David Pollard
The paper reexamines an argument by Talagrand that leads to a remarkable exponential tail bound for the concentration of probability near a set. The main novelty is the replacement of a mysterious Calculus inequality by an application of Jensen's inequality.
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This note presents a sharp transport-entropy inequality that improves on Talagrand's inequality for the Gaussian measure, arising as a dual formulation of the functional Santaló inequality. We also discuss some extensions and connections with concentration of measure.
Talagrand's inequality for independent Bernoulli random variables is extended to many interacting particle systems (IPS). The main assumption is that the IPS satisfies a log-Sobolev inequality. In this context it is also shown that a slightly stronger version of Talagrand's inequality is equivalent to a log-Sobolev inequality. Additionally we also look at a common application, the relation between the probability of increasing events and the influences on that event by changing a single spin.
This note reviews the studies of the last decades emphasizing a common principle based on entropy, logarithmic Sobolev inequality and hypercontractivity, behind four most celebrated inequalities by M. Talagrand: the convex distance inequality, the $\mathrm {L}^1$-$\mathrm {L}^2$ variance inequality, the quadratic transportation cost inequality, and the inequality on the supremum of empirical processes.
We give by simple arguments sufficient conditions, so called Lyapunov conditions, for Talagrand's transportation information inequality and for the logarithmic Sobolev inequality. Those sufficient conditions work even in the case where the Bakry-Emery curvature is not lower bounded. Several new examples are provided.
In this paper we revisit Talagrand's proof of concentration inequality for empirical processes. We give a different shorter proof of the main technical lemma that guarantees the existence of a certain kernel. Our proof provides the almost optimal value of the constant involved in the statement of this lemma.
By using a split argument due to [1], the transportation cost inequality is established on the free path space of Markov processes. The general result is applied to stochastic reaction diffusion equations with random initial values.
In this article we study generalization of the classical Talagrand transport-entropy inequality in which the Wasserstein distance is replaced by the entropic transportation cost. This class of inequalities has been introduced in the recent work [9], in connection with the study of Schrödinger bridges. We provide several equivalent characterizations in terms of reverse hypercontractivity for the heat semigroup, contractivity of the Hamilton-Jacobi-Bellman semigroup and dimension-free concentration of measure. Properties such as tensorization and relations to other functional inequalities are al…
Stochastic proof of the sharp symmetrized Talagrand inequality
该研究提出了Talagrand输运熵不等式尖锐对称化形式的全新随机证明，核心创新在于采用时间反转鞅表示诱导的特定耦合方法，为高斯泛函不等式研究提供了新的随机分析思路。
Quantum Talagrand-type Inequalities via Variance Decay
本文通过方差衰减视角，建立了量子Talagrand型不等式，推导了多种量子等周不等式和高阶影响-方差不等式，统一了第一和高阶Talagrand型现象。
We survey several Talagrand type inequalities and their application to influences with the tool of hypercontractivity for both discrete and continuous, and product and non-product models. The approach covers similarly by a simple interpolation the framework of geometric influences recently developed by N. Keller, E. Mossel and A. Sen. Geometric Brascamp-Lieb decompositions are also considered in this context.
We show that Talagrand's transport inequality is equivalent to a restricted logarithmic Sobolev inequality. This result clarifies the links between these two important functional inequalities. As an application, we give the first proof of the fact that Talagrand's inequality is stable under bounded perturbations.
We give a short proof of the free analogue of the Talagrand inequality for the transportation cost to the semicircular which was originally proved by Biane and Voiculescu. The proof is based on a convexity argument and is in the spirit of the original Talagrand's proof. We also discuss the convergence, fluctuations and large deviations of the energy of the eigenvalues of beta ensembles, which gives also yet another proof of the convergence of the eigenvalue distribution to the semicircle law.
We show that the quadratic transportation cost inequality $T_2$ is equivalent to both a Poincaré inequality and a strong form of the Gaussian concentration property. The main ingredient in the proof is a new family of inequalities, called modified quadratic transportation cost inequalities in the spirit of the modified logarithmic-Sobolev inequalities by Bobkov and Ledoux \cite{BL97}, that are shown to hold as soon as a Poincaré inequality is satisfied.
We provide deficit estimates for Nelson's hypercontractivity inequality, the logarithmic Sobolev inequality, and Talagrand's transportation cost inequality under the restriction that the inputs are semi-log-subharmonic, semi-log-convex, or semi-log-concave. In particular, our result on the logarithmic Sobolev inequality complements a recently obtained result by Eldan, Lehec and Shenfeld concerning a deficit estimate for inputs with small covariance. Similarly, our result on Talagrand's transportation cost inequality complements and, for a large class of semi-log-concave inputs, improves a defi…
The purpose of this paper is twofold. Firstly, we prove transportation inequalities ${\bf T_2}(C)$ on the space of continuous paths with respect to the uniform metric for the law of the solution to a class of non-linear monotone stochastic partial differential equations (SPDEs) driven by the Wiener noise. Furthermore, we also establish the ${\bf T_1}(C)$ property for such SPDEs but with merely locally monotone coefficients, including the stochastic Burgers type equation and stochastic $2$-D Navier-Stokes equation.
Let f be a stationary isotropic non-degenerate Gaussian field on R^2. Assume that f = q * W where q is both C^2 and L^2 and W is the L^2 white noise on R^2. We extend a result by Stephen Muirhead and Hugo Vanneuville by showing that, assuming that q * q is pointwise non-negative and has fast enough decay, the set {f > -l} percolates with probability one when l > 0 and with probability zero if l < 0 or l = 0. We also prove exponential decay of crossing probabilities and uniqueness of the unbounded cluster. To this end, we study a Gaussian field g defined on the torus and establish a superconcen…
This note is concerned with an extension, at second order, of an inequality on the discrete cube $C_n=\{-1,1\}$ (equipped with the uniform measure) due to Talagrand (\cite{TalL1L2}). As an application, the main result of this note is a Theorem in the spirit of a famous result from Kahn, Kalai and Linial (cf. \cite{KKL}) concerning the influence of Boolean functions. The notion of the influence of a couple of coordinates $(i,j)\in\{1,\ldots,n\}^2$ is introduced in section 2 and the following alternative is obtained : for any Boolean function $f\,:\, C_n\to \{0,1\}$, either there exists a coordi…
The classical correlation inequality of Harris asserts that any two monotone increasing families on the discrete cube are nonnegatively correlated. In 1996, Talagrand established a lower bound on the correlation in terms of how much the two families depend simultaneously on the same coordinates. Talagrand's method and results inspired a number of important works in combinatorics and probability theory. In this paper we present stronger correlation lower bounds that hold when the increasing families satisfy natural regularity or symmetry conditions. In addition, we present several new classes o…
We establish a dimension-free improvement of Talagrand's Gaussian transport-entropy inequality, under the assumption that the measures satisfy a Poincaré inequality. We also study stability of the inequality, in terms of relative entropy, when restricted to measures whose covariance matrix trace is smaller than the ambient dimension. In case the covariance matrix is strictly smaller than the identity, we give dimension-free estimates which depend on its eigenvalues. To complement our results, we show that our conditions cannot be relaxed, and that there exist measures with covariance larger th…
Recently Talagrand [T] estimated the deviation of a function on $\{0,1\}^n$ from its median in terms of the Lipschitz constant of a convex extension of $f$ to $\ell ^n_2$; namely, he proved that $$P(|f-M_f| > c) \le 4 e^{-t^2/4σ^2}$$ where $σ$ is the Lipschitz constant of the extension of $f$ and $P$ is the natural probability on $\{0,1\}^n$. Here we extend this inequality to more general product probability spaces; in particular, we prove the same inequality for $\{0,1\}^n$ with the product measure $((1-η)δ_0 + ηδ_1)^n$. We believe this should be useful in proofs involving random selections. …
Talagrand-type transport inequalities for path spaces over Carnot groups
研究Carnot群上布朗运动的Talagrand型运输不等式，通过增强路径空间证明并探讨非交换态空间下的新现象。
The optimal transport problem studies how to transport one measure to another in the most cost-effective way and has wide range of applications from economics to machine learning. In this paper, we introduce and study an information constrained variation of this problem. Our study yields a strengthening and generalization of Talagrand's celebrated transportation cost inequality. Following Marton's approach, we show that the new transportation cost inequality can be used to recover old and new concentration of measure results. Finally, we provide an application of this new inequality to network…
Talagrand's influence inequality revisited
本文系统研究Talagrand影响不等式，首次将其推广至Banach空间值函数，引入新的度量不变量Talagrand型，并证明其阻碍超立方体非线性商的双Lipschitz嵌入。
On the stochastic proof of the Blaschke-Santaló inequality
本文通过简化方法，直接推导出函数Santaló不等式，而非其对偶形式，基于对Borel证明的改进。
Using Talagrand's concentration inequality on the discrete cube {0,1}^m we show that given a real-valued function Z(x)on {0,1}^m that satisfies certain monotonicity conditions one can control the deviations of Z(x) above its median by a local Lipschitz norm of Z(x) at the point x. As one application, we give a simple proof of a nearly optimal deviation inequality for the number of k-cycles in a random graph.
We characterize the symmetric measures which satisfy the one dimensional convex infimum convolution inequality of Maurey. For these measures the tensorization argument yields the two level Talagrand's concentration inequalities for their products and convex sets in $\mathbb{R}^n$.
We introduce a symmetrization technique that allows us to translate a problem of controlling the deviation of some functionals on a product space from their mean into a problem of controlling the deviation between two independent copies of the functional. As an application we give a new easy proof of Talagrand's concentration inequality for empirical processes, where besides symmetrization we use only Talagrand's concentration inequality on the discrete cube {-1,+1}^n. As another application of this technique we prove new Vapnik-Chervonenkis type inequalities. For example, for VC-classes of fu…
We obtain a Bernstein type Gaussian concentration inequality for martingales. Our inequality improves the Azuma-Hoeffding inequality for moderate deviations $x$. Following the work of McDiarmid (1989), Talagrand (1996) and Boucheron, Lugosi and Massart (2000,2003), we show that our result can be applied to the concentration of random functions, Erdös-Rényi random graph, and maxima of empirical processes. Several interesting Gaussian concentration inequalities have been obtained.
We show sharp bounds for probabilities of large deviations for sums of independent random variables satisfying Bernstein's condition. One such bound is very close to the tail of the standard Gaussian law in certain case; other bounds improve the inequalities of Bennett and Hoeffding by adding missing factors in the spirit of Talagrand (1995). We also complete Talagrand's inequality by giving a lower bound of the same form, leading to an equality. As a consequence, we obtain large deviation expansions similar to those of Cramér (1938), Bahadur-Rao (1960) and Sakhanenko (1991). We also show that…
An analogue of Talagrand's convex distance for binomial and Poisson point processes is defined. A corresponding large deviation inequality is proved.
On Orlicz spaces satisfying the Hoffmann-Jørgensen inequality
本研究基于Talagrand的相关证明，完整刻画了满足Hoffmann-Jørgensen型不等式的Orlicz函数，推广了多类集中不等式与有界性结论。
In the paper we pursue the analysis from the section 5 of the Talagrand's paper "Sample boundedness of stochastic processes under increment conditions." Ann. Probab. 18, No. 1, 1-49. In particular we give the proof of some Sobolev Inequality and then apply it to obtain if and only if condition for all processes with bounded icrements to have bounded samples. The processes are defined on a compact, concave subspaces of $\R^n$ with a metric $d(s,t)=η(||s-t||)$, where $η$ is concave and $||.||$ is a norm on $\R^n$.
We develop a new technique for proving concentration inequalities which relate between the variance and influences of Boolean functions. Using this technique, we 1. Settle a conjecture of Talagrand [Tal97] proving that $$\int_{\left\{ -1,1\right\} ^{n}}\sqrt{h_{f}\left(x\right)}dμ\geq C\cdot\mathrm{var}\left(f\right)\cdot\left(\log\left(\frac{1}{\sum\mathrm{Inf}_{i}^{2}\left(f\right)}\right)\right)^{1/2},$$ where $h_{f}\left(x\right)$ is the number of edges at $x$ along which $f$ changes its value, and $\mathrm{Inf}_{i}\left(f\right)$ is the influence of the $i$-th coordinate. 2. Strengthen se…
A concentration inequality for random combinatorial optimisation problems
本文针对随机组合优化问题中的最小权重随机变量，通过定义子集族的可修补性并结合Talagrand不等式与喷洒型论证，证明了其围绕均值的尖锐浓度不等式。
Threshold phenomena are investigated using a general approach, following Talagrand [Ann. Probab. 22 (1994) 1576--1587] and Friedgut and Kalai [Proc. Amer. Math. Soc. 12 (1999) 1017--1054]. The general upper bound for the threshold width of symmetric monotone properties is improved. This follows from a new lower bound on the maximal influence of a variable on a Boolean function. The method of proof is based on a well-known logarithmic Sobolev inequality on $\{0,1\}^n$. This new bound is shown to be asymptotically optimal.
We prove that quadratic forms in isotropic random vectors $X$ in $\mathbb{R}^n$, possessing the convex concentration property with constant $K$, satisfy the Hanson-Wright inequality with constant $CK$, where $C$ is an absolute constant, thus eliminating the logarithmic (in the dimension) factors in a recent estimate by Vu and Wang. We also show that the concentration inequality for all Lipschitz functions implies a uniform version of the Hanson-Wright inequality for suprema of quadratic forms (in the spirit of the inequalities by Borell, Arcones-Giné and Ledoux-Talagrand). Previous results of …
We prove that reflected Brownian motion with normal reflections in a convex domain satisfies a dimension free Talagrand type transportation cost-information inequality. The result is generalized to other reflected diffusions with suitable drifts and diffusions. We apply this to get such an inequality for interacting Brownian particles with rank-based drift and diffusion coefficients such as the infinite Atlas model. This is an improvement over earlier dimension-dependent results.
We prove concentration inequalities for general functions of weakly dependent random variables satisfying the Dobrushin condition. In particular, we show Talagrand's convex distance inequality for this type of dependence. We apply our bounds to a version of the stochastic salesman problem, the Steiner tree problem, the total magnetisation of the Curie-Weiss model with external field, and exponential random graph models. Our proof uses the exchangeable pair method for proving concentration inequalities introduced by Chatterjee (2005). Another key ingredient of the proof is a subclass of $(a,b)$…
In this paper, we prove a Talagrand's T2 transportation cost-information inequality for the law of a stochastic wave equation in spatial dimension d=3 driven by the Gaussian random field, white in time and correlated in space, on the continuous paths space with respect to the uniform topology.
We consider a generic modified logarithmic Sobolev inequality (mLSI) of the form $\mathrm{Ent}_μ(e^f) \le \tfracρ{2} \mathbb{E}_μe^f Γ(f)^2$ for some difference operator $Γ$, and show how it implies two-level concentration inequalities akin to the Hanson--Wright or Bernstein inequality. This can be applied to the continuous (e.\,g. the sphere or bounded perturbations of product measures) as well as discrete setting (the symmetric group, finite measures satisfying an approximate tensorization property, \ldots). Moreover, we use modified logarithmic Sobolev inequalities on the symmetric group $S…
We present concentration inequalities on the multislice which are based on (modified) log-Sobolev inequalities. This includes bounds for convex functions and multilinear polynomials. As an application we show concentration results for the triangle count in the $G(n,M)$ Erdős--Rényi model resembling known bounds in the $G(n,p)$ case. Moreover, we give a proof of Talagrand's convex distance inequality for the multislice. Interpreting the multislice in a sampling without replacement context, we furthermore present concentration results for $n$ out of $N$ sampling without replacement. Based on a b…
The aim of this paper is to show that a probability measure concentrates independently of the dimension like a gaussian measure if and only if it verifies Talagrand's $\T_2$ transportation-cost inequality. This theorem permits us to give a new and very short proof of a result of Otto and Villani. Generalizations to other types of concentration are also considered. In particular, one shows that the Poincaré inequality is equivalent to a certain form of dimension free exponential concentration. The proofs of these results rely on simple Large Deviations techniques.
We extend the dimension free Talagrand inequalities for convex distance \cite{talagrand:1995} using an extension of Marton's weak transport \cite{marton:1996a} to other metrics than the Hamming distance. We study the dual form of these weak transport inequalities for the euclidian norm and prove that it implies sub-gaussianity and convex Poincaré inequality \cite{bobkov:gotze:1999a}. We obtain new weak transport inequalities for non products measures extending the results of Samson in \cite{samson:2000}. Many examples are provided to show that the euclidian norm is an appropriate metric for cl…
Initially motivated by the study of the non-asymptotic properties of non-parametric tests based on permutation methods, concentration inequalities for uniformly permuted sums have been largely studied in the literature. Recently, Delyon et al. proved a new Bernstein-type concentration inequality based on martingale theory. This work presents a new proof of this inequality based on the fundamental inequalities for random permutations of Talagrand. The idea is to first obtain a rough inequality for the square root of the permuted sum, and then, iterate the previous analysis and plug this first i…
Graph Convex Hull Bounds as generalized Jensen Inequalities
本文通过研究函数图像的凸包，为任意域上的任意函数给出期望的精确上下界，将 Jensen 不等式推广到有限/无限维及条件期望、Markov 算子情形。
In this article we derive Talagrand's $T_2$ inequality on the path space w.r.t. the maximum norm for various stochastic processes, including solutions of one-dimensional stochastic differential equations with measurable drifts, backward stochastic differential equations, and the value process of optimal stopping problems. The proofs do not make use of the Girsanov method, but of pathwise arguments. These are used to show that all our processes of interest are Lipschitz transformations of processes which are known to satisfy desired functional inequalities.
Sylvester's four point problem for ball-convex bodies
本文研究球凸体的西尔维斯特四点问题，证明Efron-Buchta恒等式成立，确定平面球凸体中三、四点的西尔维斯特概率，推广Groemer不等式并推导相关等周不等式，还给出单位圆盘下西尔维斯特概率的显式结果及外接圆半径的分布函数。
The isoperimetric problem for convex hulls and the large deviations rate functionals of random walks
该研究针对平面随机游走凸包面积大偏差的最概然轨迹渐近行为，证明其对应凸包非齐次各向异性等周问题的解，最优轨迹为凸且满足欧拉-拉格朗日方程，形状与闵可夫斯基平面等周不等式最优解相似。
Gaussian processes can be considered as subsets of a standard Hilbert space, but the geometric understanding that would relate the size of a set with the size of its convex hull is still lacking. In this work, we adopt a geometric approach to the majorizing measure problem by identifying the covering number relationships between a given space $T$ and its convex hull, represented by $T_h$. If the space $T$ is a closed bounded polyhedra in $\mathbb{R}^n$, we can evaluate the volume ratio between the space $T$ and its convex hull obtained by the Quickhull algorithm. If the space $T$ is a general …
This paper is devoted to uniform versions of the Hanson-Wright inequality for a random vector $X \in \mathbb{R}^n$ with independent subgaussian components. The core technique of the paper is based on the entropy method combined with truncations of both gradients of functions of interest and of the coordinates of $X$ itself. Our results recover, in particular, the classic uniform bound of Talagrand (1996) for Rademacher chaoses and the more recent uniform result of Adamczak (2015), which holds under certain rather strong assumptions on the distribution of $X$. We provide several applications of…
Exponential Ergodicity in Relative Entropy and $L^2$-Wasserstein Distance for non-equilibrium partially dissipative Kinetic SDEs
本文证明部分耗散动力学SDE在相对熵和L^2-Wasserstein距离下的指数遍历性，适用于非平衡情形，并推广至McKean-Vlasov及平均场粒子系统。
As a natural analog of Urysohn's inequality in Euclidean space, Gao, Hug, and Schneider showed in 2003 that in spherical or hyperbolic space, the total measure of totally geodesic hypersurfaces meeting a given convex body K is minimized when K is a geodesic ball. We present a random extension of this result by taking K to be the convex hull of finitely many points drawn according to a probability distribution and by showing that the minimum is attained for uniform distributions on geodesic balls. As a corollary, we obtain a randomized Blaschke--Santalo inequality on the sphere.
For a $d$-dimensional random vector $X$, let $p_{n, X}(θ)$ be the probability that the convex hull of $n$ independent copies of $X$ contains a given point $θ$. We provide several sharp inequalities regarding $p_{n, X}(θ)$ and $N_X(θ)$ denoting the smallest $n$ for which $p_{n, X}(θ)\ge1/2$. As a main result, we derive the totally general inequality $1/2 \le α_X(θ)N_X(θ)\le 3d + 1$, where $α_X(θ)$ (a.k.a. the Tukey depth) is the minimum probability that $X$ is in a fixed closed halfspace containing the point $θ$. We also show several applications of our general results: one is a moment-based bo…
A Geometric Perspective on the Injective Norm of Sums of Random Tensors
本文研究了张量集中不等式，为具有相关条目的随机张量的$\ell_p$注入范数建立了非渐近不等式，在特定范围内维度依赖性近乎最优，并基于覆盖数估计提供了几何证明，应用于张量PCA等问题。
Under the Ornstein-Uhlenbeck semigroup $\{U_t\}$, any non-negative measurable $f : \mathbb R^n \to \mathbb R_+$ exhibits a uniform tail bound better than that implied by Markov's inequality and conservation of mass: For every $α\geq e^3$, and $t > 0$, \[ γ_n\left(\left\{x \in \mathbb R^n : U_t f(x) > α\int f\,dγ_n\right\}\right) \leq C(t) \frac{1}α \sqrt{\frac{\log \log α}{\log α}}\] where $γ_n$ is the $n$-dimensional Gaussian measure and $C(t)$ is a constant depending only on $t$. This confirms positively the Gaussian limiting case of Talagrand's convolution conjecture (1989). This is shown t…
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