Leonid Kontorovich
We prove an elementary yet useful inequality bounding the maximal value of certain linear programs. This leads directly to a bound on the martingale difference for arbitrarily dependent random variables, providing a generalization of some recent concentration of measure results. The linear programming inequality may be of independent interest.
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We prove that a probability measure on an abstract metric space satisfies a non trivial dimension free concentration inequality for the $\ell_2$ metric if and only if it satisfies the Poincaré inequality.
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 prove a concentration inequality which asserts that, under some mild regularity conditions, every random variable defined on the product of sufficiently many probability spaces exhibits pseudorandom behavior.
Given a suitably normalized $X\in\mathbb{R}^n$ we observe that the function $θ\mapsto\mathbb{E}|X\cdotθ|$, defined for $θ\in S^{n-1}$, admits surprisingly strong concentration far surpassing what is expected on account of Lévy's isoperimetric inequality. Among the measures to which the above holds are all log-concave measures, for which a solution of the similar problem concerning the third marginal moments $θ\mapsto\mathbb{E} (X\cdot θ)^3$ would imply the hyperplane conjecture.
A note on time-uniform concentration inequality for matrix products
本文通过简单论证，将固定时间界的矩阵乘积集中性界推广到任意时间界，受Oja算法分析启发。
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.
Measure concentration for vector valued functions on Hamming cube
本文证明了Hamming立方体上取值于有限cotype Banach空间的Lipschitz函数满足测度集中不等式，推广了经典标量结果。
We are interested in Sobolev type inequalities and their relationship with concentration properties on higher dimensions. We consider unbounded spin systems on the d-dimensional lattice with interactions that increase slower than a quadratic. At first we assume that the one site measure satisfies a Modified log-Sobolev inequality with a constant uniformly on the boundary conditions and we determine conditions so that the infinite dimensional Gibbs measure satisfies a concentration as well as a Talagrand type inequality. Then a Modified Log-Sobolev type concentration property is obtained under …
The Euclidean concentration inequality states that, among sets with fixed volume, balls have $r$-neighborhoods of minimal volume for every $r>0$. On an arbitrary set, the deviation of this volume growth from that of a ball is shown to control the square of the volume of the symmetric difference between the set and a ball. This sharp result is strictly related to the physically significant problem of understanding near maximizers in the Riesz rearrangement inequality with a strictly decreasing radially decreasing kernel. Moreover, it implies as a particular case the sharp quantitative Euclidean…
We establish an Azuma type inequality under a Lipshitz condition for martingales in the framework of noncommutative probability spaces and apply it to deduce a noncommutative Heoffding inequality as well as a noncommutative McDiarmid type inequality. We also provide a noncommutative Azuma inequality for noncommutative supermartingales in which instead of a fixed upper bound for the variance we assume that the variance is bounded above by a linear function of variables. We then employ it to deduce a noncommutative Bernstein inequality and an inequality involving $L_p$-norm of the sum of a marti…
The aim of this paper is to prove an improved version of the bounded differences inequality for matrix valued functions, by developing the methods of Mackey et al.: "Matrix Concentration Inequalities via the Method of Exchangeable Pairs". Along the way, we prove new trace inequalities for the matrix exponential.
Almost minimizing Yang$-$Mills fields: log-epiperimetric inequality, non-concentration, and uniqueness of tangents
本文为任意维度Yang-Mills场建立直接的对数周长不等式，并据此证明能量极小化Yang-Mills场和$ω$-ASD连接的孤立奇点切锥唯一性，同时通过Luckhaus型引理排除blow-up序列的曲率集中。
We prove an extension of McDiarmid's inequality for metric spaces with unbounded diameter. To this end, we introduce the notion of the {\em subgaussian diameter}, which is a distribution-dependent refinement of the metric diameter. Our technique provides an alternative approach to that of Kutin and Niyogi's method of weakly difference-bounded functions, and yields nontrivial, dimension-free results in some interesting cases where the former does not. As an application, we give apparently the first generalization bound in the algorithmic stability setting that holds for unbounded loss functions…
In this note we study how a concentration phenomenon can be transmitted from one measure $μ$ to a push-forward measure $ν$. In the first part, we push forward $μ$ by $π:supp(μ)\rightarrow \Ren$, where $πx=\frac{x}{\norm{x}_L}\norm{x}_K$, and obtain a concentration inequality in terms of the medians of the given norms (with respect to $μ$) and the Banach-Mazur distance between them. This approach is finer than simply bounding the concentration of the push forward measure in terms of the Banach-Mazur distance between $K$ and $L$. As a corollary we show that any normed probability space with good…
In this paper, suppose $F: \mathbb{R}^{N} \rightarrow [0, +\infty)$ be a convex function of class $C^{2}(\mathbb{R}^{N} \backslash \{0\})$ which is even and positively homogeneous of degree 1. We establish the Lions type concentration-compactness principle of singular Trudinger-Moser Inequalities involving $N$-Finsler--Laplacian operator. Let $Ω\subset \mathbb{R}^{N}(N\geq 2)$ be a smooth bounded domain. $\{u_n\}\subset W_0^{1, N}(Ω)$ be a sequence such that anisotropic Dirichlet norm$\int_ΩF^N (\nabla u_n)dx=1$, $u_n \rightharpoonup u \not \equiv 0$ weakly in $W_0^{1, N}(Ω)$. Then for any $0 …
Concentration comparison for nonlinear diffusion on model manifolds and Pólya-Szegő inequality
本文在球对称模型流形上研究非线性扩散方程的集中性比较，证明其成立当且仅当流形支持Pólya-Szegő不等式，并给出标量曲率条件及中心等周不等式的充分条件，涵盖双曲空间与球面。
We prove a log-Sobolev inequality for a certain class of log-concave measures in high dimension. These are the probability measures supported on the unit cube in R^n whose density takes the form exp(-H) where the function H is assumed to be convex (but not strictly convex) with bounded pure second derivatives. Our argument relies on a transportation-cost inequality a la Talagrand.
We prove a tight subspace concentration inequality for the dual curvature measures of a symmetric convex body.
Building on the inequalities for homogeneous tetrahedral polynomials in independent Gaussian variables due to R. Latała we provide a concentration inequality for non-necessarily Lipschitz functions $f\colon \R^n \to \R$ with bounded derivatives of higher orders, which hold when the underlying measure satisfies a family of Sobolev type inequalities $\|g- \E g\|_p \le C(p)\|\nabla g\|_p.$ Such Sobolev type inequalities hold, e.g., if the underlying measure satisfies the log-Sobolev inequality (in which case $C(p) \le C\sqrt{p}$) or the Poincaré inequality (then $C(p) \le Cp$). Our concentration …
Operator capacity, the Brascamp--Lieb inequality and geometric programming
本文通过几何规划视角探讨算子容量与布拉斯坎普-利布常数，提出新的研究成果，包括算子容量的近似极小值和局部霍尔德正则性，以及推广到有向图数据容量的可能。
We propose a power type approximation of the Moser--Trudinger functional and show that its concentration level converges to the Carleson--Chang limit.
In a previous work, the authors introduced a Lin-Lu-Yau type Ricci curvature for directed graphs referring to the formulation of the Chung Laplacian. The aim of this note is to provide a von Renesse-Sturm type characterization of our lower Ricci curvature bound via a gradient estimate for the heat semigroup, and a transportation inequality along the heat flow. As an application, we will conclude a concentration of measure inequality for directed graphs of positive Ricci curvature.
Concentration of invariant means and dynamics of chain stabilizers in continuous geometries
本研究针对适配顺从拓扑子群链的不变均值证明了基于Azuma鞅不等式的集中不等式，提供建立极端顺从性的方法，构造了源自von Neumann连续几何的新极端顺从群例子，并解答了Pestov关于顺从群直积动力学集中性的问题。
We prove concentration results for $\ell_p^n$ operator norms of rectangular random matrices and eigenvalues of self-adjoint random matrices. The random matrices we consider have bounded entries which are independent, up to a possible self-adjointness constraint. Our results are based on an isoperimetric inequality for product spaces due to Talagrand.
We prove a version of Talagrand's concentration inequality for subordinated sub-Laplacian on a compact Riemannian manifold using tools from noncommutative geometry. As an application, motivated by quantum information theory, we show that on a finite dimensional matrix algebra the set of self-adjoint generators satisfying a tensor stable modified logarithmic Sobolev inequality is dense.
Quantum functional inequalities (e.g. the logarithmic Sobolev- and Poincaré inequalities) have found widespread application in the study of the behavior of primitive quantum Markov semigroups. The classical counterparts of these inequalities are related to each other via a so-called transportation cost inequality of order 2 (TC2). The latter inequality relies on the notion of a metric on the set of probability distributions called the Wasserstein distance of order 2. (TC2) in turn implies a transportation cost inequality of order 1 (TC1). In this paper, we introduce quantum generalizations of …
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