Egor Kosov
The paper provides an estimate of the total variation distance between distributions of polynomials defined on a space equipped with a logarithmically concave measure in terms of the $L^2$-distance between these polynomials.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
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 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.
On measures strongly log-concave on a subspace
研究仅在方向子空间上弯曲的对数凹测度的集中性质，采用适配的随机局部化过程进行证明。
We establish upper bounds for tails of order statistics of isotropic log-concave vectors and apply them to derive a concentration of l_r norms of such vectors.
We study the log-concave measures, their characterization via the Prékopa-Leindler property and also define a subset of it whose elements are called super log-concave measures which have the property of satisfying a logarithmic Sobolev inequality. We give some results about their stability. Certain relations with measure transportation of Monge-Kantorovitch and the Monge-Ampére equation are also indicated with applications.
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 formulate and discuss a conjecture concerning lower bounds for norms of log-concave vectors, which generalizes the classical Sudakov minoration principle for Gaussian vectors. We show that the conjecture holds for some special classes of log-concave measures and some weaker forms of it are satisfied in the general case. We also present some applications based on chaining techniques.
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.
A concentration property of the functional ${-}\log f(X)$ is demonstrated, when a random vector X has a log-concave density f on $\mathbb{R}^n$. This concentration property implies in particular an extension of the Shannon-McMillan-Breiman strong ergodic theorem to the class of discrete-time stochastic processes with log-concave marginals.
A Chaining Approach to Canonical Processes with Log-Concave Tails
本文证明了无需Δ2条件的对数凹尾典型过程的分解定理，采用基于可容许划分和切割映射的经典链式方法，并恢复了控制测度刻画。
A note on the spectral gap for log-concave probability measures on convex bodies
本文研究了凸体上对数凹概率测度的谱隙问题，提出了基于分布参数、维度、域几何特征和原点位置等量的显式下界，并通过经典与非经典例子验证了结果。
We discuss situations where perturbing a probability measure on $\mathbb{R}^n$ does not deteriorate its Poincaré constant by much. A particular example is the symmetric exponential measure in $\mathbb{R}^n$, even log-concave perturbations of which have Poincaré constants that grow at most logarithmically with the dimension. This leads to estimates for the Poincaré constants of $(n/2)$-dimensional sections of the unit ball of $\ell_p^n$ for $1 \leq p \leq 2$, which are optimal up to logarithmic factors. We also consider symmetry properties of the eigenspace of the Laplace-type operator associat…
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.
An example of a non-log-concave distribution where the difference has a log-concave density
本文通过具体例子证明，非对数凹分布的差值不一定具有对数凹密度，推翻了Prékopa-Leindler不等式的逆向结论。
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 prove a sharp moment inequality for a log-concave or a log-convex function, on Gaussian random vectors. As an application we take a stability result for the classical logarithmic Sobolev inequality of L. Gross in the case where the function is log-concave.
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 explore the validity of the circular law for random matrices with non i.i.d. entries. Let A be a random n \times n real matrix having as a random vector in R^{n^2} a log-concave isotropic unconditional law. In particular, the entries are uncorellated and have a symmetric law of zero mean and unit variance. This allows for some dependence and non equidistribution among the entries, while keeping the special case of i.i.d. standard Gaussian entries. Our main result states that as n goes to infinity, the empirical spectral distribution of n^{-1/2}A tends to the uniform law on the unit disc of …
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.
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.
We establish new tail estimates for order statistics and for the Euclidean norms of projections of an isotropic log-concave random vector. More generally, we prove tail estimates for the norms of projections of sums of independent log-concave random vectors, and uniform versions of these in the form of tail estimates for operator norms of matrices and their sub-matrices in the setting of a log-concave ensemble. This is used to study a quantity $A_{k,m}$ that controls uniformly the operator norm of the sub-matrices with $k$ rows and $m$ columns of a matrix $A$ with independent isotropic log-con…
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).
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.
On the Log-Sobolev Constant of Log-Concave Vectors
本文研究了在对数-凹假设下，对数-赛博斯坦常数的逆推问题，并改进了Bobkov的结果，给出了次高斯对数-凹测度的最佳维度依赖界。
A Note on Statistical Distances for Discrete Log-Concave Measures
研究离散对数凹分布上总变差、Wasserstein与$f$-散度等标准统计距离的等价性。
An invariant variational proof for canonical processes with independent symmetric log-concave tails
本文通过不变变分方法，证明了无需加倍假设的独立对称对数凹尾典范过程的指定指数律比较。
Sharp estimates for the Cramér transform of log-concave measures and geometric applications
该研究建立了对数凹概率分布的累积量生成函数Legendre变换与半空间深度的新比较关系，结合旋转不变测度Cramér变换的精确估计，得到随机多胞形期望测度渐近性的新相变结果，并解决了对称对数凹分布的精确指数可分常数问题。
We study how well moments of sums of independent symmetric random variables with logarithmically concave tails may be approximated by moments of Gaussian random variables.
We derive two-sided bounds for moments of linear combinations of coordinates od unconditional log-concave vectors. We also investigate how well moments of such combinations may be approximated by moments of Gaussian random variables.
Stability version of the Prekopa-Leindler inequality for log-concave functions on the n-dimensional Euclidean space is established.
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…
One-sided concentration near the mean of log-concave distributions
本文研究了对数凹分布均值附近的一侧浓度特性，得出了在EX=0和EX²=1条件下概率P(0<X<δ)的下界。
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。
We show that the conjecture of Kannan, Lovász, and Simonovits on isoperimetric properties of convex bodies and log-concave measures, is true for log-concave measures of the form $ρ(|x|_B)dx$ on $\mathbb{R}^n$ and $ρ(t,|x|_B) dx$ on $\mathbb{R}^{1+n}$, where $|x|_B$ is the norm associated to any convex body $B$ already satisfying the conjecture. In particular, the conjecture holds for convex bodies of revolution.
We show that a measure on the real line that is the image of a log-concave measure under a polynomial of degree $d$ possesses a density from the Nikol'skii--Besov class of fractional order $1/d$. This result is used to prove an estimate of the total variation distance between such measures in terms of the Fortet--Mourier distance.
We show that for every mean zero log-concave real random variable $X$ one has $\|X\|_p \leq \frac{p}{q} \|X\|_q$ for $p \geq q \geq 1$, going beyond the well-known case of symmetric random variables. We also prove that in the class of arbitrary log-concave real random variables for $p>q > 0$ the quantity $\|X\|_p / \|X\|_q$ is maximized for some shifted exponential distribution. Building upon this we derive the bound $\|X\|_p \leq C_0 \frac{p}{q} \|X\|_q$ for arbitrary log-concave $X$, with best possible absolute constant $C_0=e^{W(1/e)} \approx 1.3211$ in front of $\frac{p}{q}$, where $W$ sta…
We give a self-contained proof of the strongest version of Mason's conjecture, namely that for any matroid the sequence of the number of independent sets of given sizes is ultra log-concave. To do this, we introduce a class of polynomials, called completely log-concave polynomials, whose bivariate restrictions have ultra log-concave coefficients. At the heart of our proof we show that for any matroid, the homogenization of the generating polynomial of its independent sets is completely log-concave.
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.
We present two-sided estimates of moments and tails of polynomial chaoses of order at most three generated by independent symmetric random variables with log-concave tails as well as for chaoses of arbitrary order generated by independent symmetric exponential variables. The estimates involve only deterministic quantities and are optimal up to constants depending only on the order of the chaos variable.
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。
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。
Bi-log-concavity of probability measures is a univariate extension of the notion of log-concavity that has been recently proposed in a statistical literature. Among other things, it has the nice property from a modelisation perspective to admit some multimodal distributions, while preserving some nice features of log-concave measures. We compute the isoperimetric constant for a bi-log-concave measure, extending a property available for log-concave measures. This implies that bi-log-concave measures have exponentially decreasing tails. Then we show that the convolution of a bi-log-concave measu…
A note on the Littlewood-Offord problem for discrete log-concave distributions
本文扩展了Littlewood-Offord问题，研究离散对数凹分布下的熵功率不等式，并探讨了算术级数和熵版本的问题。
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-w…
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.
A note on the maximum probability of ultra log-concave distributions
本文指出，超对数凹分布的最大概率不小于同均值泊松分布最大概率的结论，在均值大于1时并非普遍成立。
In this paper, we consider the log-concave ensemble of random matrices, a class of covariance-type matrices $XX^*$ with isotropic log-concave $X$-columns. A main example is the covariance estimator of the uniform measure on isotropic convex body. Non-asymptotic estimates and first order asymptotic limits for the extreme eigenvalues have been obtained in the literature. In this paper, with the recent advancements on log-concave measures \cite{chen, KL22}, we take a step further to locate the eigenvalues with a nearly optimal precision, namely, the spectral rigidity of this ensemble is derived. …
Mixture and separation of log-concave measures
研究双组分对数凹混合模型，探讨给定样本时各组分分布参数的精确恢复程度，识别出混合分布的两种情况，提出用二次分类器分离组分分布的方法。
We show the existence of the strong solutions of the Monge-Ampere equation for the log-concave measures without any regularity assumption. In particular the measures with density of the form $\exp -f$, where $f$ is an $H$-convex function which may take the value $\infty$ on a set of positive measure are included.
The first step to study lower bounds for a stochastic process is to prove a special property - Sudakov minoration. The property means that if a certain number of points from the index set are well separated then we can provide an optimal type lower bound for the mean value of the supremum of the process. Together with the generic chaining argument the property can be used to fully characterize the mean value of the supremum of the stochastic process. In this article we prove the property for canonical processes based on radial-type log concave measures.
An elementary proof is provided of sharp bounds for the varentropy of random vectors with log-concave densities, as well as for deviations of the information content from its mean. These bounds significantly improve on the bounds obtained by Bobkov and Madiman ({\it Ann. Probab.}, 39(4):1528--1543, 2011).
A reverse entropy power inequality for i.i.d. log-concave random variables
研究独立同分布对数凹随机变量的反向熵幂不等式，通过递减重排、控制和测度变换等工具，证明\(h_\infty(X + Y) \leq h_\infty(Z + W)\)，并得到整数值单调对数凹随机变量类似结果。
Convolution of a symmetric log-concave distribution and a symmetric bimodal distribution can have any number of modes
本文证明了离散及光滑情形下，对称对数凹分布与对称双峰分布的卷积可产生任意正数量的众数。
We establish two-sided bounds for expectations of order statistics ($k$-th maxima) of moduli of coordinates of centered log-concave random vectors with uncorrelated coordinates. Our bounds are exact up to multiplicative universal constants in the unconditional case for all $k$ and in the isotropic case for $k \leq n-cn^{5/6}$. We also derive two-sided estimates for expectations of sums of $k$ largest moduli of coordinates for some classes of random vectors.
We establish uniform bounds on the low-order derivatives of Stein equation solutions for a broad class of multivariate, strongly log-concave target distributions. These "Stein factor" bounds deliver control over Wasserstein and related smooth function distances and are well-suited to analyzing the computable Stein discrepancy measures of Gorham and Mackey. Our arguments of proof are probabilistic and feature the synchronous coupling of multiple overdamped Langevin diffusions.
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。
From Triangular Array Progression to Near-Log-Concave State Transitions
本文提出一种生成三角阵列的无限整数序列，该阵列具有近对数凹性等特性，可用于物理建模电子跃迁系统，还能生成满足特定状态转移特征的右随机矩阵族。
On a Conjecture of Feige for Discrete Log-Concave Distributions
本文针对Feige猜想，在离散对数凹分布类上证明了其加强版本，即独立离散对数凹随机变量之和小于其期望加1的概率至少为1/e，且对任意期望均成立。
We show that for $p\geq 1$ and $r\geq 1$ the $p$-th moment of the $l_r$-norm of a log-concave random vector is comparable to the sum of the first moment and the weak $p$-th moment up to a constant proportional to $r$. This extends the previous result of Paouris concerning Euclidean norms.
We prove an isoperimetric inequality for probability measures $μ$ on $\mathbb{R}^n$ with density proportional to $\exp(-ϕ(λ| x|))$, where $|x|$ is the euclidean norm on $\mathbb{R}^n$ and $ϕ$ is a non-decreasing convex function. It applies in particular when $ϕ(x)=x^α$ with $α\ge1$. Under mild assumptions on $ϕ$, the inequality is dimension-free if $λ$ is chosen such that the covariance of $μ$ is the identity.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。