Mark Rudelson, Roman Vershynin
In this expository note, we give a modern proof of Hanson-Wright inequality for quadratic forms in sub-gaussian random variables. We deduce a useful concentration inequality for sub-gaussian random vectors. Two examples are given to illustrate these results: a concentration of distances between random vectors and subspaces, and a bound on the norms of products of random and deterministic matrices.
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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 …
A note on the improved sparse Hanson-Wright inequalities
本文为稀疏 α-次指数随机向量的二次型建立并改进 Hanson-Wright 不等式，在 α≤1 时得到典型模型下的最优界，并展示其在协方差估计、低秩逼近和集中不等式中的应用。
The Hanson-Wright inequality is an upper bound for tails of real quadratic forms in independent subgaussian random variables. In this work, we extend the Hanson-Wright inequality for the maximum eigenvalue of the quadratic sum of random Hermitian tensors under Einstein product. We first prove Weyl inequality for tensors under Einstein product and apply this fact to separate the quadratic form of random Hermitian tensors into diagonal sum and coupling (non-diagonal) sum parts. For the diagonal part, we can apply Bernstein inequality to bound the tail probability of the maximum eigenvalue of the…
A Gaussian Chain Rule Proof of the Banach Space Hanson Wright Bound
该论文提出高斯链式法则证明，用矩阵范数和混沌上界界定有限实矩阵像椭球并集的高斯宽度。
The Hanson-Wright inequality is an upper bound for tails of real quadratic forms in independent random variables. In this work, we extend the Hanson-Wright inequality for the Ky Fan k-norm for the polynomial function of the quadratic sum of random tensors under Einstein product. We decompose the quadratic tensors sum into the diagonal part and the coupling part. For the diagonal part, we can apply the generalized tensor Chernoff bound directly. But, for the coupling part, we have to apply decoupling method first, i.e., decoupling inequality to bound expressions with dependent random tensors wi…
Uniform Hanson-Wright Type Deviation Inequalities for $α$-Subexponential Random Vectors
本文提出独立中心化 $α$-次指数随机向量的一致 Hanson-Wright 型偏差不等式，借助新型解耦及强弱矩比较，并用于证明部分随机循环矩阵的 RIP。
An Elementary Proof of the Hanson-Wright Inequality
本文针对独立次高斯向量二次型的Hanson-Wright不等式，提出一种无需Bourgain凸解耦、仅利用次高斯性质与高斯旋转不变性的更初等证明，并获得改进常数。
We provide moment bounds for expressions of the type $(X^{(1)} \otimes \dots \otimes X^{(d)})^T A (X^{(1)} \otimes \dots \otimes X^{(d)})$ where $\otimes$ denotes the Kronecker product and $X^{(1)}, \dots, X^{(d)}$ are random vectors with independent, mean 0, variance 1, subgaussian entries. The bounds are tight up to constants depending on $d$ for the case of Gaussian random vectors. Our proof also provides a decoupling inequality for expressions of this type. Using these bounds, we obtain new, improved concentration inequalities for expressions of the form $\|B (X^{(1)} \otimes \dots \otimes…
On The Absolute Constant in Hanson-Wright Inequality
本研究重新审视并微调高斯Hanson-Wright不等式的证明，重点追踪该不等式表述中的绝对常数，推进相关概率不等式的常数精细化研究。
We derive a dimension-free Hanson-Wright inequality for quadratic forms of independent sub-gaussian random variables in a separable Hilbert space. Our inequality is an infinite-dimensional generalization of the classical Hanson-Wright inequality for finite-dimensional Euclidean random vectors. We illustrate an application to the generalized $K$-means clustering problem for non-Euclidean data. Specifically, we establish the exponential rate of convergence for a semidefinite relaxation of the generalized $K$-means, which together with a simple rounding algorithm imply the exact recovery of the t…
We discuss two-sided bounds for moments and tails of quadratic forms in Gaussian random variables with values in Banach spaces. We state a natural conjecture and show that it holds up to additional logarithmic factors. Moreover in a certain class of Banach spaces (including $L_r$-spaces) these logarithmic factors may be eliminated. As a corollary we derive upper bounds for tails and moments of quadratic forms in subgaussian random variables, which extend the Hanson-Wright inequality.
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…
Concentration of measure and generalized product of random vectors with an application to Hanson-Wright-like inequalities
本文基于测度集中假设，提出了泛函在随机向量乘积上的集中度表达式，并通过推广Hanson-Wright不等式及分析随机矩阵预解式，证明了其在统计机器学习中的重要性。
We prove extensions of classical concentration inequalities for random variables which have $α$-subexponential tail decay for any $α\in (0,2]$. This includes Hanson--Wright type and convex concentration inequalities. We also provide some applications of these results. This includes uniform Hanson--Wright inequalities and concentration results for simple random tensors in the spirit of previous work by Klochkov--Zhivotovskiy and Vershynin.
In this paper, we provide a proof for the Hanson-Wright inequalities for sparsified quadratic forms in subgaussian random variables. This provides useful concentration inequalities for sparse subgaussian random vectors in two ways. Let $X = (X_1, \ldots, X_m) \in \mathbb{R}^m$ be a random vector with independent subgaussian components, and $ξ=(ξ_1, \ldots, ξ_m) \in \{0, 1\}^m$ be independent Bernoulli random variables. We prove the large deviation bound for a sparse quadratic form of $(X \circ ξ)^T A (X \circ ξ)$, where $A \in \mathbb{R}^{m \times m}$ is an $m \times m$ matrix, and random vect…
Sparse Hanson-Wright Inequalities with Applications
本文提出稀疏Hanson-Wright不等式，用于稀疏亚指数随机矩阵的局部定律和本征向量去局部化，以及线性变换的范数集中，扩展了经典不等式并改进了现有结果。
The Restricted Isometry Property of Block Diagonal Matrices Generated by $φ$-Sub-Gaussian Variables
本文证明了元素来自 $φ$-次高斯变量的块对角随机矩阵的限制等距性质，推广了次高斯情形的已知结果，其证明关键在于一个改进的一致 Hanson-Wright 偏差不等式。
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…
In this paper I prove good estimates on the moments and tail distribution of $k$-fold Wiener--Itô integrals and also present their natural counterpart for polynomials of independent Gaussian random variables. The proof is based on the so-called diagram formula for Wiener--Itô integrals which yields a good representation for their products as a sum of such integrals. I intend to show in a subsequent paper that this method also yields good estimates for degenerate $U$-statistics. The main result of this paper is a generalization of the estimates of Hanson and Wright about bilinear forms of indep…
In this work we derive multi-level concentration inequalities for polynomial functions in independent random variables with a $α$-sub-exponential tail decay. A particularly interesting case is given by quadratic forms $f(X_1, \ldots, X_n) = \langle X,A X \rangle$, for which we prove Hanson-Wright-type inequalities with explicit dependence on various norms of the matrix $A$. A consequence of these inequalities is a two-level concentration inequality for quadratic forms in $α$-sub-exponential random variables, such as quadratic Poisson chaos. We provide various applications of these inequalities…
We present precise multilevel exponential concentration inequalities for polynomials in Ising models satisfying the Dobrushin condition. The estimates have the same form as two-sided tail estimates for polynomials in Gaussian variables due to Latała. In particular, for quadratic forms we obtain a Hanson-Wright type inequality. We also prove concentration results for convex functions and estimates for nonnegative definite quadratic forms, analogous as for quadratic forms in i.i.d. Rademacher variables, for more general random vectors satisfying the approximate tensorization property for entropy…
Starting from a sequence of independent Wright-Fisher diffusion processes on $[0,1]$, we construct a class of reversible infinite dimensional diffusion processes on $\DD_\infty:= \{{\bf x}\in Let $M$ be a complete Riemnnian manifold and $μ$ the distribution of the diffusion process generated by $\ff 1 2\DD+Z$ where $Z$ is a $C^1$-vector field. When $\Ric-\nn Z$ is bounded below and $Z$ has, for instance, linear growth, the transportation-cost inequality with respect to the uniform distance is established for $μ$ on the path space over $M$. A simple example is given to show the optimality of th…
Consider a sequence of polynomials of bounded degree evaluated in independent Gaussian, Gamma or Beta random variables. We show that, if this sequence converges in law to a nonconstant distribution, then (i) the limit distribution is necessarily absolutely continuous with respect to the Lebesgue measure and (ii) the convergence automatically takes place in the total variation topology. Our proof, which relies on the Carbery-Wright inequality and makes use of a diffusive Markov operator approach, extends the results of \cite{NP} to the Gamma and Beta cases.
Sparsity-adaptive concentration inequalities for random polynomials
针对独立稀疏α-次指数随机变量的多项式，建立了基于稀疏加权期望导数张量划分范数的L_r矩界，捕获了集体波动和极端坐标区域的稀疏缩放，推广了Götze等人的多项式集中不等式和稀疏Hanson-Wright界。
On Pseudospectral Concentration for Rank-1 Sampling
本文研究秩-1随机采样扰动下复矩阵的伪谱集中性，分别为正规矩阵、幂零Jordan块及上三角Toeplitz矩阵建立分离半径缩放规律与概率集中框架。
Concentration of a high dimensional sub-gaussian vector
本文研究高维次高斯向量经线性算子作用后平方范数的集中现象，利用最新 Laplace 逼近方法将高斯偏差界推广到非高斯情形，并给出独立同分布和情形的说明。
Deviation bounds for the norm of a random vector under exponential moment conditions with applications
本文在指数矩条件下推广 Hanson-Wright 不等式，给出随机向量范数的上分位函数，并揭示相变效应，应用于伯努利向量和与协方差估计。
Simple, unified analysis of Johnson-Lindenstrauss with applications
本文提出Johnson-Lindenstrauss引理的简化统一分析，统一多种构造并首次严格证明球形构造，扩展Hanson-Wright不等式至高维，去除独立性假设，增强理论并拓展实际应用。
Sphere Valued Noise Stability and Quantum MAX-CUT Hardness
该研究证明了球面值高斯噪声稳定性不等式，并在唯一博弈猜想下得出乘积态量子MAX-CUT的NP难近似下界为0.9859，对比已知多项式算法近似比0.956。
Operation with Concentration Inequalities
本文提出了一种基于集中函数的运算框架，用于研究随机变量的集中性，并推导了与重尾分布相关的Hanson-Wright不等式变体。
We develop a new method for bounding the relative entropy of a random vector in terms of its Stein factors. Our approach is based on a novel representation for the score function of smoothly perturbed random variables, as well as on the de Bruijn's identity of information theory. When applied to sequences of functionals of a general Gaussian field, our results can be combined with the Carbery-Wright inequality in order to yield multidimensional entropic rates of convergence that coincide, up to a logarithmic factor, with those achievable in smooth distances (such as the 1-Wasserstein distance)…
Refining Concentration Bounds for Gaussian Quadratic Chaos with Applications
本文改进了高斯二次混沌的非渐近浓度不等式，包括Hanson-Wright不等式和Laurent-Massart不等式，并提出了基于Schatten范数的新不等式族，应用于声纳和无线通信信号检测。
Quadratic form of heavy-tailed self-normalized random vector with applications in $α$-heavy Marčenko--Pastur law
研究重尾自归一化随机向量的二次型分布，推导α-重尾Marčenko-Pastur定律的隐式残差表示，并证明其无原子特性。
Let $A$ be an $n \times n$ matrix, $X$ be an $n \times p$ matrix and $Y = AX$. A challenging and important problem in data analysis, motivated by dictionary learning and other practical problems, is to recover both $A$ and $X$, given $Y$. Under normal circumstances, it is clear that this problem is underdetermined. However, in the case when $X$ is sparse and random, Spielman, Wang and Wright showed that one can recover both $A$ and $X$ efficiently from $Y$ with high probability, given that $p$ (the number of samples) is sufficiently large. Their method works for $p \ge C n^2 \log^ 2 n$ and the…
Deformed semicircle law and concentration of nonlinear random matrices for ultra-wide neural networks
本文研究超宽两层神经网络中经验共轭核与神经正切核的谱分布，证明其极限服从变形半圆律，并利用非线性 Hanson-Wright 不等式建立谱范数集中性，进而说明随机特征回归与极限核回归具有相同的渐近性能。
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