Central and $L^p$-concentration of 1-Lipschitz maps into $\mathbb{R}$-trees

Log concavity and concentration of Lipschitz functions on the Boolean hypercube

It is well-known that measures whose density is the form $e^{-V}$ where $V$ is a uniformly convex potential on $\RR^n$ attain strong concentration properties. In search of a notion of log-concavity on the discrete hypercube, we consider measures on $\{-1,1\}^n$ whose multi-linear extension $f$ satisfies $\log \nabla^2 f(x) \preceq β\Id$, for $β\geq 0$, which we refer to as $β$-semi-log-concave. We prove that these measures satisfy a nontrivial concentration bound, namely, any Hamming Lipchitz test function $φ$ satisfies $\Var_ν[φ] \leq n^{2-C_β}$ for $C_β>0$. As a corollary, we prove a concent…

Variations and extensions of the Gaussian concentration inequality, Part I

The classical Gaussian concentration inequality for Lipschitz functions is adapted to a setting where the classical assumptions (i.e. Lipschitz and Gaussian) are not met. The theory is more direct than much of the existing theory designed to handle related generalizations. An application is presented to linear combinations of heavy tailed random variables.

Optimal Sample Complexity of Subgradient Descent for Amplitude Flow via Non-Lipschitz Matrix Concentration

We consider the problem of recovering a real-valued $n$-dimensional signal from $m$ phaseless, linear measurements and analyze the amplitude-based non-smooth least squares objective. We establish local convergence of subgradient descent with optimal sample complexity based on the uniform concentration of a random, discontinuous matrix-valued operator arising from the objective's gradient dynamics. While common techniques to establish uniform concentration of random functions exploit Lipschitz continuity, we prove that the discontinuous matrix-valued operator satisfies a uniform matrix concentr…

通过 Föllmer 流构造 Lipschitz 传输映射

Lipschitz Transport Maps via the Follmer Flow

本文受 Föllmer 过程启发构造了欧几里得空间上的 Föllmer 流，证明了其适定性及时间 1 处流映射的 Lipschitz 性质，并利用该映射推导出经验测度的与维度无关的泛函不等式和集中不等式。

A variance-sensitive Gaussian concentration inequality

In this note, we obtain a Gaussian concentration inequality for a class of non-Lipschitz functions. In the one-dimensional case, our results supplement those established by Paouris and Valettas in [8].

Concentration inequalities for non-Lipschitz functions with bounded derivatives of higher order

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 …

Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps

We consider the Markov chain $\{X_n^x\}_{n=0}^\infty$ on $\R^d$ defined by the stochastic recursion $X_{n}^{x}=\p_{θ_{n}}(X_{n-1}^{x})$, starting at $x\in\R^d$, where $θ_{1}, θ_{2},...$ are i.i.d. random variables taking their values in a metric space $(Θ, \mathfrak{r}),$ and $\p_{θ_{n}}:\R^d\mapsto\R^d$ are Lipschitz maps. Assume that the Markov chain has a unique stationary measure $ν$. Under appropriate assumptions on $\p_{θ_n}$, we will show that the measure $ν$ has a heavy tail with the exponent $α>0$ i.e. $ν(\{x\in\R^d: |x|>t\})\asymp t^{-α}$. Using this result we show that properly norm…

A Measure on the space of Lipschitz isometric maps of a compact 1-manifold into $\mathbb R^2$

Let $M$ be a compact 1-manifold. Given a continuous function $g:M\to \mathbb R_+$ we consider the following ordinary differential equation: $\|\dot{f}(t)\|=g(t)$, where $f:M\to \mathbb R^2$. We construct a probability measure on the space of almost everywhere differentiable solutions of this differential equation and study this measure. A solution of this equation can be viewed as an isometric immersion of a compact 1-manifold into $\mathbb R^2$. Nash's convergence technique in the proof of isometric $C^1$-immersion theorem plays an important role in the construction.

Lipschitz传输映射的耦合方法

A coupling approach to Lipschitz transport maps

本文提出一种基于耦合技术的概率方法，用于界定Langevin流映射的Lipschitz常数，并成功将其应用于构造对数凹测度到对数Lipschitz扰动的传输映射，同时放宽了凸性条件并移除了势能三阶导数界限。

热流、对数凹性与Lipschitz传输映射

Heat flow, log-concavity, and Lipschitz transport maps

本文研究热方程解的log-Hessian估计，对强对数凹测度的对数Lipschitz扰动给出显式一致下界，并据此估计从标准高斯到该类测度的Lipschitz传输映射常数，同时讨论与分数扩散模型及对数Sobolev不等式的联系。

Hamming立方体上向量值函数的测度集中

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。

Concentration of measure, classification of submeasures, and dynamics of $L_{0}$

Exhibiting a new type of measure concentration, we prove uniform concentration bounds for measurable Lipschitz functions on product spaces, where Lipschitz is taken with respect to the metric induced by a weighted covering of the index set of the product. Our proof combines the Herbst argument with an analogue of Shearer's lemma for differential entropy. We give a quantitative "geometric" classification of diffuse submeasures into elliptic, parabolic, and hyperbolic. We prove that any non-elliptic submeasure (for example, any measure, or any pathological submeasure) has a property that we call…

Concentration Inequalities in Locally Dependent Spaces

This paper studies concentration inequalities for functions of locally dependent random variables. We show that the usual definition of local dependence does not imply concentration for general Hamming Lipschitz functions. We define hypergraph dependence, which is a special case of local dependence, and show that it implies concentration if the maximal neighborhood size is small. We prove concentration in Hamming distance, Talagrand distance, and for self-bounding functions of a particular type under this dependence structure.

Median, Concentration and Fluctuation for Lévy Processes

We estimate a median of $f(X_t)$ where $f$ is a Lipschitz function, $X$ is a Lévy process and $t$ an arbitrary time. This leads to concentration inequalities for $f(X_t)$. In turn, corresponding fluctuation estimates are obtained under assumptions typically satisfied if the process has a regular behavior in small time and a, possibly different, regular behavior in large time.

On finite range stable type concentration

The purpose of these notes is to further complete our understanding of the stable concentration phenomenon, by obtaining the finite range behavior of $P(F-E[F]\geq x)$, with $F=f(X)$ where $f$ is a Lipschitz function and $X$ is a stable random vector or with $F$ a stochastic functional on the Poisson space equipped with a stable Lévy measure.

乘积空间中凸Lipschitz函数的维数依赖型集中性研究

On Dimension-dependent concentration for convex Lipschitz functions in product spaces

该研究针对乘积空间中分量为独立K--次高斯变量的随机向量，推导了1--Lipschitz凸函数的维数依赖型集中不等式，证明了估计的最优性，并对比了有限ψ_p范数变量的情形。

到对数-Lipschitz扰动的运输

Transportation onto log-Lipschitz perturbations

该研究建立概率测度与其对数-Lipschitz扰动间具维度无关界的全局Lipschitz运输映射存在的充分条件，证明非欧几里得情形下首批此类映射的最优性，并将其应用于函数不等式得到新的高斯等周不等式。

Concentration of Broadcast Models on Trees

An inequality of K. Marton shows that the joint distribution of a Markov chain with uniformly contracting transition kernels exhibits concentration. We prove an analogous inequality for broadcast models on finite trees. We use this inequality to develop a condition for the sequence of depth-$k$ marginals of a broadcast model on a rooted infinite tree to form a normal Lévy family in terms of the Lipschitz constants of the transition kernels and the growth rate of the tree.

通过集中不等式估计最优传输映射的增长

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。