functions are not extremal in Maurey's property $(τ)$. Using transport of measure we use this result to better constants in the IC inequalities for product symmetric log-concave measures as well as in the Talagrand's two level concentration inequality for the exponential distribution.
Let $μ$ be a probability measure on $\rr^n$ ($n \geq 2$) with Lebesgue density proportional to $e^{-V (\Vert x\Vert )}$, where $V : \rr_+ \to \rr$ is a smooth convex potential. We show that the associated spectral gap in $L^2 (μ)$ lies between $(n-1) / \int_{\rr^n} \Vert x\Vert ^2 μ(dx)$ and $n / \int_{\rr^n} \Vert x\Vert ^2 μ(dx)$, improving a well-known two-sided estimate due to Bobkov. Our Markovian approach is remarkably simple and is sufficiently robust to be extended beyond the log-concave case, at the price of potentially modifying the underlying dynamics in the energy, leading to weigh…
In a recent paper A. Cianchi, N. Fusco, F. Maggi, and A. Pratelli have shown that, in the Gauss space, a set of given measure and almost minimal Gauss boundary measure is necessarily close to be a half-space. Using only geometric tools, we extend their result to all symmetric log-concave measures μon the real line. We give sharp quantitative isoperimetric inequalities and prove that among sets of given measure and given asymmetry (distance to half line, i.e. distance to sets of minimal perimeter), the intervals or complements of intervals have minimal perimeter.
High-Dimensional Ultra-Log-Concave Distributions
本文提出高维超对数凹分布的定量推广（$\delta$-超对数凹性），建立函数不等式、集中性与最大熵原理，并应用于多个模型。
In this paper we present new versions of the classical Brunn-Minkowski inequality for different classes of measures and sets. We show that the inequality \[ μ(λA + (1-λ)B)^{1/n} \geq λμ(A)^{1/n} + (1-λ)μ(B)^{1/n} \] holds true for an unconditional product measure $μ$ with decreasing density and a pair of unconditional convex bodies $A,B \subset \mathbb{R}^n$. We also show that the above inequality is true for any unconditional $\log$-concave measure $μ$ and unconditional convex bodies $A,B \subset \mathbb{R}^n$. Finally, we prove that the inequality is true for a symmetric $\log$-concave measu…
In this work, we study the validity of Entropy inequalities under decorrelation assumptions at large-scales, and also we study the problem of Weighted Meyers estimate in perforated domains. As a consequence, we obtain the hypercontractivity of correctors with Neumann boundary conditions.
We obtain several quantitative bounds on the mixing properties of the Hamiltonian Monte Carlo (HMC) algorithm for a strongly log-concave target distribution $π$ on $\mathbb{R}^{d}$, showing that HMC mixes quickly in this setting. One of our main results is a dimension-free bound on the mixing of an "ideal" HMC chain, which is used to show that the usual leapfrog implementation of HMC can sample from $π$ using only $\mathcal{O}(d^{\frac{1}{4}})$ gradient evaluations. This dependence on dimension is sharp, and our results significantly extend and improve previous quantitative bounds on the mixin…
We prove that the (B) conjecture and the Gardner-Zvavitch conjecture are true for all log-concave measures that are rotationally invariant, extending previous results known for Gaussian measures. Actually, our result apply beyond the case of log-concave measures, for instance to Cauchy measures as well. For the proof, new sharp weighted Poincaré inequalities are obtained for even probability measures that are log-concave with respect to a rotationally invariant measure.
On Log-Concave-Tailed Chaoses and the Restricted Isometry Property
本文建立对数凹尾非齐次混沌过程上确界的p阶矩界及新型解耦不等式，并据此推导α-次指数元素的Hanson-Wright型偏差不等式，证明部分随机循环矩阵与时频结构随机矩阵的限制等距性质。
The optimal transport map between the standard Gaussian measure and an $α$-strongly log-concave probability measure is $α^{-1/2}$-Lipschitz, as first observed in a celebrated theorem of Caffarelli. In this paper, we apply two classical covariance inequalities (the Brascamp-Lieb and Cramér-Rao inequalities) to prove a sharp bound on the Lipschitz constant of the map that arises from entropically regularized optimal transport. In the limit as the regularization tends to zero, we obtain an elegant and short proof of Caffarelli's original result. We also extend Caffarelli's theorem to the setting …
We establish some quantitative concentration estimates for the empirical measure of many independent variables, in transportation distances. As an application, we provide some error bounds for particle simulations in a model mean field problem. The tools include coupling arguments, as well as regularity and moments estimates for solutions of certain diffusive partial differential equations.
Quantitative merging for time-inhomogeneous Markov chains in non-decreasing environments via functional inequalities
利用Poincaré、Nash和对数Sobolev不等式，研究时间非齐次马尔可夫链在不变测度序列非递减条件下的渐近行为，给出从任意两点出发的分布合并时间的定量界。
We establish a discrete analog of the Rényi entropy comparison due to Bobkov and Madiman. For log-concave variables on the integers, the min entropy is within log e of the usual Shannon entropy. Additionally we investigate the entropic Rogers-Shephard inequality studied by Madiman and Kontoyannis, and establish a sharp Rényi version for certain parameters in both the continuous and discrete cases
The isoperimetric inequalities for the expected lifetime of Brownian motion state that the $L^p$-norms of the expected lifetime in a bounded domain for $1\leq p\leq \infty$ are maximized when the region is a ball with the same volume. In this paper, we prove quantitative improvements of the inequalities. Since the isoperimetric properties hold for a wide class of Lévy processes, many questions arise from these improvements.
We establish quantitative stability results for the entropy power inequality (EPI). Specifically, we show that if uniformly log-concave densities nearly saturate the EPI, then they must be close to Gaussian densities in the quadratic Wasserstein distance. Further, if one of the densities is log-concave and the other is Gaussian, then the deficit in the EPI can be controlled in terms of the $L^1$-Wasserstein distance. As a counterpoint, an example shows that the EPI can be unstable with respect to the quadratic Wasserstein distance when densities are uniformly log-concave on sets of measure arb…
Reweighted information inequalities
本文建立了一种混合分布的log-Sobolev和传输-信息不等式变体，通过重新加权解释了非log-凹测度的Fisher信息界，并解释了Langevin蒙特卡洛方法在多模分布分析中的现象。
New Brunn--Minkowski and functional inequalities via convexity of entropy
本文研究了测度的凹性与相对熵的凸性之间的联系，证明了新的中心星形体的Brunn-Minkowski不等式，并通过Gaussian相对熵的凸性得到新的函数不等式。
We compare weighted sums of i.i.d. positive random variables according to the usual stochastic order. The main inequalities are derived using majorization techniques under certain log-concavity assumptions. Specifically, let $Y_i$ be i.i.d. random variables on $\mathbf{R}_+$. Assuming that $\log Y_i$ has a log-concave density, we show that $\sum a_iY_i$ is stochastically smaller than $\sum b_iY_i$, if $(\log a_1,...,\log a_n)$ is majorized by $(\log b_1,...,\log b_n)$. On the other hand, assuming that $Y_i^p$ has a log-concave density for some $p>1$, we show that $\sum a_iY_i$ is stochasticall…
We study stability of the sharp Poincar{é} constant of the invariant probability measure of a reversible diffusion process satisfying some natural conditions. The proof is based on the spectral interpretation of Poincar{é} inequalities and Stein's method. In particular, these results are applied to the gamma distributions and to strictly log-concave measures in dimension one, giving stability for Brascamp-Lieb inequalities.
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