Transport Inequalities. A Survey
Nathael Gozlan, Christian Léonard
这是一份关于运输不平等领域最新动态的概览。我们调查其后果的浓度和偏差的不平等和素描他们的联系与其他功能的不平等和大偏差理论。
This is a survey of recent developments in the area of transport inequalities. We investigate their consequences in terms of concentration and deviation inequalities and sketch their links with other functional inequalities and also large deviation theory.
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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.
Characterization of Talagrand's transport-entropy inequalities in metric spaces
给出了度量空间中输运熵不等式的一个刻划。作为应用，我们证明了这类不等式在有界扰动下是稳定的（Holley-Stroock扰动引理）.
Transport-entropy inequalities on the line
给出了一维输运熵不等式成立的一个充要条件。作为应用，我们构造了一个新的概率分布的例子来验证Talagrand的T2不等式，而不是对数Sobolev不等式.
Transport proofs of weighted Poincaré inequalities for log-concave distributions
利用最优输运工具证明了对数凹随机向量满足中心条件的加权Poincar不等式.通过这种方法，我们恢复了Klartag和Barthe-Cordero-Erausquin对具有对称性的对数凹随机向量的类似结果。此外，我们还证明了方差猜想对对数凹鞅的增量是成立的。
We study an optimal weak transport cost related to the notion of convex order between probability measures. On the real line, we show that this weak transport cost is reached for a coupling that does not depend on the underlying cost function. As an application, we give a necessary and sufficient condition for weak transport-entropy inequalities in dimension one. In particular, we obtain a weak transport-entropy form of the convex Poincar{é} inequality in dimension one.
Transport inequalities for random point measures
本文研究随机点测度的传输-熵不等式，证明满足Talagrand传输不等式的有限强度测度对应的点过程也满足该类不等式，且任意σ-有限强度的泊松点过程满足Marton型普适不等式，拓展了Reitzner的偏差不等式。
Hamilton Jacobi equations on metric spaces and transport-entropy inequalities
本文证明了一般度量空间上一类Hamilton- Jacobi方程解的Hopf-Lax-Oleinik公式。作为第一个结果，我们在全通性中证明了log-Sobolev不等式等价于Hamilton-Jacobi半群的超压缩性。作为第二个推论，我们证明了度量空间中的Talagrand输运熵不等式可用限制在c-凸函数类上的log-Sobolev不等式来刻画.
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…
Following Talagrand's concentration results for permutations picked uniformly at random from a symmetric group [Tal95], Luczak and McDiarmid have generalized it to more general groups G of permutations which act suitably 'locally'. Here we extend their results by setting transport-entropy inequalities on these permutations groups. Talagrand and Luczak-Mc-Diarmid concentra- tion properties are consequences of these inequalities. The results are also gen- eralised to a larger class of measures including Ewens distributions of arbitrary parameter $θ$ on the symmetric group. By projection, we deri…
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。
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 …
Properties of Isoperimetric, Functional and Transport-Entropy Inequalities Via Concentration
研究了带有测度的黎曼流形上的等周不等式、泛函不等式、输运熵不等式和浓度不等式的各种性质，其中广义里奇曲率是下有界的.首先，得到了这些不等式关于测度扰动的稳定性。扰动的程度是用扰动和原始测量之间的几个不同的距离来测量的，如单侧$L^\infty$约束在它们的密度之间的比率，Wasserstein距离和Kullback-Leibler发散。特别地，得到了log-Sobolev不等式的Holley-Stroock扰动引理的一个推广，并将其对扰动参数的依赖性从线性改进为对数.其次，通过得到一个逆Jensen型不等式，证明了不同代价函数下的输运熵不等式的等价性。所使用的主要工具是以前的精确结果之间的等价性浓度和等周不等式在所述的设置。独立的兴趣是一个新的维数独立的特征的输运熵不等式的1-Wasserstein距离，它不承担任何曲率下界。
Log-Sobolev, isoperimetry and transport inequalities on graphs
本文利用（随机）路方法研究了有限连通图上可逆最近邻马氏过程的一些函数不等式（如Poincaré不等式、对数Sobolev不等式、广义Cheeger等周不等式、迁移信息不等式和迁移熵不等式）.我们提供了所涉及的常数的估计。
Mass transport and variants of the logarithmic Sobolev inequality
我们发展了修正的log-Sobolev不等式和等周不等式的最优运输方法。给出了这类不等式成立的各种充分条件。其中一些是新的，即使在经典的log-Sobolev的情况下。许多这些条件背后的想法是，具有非凸势的测度可以享受这样的函数不等式，只要它们具有强的可积性，可以平衡凸性的缺乏。此外，几个已知的标准恢复在一个简单的统一的方式通过运输的方法和推广到黎曼设置。
Curvature and transport inequalities for Markov chains in discrete spaces
在三种不同的Ricci曲率概念下研究了离散情形下的各种输运信息不等式：Bakry和Émery的曲率-维数条件，Bauer的指数曲率-维数条件\textit{et al.}和Ollivier的粗里奇曲率.证明了在曲率维数条件或粗里奇曲率条件下，一个$L_1$输运信息不等式成立，而在指数曲率维数条件下，一些弱输运信息不等式成立.作为应用，我们在Bakry和Emery的曲率维数条件CD $(κ,\infty)$下建立了一个Bonnet-Meyer定理.
Stein's method, logarithmic Sobolev and transport inequalities
通过引入一类新的函数不等式，我们建立了Stein近似方法、对数Sobolev和输运不等式之间的联系，这些函数不等式涉及相对熵、Stein核、相对Fisher信息和Wasserstein距离，它们与$\mathbb{R}^d$上的给定参考分布有关.对于高斯模型，所得结果改进了经典的对数Sobolev不等式和Talagrand二次运输费用不等式。进一步的例子包括多维Γ分布、β分布以及对数凹密度族。作为一个副产品，新的不等式被证明是相关的收敛到平衡，浓度不等式和熵收敛的Stein内核表示。该工具依赖于半群插值和边界，特别是通过迭代梯度的马尔可夫生成器与不变测度的分布考虑。在第二部分中，由Nourdin，Peccati和天鹅的维纳混沌最近的调查的动机，我们解决了这个问题的熵界的多维泛函$F$与斯坦内核通过一组数据上的$F$和它的梯度，而不是Fisher信息的密度。马尔可夫三重结构$(E, μ, Γ)$给出了一个自然的框架，其中抽象的Malliavin型参数可以被开发并扩展Wiener混沌设置。
We relate transport-entropy inequalities to the study of critical points of functionals defined on the space of probability measures. This approach leads in particular to a new proof of a result by Otto and Villani [43] showing that the logarithmic Sobolev inequality implies Talagrand's transport inequality.
This paper explores the connection between a generalized Riesz electric energy and norms on the set of probability measures defined in terms of duality. We derive functional inequalities linking these two notions, recovering and generalizing existing Coulomb transport inequalities. We then use them to prove concentration of measure around the equilibrium and thermal equilibrium measures. Finally, we leverage these concentration inequalities to obtain Moser-Trudinger-type inequalities, which may also be interpreted as bounds on the Laplace transform of fluctuations.
A large deviation approach to some transportation cost inequalities
利用初等的大偏差推理，导出了新的运输费用不等式。证明了它们的对偶特征，推广了S. Bobkov和F.格策研究了它们的张量化性质。这些不等式的充分条件（和必要条件太）的参考措施的可积性。应用这些结果导致新的偏差结果：测量的集中和经验过程的偏差。
Functional Inequalities via Lyapunov conditions
我们在这里回顾了作者最近的一些结果，和各种合作者，（弱，超）Poincaré不等式，运输信息不等式或对数Sobolev不等式通过一个非常简单和有效的技术：李雅普诺夫条件。
Weak transport inequalities and applications to exponential inequalities and oracle inequalities
我们推广了无量纲Talagrand不等式的凸距离\cite{talagrand:1995}使用扩展的Marton的弱输运\cite{marton:1996a}的其他度量比汉明距离。研究了这类弱迁移不等式在欧氏范数下的对偶形式，证明了它蕴含次高斯性和凸Poincaré不等式\cite{bobkov:gotze:1999a} .我们得到了非乘积测度的弱迁移不等式，推广了Samson在\cite{samson:2000}中的结果.许多例子表明，欧几里德范数是一个适当的度量经典时间序列。我们的方法，基于轨迹耦合，是更有效地获得无量纲浓度比现有的收缩假设\cite{djellout:guillin:wu:2004,marton:2004}。将普通最小二乘估计的浓度性质表示为一个条件弱迁移问题，得到了新的具有快速收敛速度的Oracle不等式。
Functional inequalities for Gaussian convolutions of compactly supported measures: explicit bounds and dimension dependence
本文的目的是建立各种功能不等式的卷积的一个紧支撑的措施和一个标准的高斯分布的Rd。我们特别关注于获得常数对维数的良好依赖性。证明了Poincar{é}不等式在无量纲界下成立.对于对数Sobolev不等式，我们通过得到随维数线性增长的界来改进最著名的结果（Zimmermann，JFA 2013）。我们还建立了各种运输成本的运输熵不等式。
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。
The purpose of this note is to show how simple Optimal Transport arguments, on the real line, can be used in Superconcentration theory. This methodology is efficient to produce sharp non-asymptotic variance bounds for various functionals (maximum, median, $l^p$ norms) of standard Gaussian random vectors in $\R^n$. The flexibility of this approach can also provide exponential deviation inequalities reflecting preceding variance bounds. As a further illustration, usual laws from Extreme theory and Coulomb gases are studied.
A new characterization of Talagrand's transport-entropy inequalities and applications
我们证明了Talagrand的输运不等式等价于一个限制的对数Sobolev不等式。这一结果澄清了这两个重要的函数不等式之间的联系。作为应用，我们首次证明了Talagrand不等式在有界扰动下是稳定的。
We introduce a general notion of transport cost that encompasses many costs used in the literature (including the classical one and weak transport costs introduced by Talagrand and Marton in the 90's), and prove a Kantorovich type duality theorem. As a by-product we obtain various applications in different directions: we give a short proof of a result by Strassen on the existence of a martingale with given marginals, we characterize the associated transport-entropy inequalities together with the log-Sobolev inequality restricted to convex/concave functions. Some explicit examples of discrete m…
On Sobolev regularity of mass transport and transportation inequalities
研究了概率测度$μ=e^{-V} \ dx$和$\R^d$上的$ν=e^{-W} \ dx$之间的最优运输$T = \nabla Φ$的Sobolev先验估计.假设均匀凸的潜在$W$，我们表明，$\int \| D^2 Φ\|^2_{HS} \ dμ$，其中$\|\cdot\|_{HS}$是希尔伯特-施密特范数，是由Fisher信息的$μ$。此外，我们证明了$\|D^2 Φ\|$的$L^p(μ)$ -模的类似估计，并得到了著名的Caffarelli压缩定理的一些$L^p$ -推广.我们建立了一个连接，我们的结果与Talagrand运输不等式。我们还证明了一个相应的无量纲版本的相对Fisher信息相对于高斯测度。
We study the non-asymptotic behavior of Coulomb gases in dimension two and more. Such gases are modeled by an exchangeable Boltzmann-Gibbs measure with a singular two-body interaction. We obtain concentration of measure inequalities for the empirical distribution of such gases around their equilibrium measure, with respect to bounded Lipschitz and Wasserstein distances. This implies macroscopic as well as mesoscopic convergence in such distances. In particular, we improve the concentration inequalities known for the empirical spectral distribution of Ginibre random matrices. Our approach is re…
Shifted Composition I: Harnack and Reverse Transport Inequalities
提出移位复合规则这一信息论新原理，通过辅助移位界定随机过程分布间的散度，借此证明扩散过程的反向传输不等式并蕴含维度无关的 Harnack 不等式，同时连接了连续与离散时间方法。
Free transport-entropy inequalities for non-convex potentials and application to concentration for random matrices
塔拉格兰德不等式将概率的两个基本概念联系起来：测度的传输和熵。在自由概率的背景下，这些不等式的对应物的研究由Biane和Voiculescu发起，后来由Hiai，Petz和Ueda扩展为凸势。在这项工作中，我们证明了一个自由模拟的结果Bobkov和Götze在经典的设置，从而提供了一个非常自然的一类措施出现在随机矩阵理论的自由运输熵不等式。这些不等式比Hiai，Petz和Ueda的不等式弱，但在凸情形之外仍然成立。然后，我们使用这个结果得到一个浓度估计$β$ -合奏温和的假设下的潜力。
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 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。
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.
Transport-entropy inequalities and deviation estimates for stochastic approximation schemes
我们得到了新的输运熵不等式，作为副产品，新的偏差估计的两种离散随机逼近计划的法律。第一个是指在一个固定的确定性的日期和第二个涉及的法律的随机逼近算法在一个给定的时间步长的扩散过程的欧拉样离散计划的法律。我们的结果显着改善和完善[Frikha，Menozzi，2012]中获得的结果。关键是要适当地量化的扩散项的浓度制度的贡献。我们还得到了一个一般的非渐近偏差界的连续欧拉计划的轨迹与扩散过程和它的平均值之间的差异的功能。最后，我们得到了轨迹平均的随机逼近的非渐近界，特别是我们证明了，平均一个随机逼近算法与一个缓慢下降的步骤序列产生的最佳浓度率。
Global solutions of a free boundary problem via mass transport inequalities
我们研究了一个自由边界问题，它是在傅立叶定律的背景下随机粒子系统的连续形式。经典解的局部存在唯一性在自由边界问题的文献中是众所周知的。我们引入了广义解的概念（当经典解存在时，它是经典解的推广），并证明了对于一大类初始数据，广义解的整体存在性和唯一性.证明是通过表征广义解的唯一元素，它分离适当定义的上下障碍的意义上的质量传输不等式。
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 …
Smoothing inequalities for transport metrics in compact spaces
本文在紧致流形上建立了Wasserstein度量与傅里叶变换之间的上界估计，首次覆盖$p$全范围，并应用于证明球面设计的最优均匀性。
On maximal inequalities for purely discontinuous martingales in infinite dimensions
本文研究了无穷维空间中关于Poisson测度的一类极大不等式，并给出了关于Poisson测度的随机积分和卷积的极大不等式.这类极大不等式在带跳型噪声的随机偏微分方程的研究中具有重要意义。
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。
A Contraction Theory for Sinkhorn and Schrodinger Bridges via Log-Sobolev Inequalities
本文基于Log-Sobolev不等式，提出了一种用于Sinkhorn和Schrodinger桥的收缩理论，通过运输成本不等式和Riccati方程获得熵和Wasserstein收缩界，并在高斯参考转移下得到精确的收敛估计。
Non-local Wasserstein Geometry, Gradient Flows, and Functional Inequalities for Stationary Point Processes
该研究在平稳点过程空间构造非局部Benamou-Brenier型传输距离，证明其为[DSHS24]中传输距离的变体，揭示Ornstein-Uhlenbeck半群是相对熵的梯度流，还建立了平稳测地线存在性、相对熵的1-测地凸性及泛函不等式的平稳类似物。
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…
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