Sharp variance-entropy comparison for nonnegative Gaussian quadratic forms

基于方差的相依敏感性指数与Shapley效应之间的尖锐不等式：上界

Sharp inequalities between variance-based dependent sensitivity indices and Shapley effects: upper-bounds

研究输入相依时基于方差的Shapley效应与相依敏感性指数之间的不等式关系，证明Shapley效应介于主效应与总效应之间，并给出多种上界以简化高维非相关输入识别。

Sharp Lower and Upper Bounds for the Covariance of Bounded Random Variables

In this paper we derive sharp lower and upper bounds for the covariance of two bounded random variables when knowledge about their expected values, variances or both is available. When only the expected values are known, our result can be viewed as an extension of the Bhatia-Davis Inequality for variances. We also provide a number of different ways to standardize covariance. For a binary pair random variables, one of these standardized measures of covariation agrees with a frequently used measure of dependence between genetic variants.

锐利的矩阵经验伯恩斯坦不等式

Sharp Matrix Empirical Bernstein Inequalities

提出了两个针对具有有界特征值的对称随机矩阵的锐利闭式经验伯恩斯坦不等式，其偏差能紧凑适应未知方差，分别适用于独立矩阵样本均值和停时下鞅依赖的均值估计量。

Multivariate Concentration Inequalities with Size Biased Couplings

Let $\mathbf{W}=(W_1,W_2,...,W_k)$ be a random vector with nonnegative coordinates having nonzero and finite variances. We prove concentration inequalities for $\mathbf{W}$ using size biased couplings that generalize the previous univariate results. Two applications on local dependence and counting patterns are provided.

从柯西到高斯：尖锐下偏差与埃哈德型不等式

From Cauchy to Gaussian: Sharp Lower Deviations and Ehrhard-Type Inequality

该研究针对球对称柯西型测度下的凸函数建立了与维度无关的尖锐下偏差界，证明了柯西型测度下凸集的埃哈德型不等式，解决了相关猜想，强化了高斯情形的已有结果。

Critical points of multidimensional random Fourier series: variance estimates

To any positive number $\varepsilon$ and any nonnegative even Schwartz function $w:\mathbb{R}\to\mathbb{R}$ we associate the random function $u^\varepsilon$ on the $m$-torus $T^m_\varepsilon:=\mathbb{R}^m/(\varepsilon^{-1}\mathbb{Z})^m$ defined as the real part of the random Fourier series $$ \sum_{ν\in\mathbb{Z}^m} X_{ν,\varepsilon} \exp\bigl(\; 2π\varepsilon \sqrt{-1} \;(ν\cdot θ)\;\bigr),$$ where $X_{ν,\varepsilon}$ are complex independent Gaussian random variables with variance $w(\varepsilon|ν|)$. Let $N^\varepsilon$ denote the number of critical points of $u^\varepsilon$. We describe exp…

独立均匀随机变量和的高斯尾界

A Sharp Gaussian Tail Bound for Sums of Uniforms

基于独立均匀随机变量和的高斯尾界，证明其尾概率受高斯尾约束，并确定最优常数。

Bounds for covariances and variances of truncated random variables

We show that a lower bound for covariance of $\min(X_1,X_2)$ and $\max(X_1,X_2)$ is $\cov{X_1}{X_2}$ and an upper bound for variance of \\ $\min(X_2,\max(X,X_1))$ is $\var{X} + \var{X_1} +\var{X_2}$ generalizing previous results. We also characterize the cases where these bounds are sharp.

Variance linearity for real Gaussian zeros

We investigate the zero set of a stationary Gaussian process on the real line, and in particular give lower bounds for the variance of the number of points on a large interval, in all generality. We prove that this point process is never hyperuniform, i.e. the variance is at least linear, and give necessary conditions to have linear variance, which are close to be sharp. We study the class of symmetric Bernoulli convolutions and give an example where the zero set is super rigid, weakly mixing, and not hyperuniform.

Estimating a sharp convergence bound for randomized ensembles

When randomized ensembles such as bagging or random forests are used for binary classification, the prediction error of the ensemble tends to decrease and stabilize as the number of classifiers increases. However, the precise relationship between prediction error and ensemble size is unknown in practice. In the standard case when classifiers are aggregated by majority vote, the present work offers a way to quantify this convergence in terms of "algorithmic variance," i.e. the variance of prediction error due only to the randomized training algorithm. Specifically, we study a theoretical upper …

精确的集中不等式：Orlicz尾部的相变与混合

Sharp Concentration Inequalities: Phase Transition and Mixing of Orlicz Tails with Variance

本文研究了子Weibull随机变量的精确集中不等式，揭示了在α=2时的相变现象，并通过min和max函数结合方差和OrliczΨ_α尾部，提出新的改进不等式，应用于各种统计和机器学习场景。