Questions tagged [measure-concentration]
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397 questions
1
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Given iid $w_1,\dotsc,w_N \sim N(0,1/d)$ iid, find a simple matrix $A$ s.t $\|aa^T-A\|_\text{op}\to0$, where $a_i := E_{G \sim N(0,1)}[f(\|w_i\| G)]$
Let $d$ and $N$ be two large comparable integers, for example assume
$$
N,d \to \infty, \quad d/N \to \gamma \in (0,\infty).
$$
Let $w_1,\dotsc,w_N$ be iid from $N(0,(1/d)I_d)$ and let $f:\mathbb R \...
- 6,853
6
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1
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306
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Weak concentration bounds for averages of independent random variables in Orlicz spaces
Let $\phi$ be an $N$-function, (i.e. $\phi : \mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0}$ is convex and satisfies $\lim_{t \to 0} \frac{\phi(t)}{t} = 0, \lim_{t\to \infty} \frac{\phi(t)}{t} = \infty$)....
- 1,190
3
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1
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530
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Wasserstein-type concentration inequalities for empirical measures on polish spaces
Let $(\mathcal{X},d)$ be a Polish (metric) space and let $\{X_n\}_{n=1}^{\infty}$ be a sequence of i.i.d. $\mathcal{X}$-valued random elements defined on a common complete (standard) probability space ...
- 5,405
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2
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204
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What is the limiting marginal distribution of a fixed number of coordinates of a random point drawn uniformly on large-dimensional sphere?
Let $X=(X_1,\ldots,X_d)$ be uniformly-distributed on the sphere of radius $\sqrt{d}$ in $\mathbb R^d$. It is well-known that in the limit $d \to \infty$, the marginal distribution of $X_1$ converges ...
- 6,853
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0
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351
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Tail bounds for random Gaussian chaos?
Let $g = (g_1, \dots, g_d)$ be a sequence of independent standard Normal random variables, and suppose $\Sigma$ is a $d \times d$ (deterministic), real, symmetric, positive definite matrix. The Hanson-...
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1
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1
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415
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Approximate the singular values of a certain random dot-product kernel matrix (in the sense of El Karoui, Cheng-Singer, etc.)
Let $g:\mathbb R \to \mathbb R $ be a continuous function which is
"sufficiently smooth" (e.g $\mathcal C^3$) around $0$, and
"sufficiently integrable" (e.g integrable w.r.t $N(0,...
- 6,853
2
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1
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90
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Asymptotics of $w^\top G^2 w$, where $w$ is a unit-vector, $G:=X^T(XX^T+t I_n)^{-1}X$, $t > 0$, and $X$ is an $n\times d$ gaussian random matrix
Let $X$ be an random $n \times d$ matrix with entries drawn iid from $N(0,1/d)$ and let $w$ be a unit-vector in $\mathbb R^d$. With $\lambda>0$, and define $G:=X^\top(XX^\top + \lambda I_n)^{-1}X$. ...
- 6,853
2
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1
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116
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A question on the applicability Chebyshev inequality for sequence of random quantities
Let $(X_n)_n$ and $(Y_n)_n$ be two mutually independent sequences of random tensors (i.e scalars, vectors, matrices, etc.) defined on the same probability space, and let $f$ be a measurable function.
...
- 6,853
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0
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86
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Expected diameter of a random point set
General problem: For a point set $S\subset X$ in a metric space $(X,d)$, let $\text{diam}(S)=\max_{x,y\in S}d(x,y)$. Given a distribution $P$ on $X$ and $m$ i.i.d. points $x_1,\ldots,x_m\sim P$, what ...
- 19
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1
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304
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An approximation problem w.r.t marginal distribution of coordinates of uniform random vector on high-dimensional unit-sphere
Let $X=(X_1,\ldots,X_d)$ be uniformly distributed on the sphere of radius $\sqrt{d}$ in $\mathbb R^d$. Fix a "sufficiently integrable" function $h:\mathbb R \to \mathbb R$, and define ...
- 6,853
2
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1
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187
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Compute the limit of trace of inverse of square of rank-1 perturbation of Wishart matrix
Let $a \ge 0$, $b,c>0$ be fixed constants, and let $X$ be an $m \times d$ random matrix with entries drawn iid from $N(0,1/d)$. Consider the random psd matrix $S := a 1_m 1_m^\top + b XX^\top + c ...
- 6,853
4
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2
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343
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Concentration of $k$-th pairwise distance of random points in a unit square
For $1\leq i \leq n$, let $X_i\sim \text{Uniform}(0,1)$, $Y_i \sim \text{Uniform}(0,1)$ be $n$ points chosen uniformly in the unit square. Denote the $k$-th smallest pairwise distances across the $n$ ...
- 662
2
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1
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99
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Concentration Inequality for Bounding Lipschitz Empirical Lass
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, $(X_n:\Omega\rightarrow \mathbb{R}^m)_n$ be a sequence of i.i.d. random variables and let $L:\mathbb{R}^m\rightarrow [0,\infty)$ be ...
- 77
1
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1
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178
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Tail bound on the RKHS norm of a zero-mean Gaussian process
Let $f \sim \mathcal{GP}(0, K)$ be a zero-mean Gaussian process defined on a compact set $\mathcal{D} \subset \mathbb{R}^d$, where $K \colon \mathcal{D} \times \mathcal{D} \rightarrow \mathbb{R} $ is ...
- 1,127
1
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1
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352
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Anti-concentration of inner product of a uniform unit norm vector with another vector
I have a question, which is necessary at one step of my research. Suppose that $X$ is a uniform random vector on the unit sphere $$S^{d-1} := \{x \in \mathbb{R}^d: \|x\|_2 = 1\}~.$$ Is there any ...
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168
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Do compact universal covers have concentration of measure phenomenon?
$\DeclareMathOperator\vol{vol}\DeclareMathOperator\diam{diam}$I have a sequence of compact Riemannian manifolds $M_n$ with $\diam (M_n) \to 0$ and finite fundamental groups $\pi_1 (M_n)$ so that their ...
- 686
1
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78
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Cosine function evaluations are linearly independent? [closed]
Let $\{x_1,\ldots,x_n\}$ be distinct points in $\mathbb{R}^d$, and consider the functions $f_j(x) = cos(w_j^T x + b_j)$ for $w_j \in\mathbb{R}^d$, $b_j\in\mathbb{R}$, $j=1\ldots,m$, and let $m\ge n$. ...
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3
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436
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Use statistical physics ideas ("replica trick") to compute asymptotic value of $\inf_{\|w\| \le r} (1/n)\|Xw-y\|^2$ for random $X$ and $y$
I'm trying to get my head around the "replica trick" and it's mathematically rigorous formulations (due to Talagrand, Parchenko, etc.). I was wondering to myself that a solution or insight ...
- 6,853
2
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1
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114
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Reweighting probability measures by convex potentials, and contraction in transport distance
Let $W: \mathbf{R}^d \to \mathbf{R}$ be a convex function such that $\int \exp(-W) = 1$, and define probability measures $\mu_y$ by
$$\mu_y (dx) = \exp( - W (x - y)) \,dx,$$
i.e. each $\mu_y$ is a ...
- 801
1
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0
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57
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Concentration inequality for matrix martingale with dynamic upper bounds
Consider a sequence of stochastic PSD matrices $X_1, X_2, \dots, X_n \in \mathbb{R}^{d\times d}$. Let $\mathcal{F}_k = \sigma(X_1, X_2, \dots, X_{k-1})$ be the natural filtration and $Y_k = \mathbb{E}[...
- 11
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273
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Question about size-biased couplings and concentration of the number of collisions
Edit/Update: I was indeed missing something quite obvious. I assumed the notion of collision used was the one I am used to (number of pairwise collisions, so if a bin received $k$ balls then this ...
- 1,372
5
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1
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225
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Anti-concentration of Gaussian when conditioning on event
Let $v$ be a given vector with $\|v\|_{\Sigma^{-1}} \leq 1$, where $\Sigma$ is a positive semi-definite matrix and $\|v\|_{\Sigma^{-1}} = \sqrt{v^\top\Sigma v}$. Meanwhile, let $u$ be a random vector ...
- 1,127
3
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83
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Ornstein-Ulhenbeck like semigroup for the orthogonal group with a hypercontractive inequality
I am looking for an Ornstein-Uhlenbeck like semigroup $P_t$ and associated generator $\mathcal{L}$ on $G = \operatorname{SO}(n)$ or $\operatorname{O}(n)$ that has a hypercontractive inequality with a ...
- 941
2
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1
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240
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Lipschitz condition with respect to operator norm of a Gaussian matrix with iid entries. Improved Gaussian Poincare Inequality?
The Gaussian Poincare inequality says that if $q:R^n\to R$ is Lipschitz (for simplicity you may additionally assume smooth with compact support), then $Var[f(X)] \le L^2$
for $X\sim N(0,I_q)$.
Now ...
- 1,724
2
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1
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302
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Concentration on discrete probability estimator
Let $t>1$ and $X_1,..., X_t$ a set of real random variables from a discrete distribution, whose pmf is $p(x)$, supported on the points $1,...,k$.
Let $N_t(x) = \sum_{i = 1}^t \mathbb{1}_{X_i =\, x}....
- 245
4
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2
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175
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Almost independence of $x^\top a$ and $x^\top b$ for $x$ uniform on the sphere in $\mathbb R^d$ and $a,b \in \mathbb R^d$ with $a^\top b = 0$
Let $d$ be a large positive integer. Let $x$ be uniformly distributed on the unit-sphere in $\mathbb R^d$ and let $a$ and $b$ be perpendicular vectors in $\mathbb R^d$, i.e such that $a^\top b=0$. Let ...
- 6,853
1
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1
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476
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Convergence of quadratic form $y^T Q y$ where $y$ is a random iid sequence of length $n$ and $Q$ is an $n \times n$ random matrix independent of $y$
For each positive integer, let $Q_n=(q_{i,j})_{i,j \in [n]}$ be a random $n \times n$ psd matrix. In the limit $n \to \infty$, suppose the eigenvalues of this sequence of matrices are uniformly ...
- 6,853
2
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0
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172
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Asymptotic lower and upper bounds for the eigenvalues of hadamard product $W \circ W$, where $W$ is a large Wishart matrix
Let $n$ and $d$ be large positive integers with $n,d \to \infty$ such that $n/d \to \gamma \in (0,\infty)$. Let $X$ be a random $n \times d$ random matrix with iid copies of log-concave isotropic ...
- 6,853
3
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226
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Eigenvalues of Hadamard product of two Wishart-type matrices
Given two independent Gaussian matrices with i.i.d. entries: $A\in\mathbb{R}^{n\times p}$ and $B\in\mathbb{R}^{n\times q}$, where and $A_{i,j},B_{i,j}\sim\mathcal{N}(0,1)$. Assume that $\max(p,q)<n....
- 31
2
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1
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416
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High-probability lower bound for norm of least squares solution when both design matrix $X$ and response vector $y$ are random (and independent)
Let $n,d \to \infty$ with $n/d \to \gamma \in (0,\infty)$. Let $X$ be a random $n \times d$ matrix independent rows uniformly distributed on the the unit-sphere in $\mathbb R^d$ and let $y$ be a ...
- 6,853
3
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1
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555
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How did the story of Kim-Vu type inequalities continue?
I am interested in the concentration of polynomials of random variables. I have been reading Boucheron, Lugosi, and Massart's "Concentration inequalities" and they give some references. ...
- 233
1
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1
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144
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Bounds for the extreme singular-values of random matrix with thresholded entries
Let $n,d,k$ be large positive integers such that $\max(n/d,k/d) =: \lambda < 1$. Let $X$ be a random $n \times d$ matrix with entries drawn iid from $N(0,1/d)$ and let $W$ be a $k \times d$ random ...
- 6,853
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155
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Relation between the class $\mathcal{M}(m,\sigma)$ and subgaussianity
In this paper, Adamczak defines, for $m>0$ and $\sigma\geq 0$, the class of probability distributions $\mathcal{M}(m,\sigma)$ over $\mathbb{R}$ as those $\mu$ satisfying the tail conditions
$$\nu^+(...
- 1,372
2
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1
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295
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A result about sub-exponential random variables
I am reading the proof of Theorem 1(a) in the paper that proposed the CLIME method for estimating precision matrix. I am puzzled by an inequality on Page 605 three lines above formula (29). I isolate ...
- 56
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0
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83
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Concentration inequalities for sets
Assume that we have a random set $B$ which is constructed by selecting elements from $U = \{ X_1, \dots, X_n \}$ where $X_i$ are independent samples from Gaussians with means $\mu_i$ and variances $\...
- 143
2
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1
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669
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Lower-bound for smallest eigenvalue of random $k \times $k matrix $C(W)$ defined by $C(W)_{i,j} := 2(w_i^\top w_j)^2 + \|w_i\|^2\|w_j\|^2$
Let $k$ and $d$ be positive integers such that $d/k:=\lambda > 1$. Let $W$ be $k \times d$ random matrix with rows $w_1,\ldots,w_k \in \mathbb R^d$ drawn iid from $N(0,(1/d)I_d)$, and define the $k ...
- 6,853
1
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0
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96
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Concentration for $\sum_{i=1}^n y_i \psi(x_i^\top u)$, for $y_1,\ldots,y_n \sim \{\pm 1\}$ and $x_1,\ldots,x_n$ uniform iid on hypersphere
Let $y_1,\ldots,y_n$ be drawn iid uniformly from $\{\pm 1\}$ and let $x_1,\ldots,x_n$ be drawn iid uniformly from the unit-sphere $(d-1)$-dimensional sphere $\mathbb S_{d-1}$, and independently from ...
- 6,853
3
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1
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379
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Concentration inequality for norm of solution to nonlinear least-squares problem
Define the piecewise-linear function $\psi(t):=\max(t,0)$ for all $t \in \mathbb R$.
Let $d,n,k \to \infty$ at the same rate (i.e $n \asymp k \asymp d$).
Let $y_1,\ldots,y_n \in \{-1,1\}$ uniformly ...
- 6,853
1
vote
1
answer
508
views
Polynomial Markov versus Chernoff Bound for random variables
Suppose that $X\geq0$, and that the moment generating function of $X$ exists in an interval around 0. Given some $\delta>0$ and integer $k=1,2,...$, show that
$$\inf_{k=0,1,...}\frac{E(|X|^k)}{\...
- 100
1
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0
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68
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(Anti-)concentration of gap between largest and second largest component of multivariate random gaussian vector
Let $n$ be a large positive integer and let $Y=(Y_1,\ldots,Y_n)$ be a zero-centered random $n$-dmensional real vector with covariance matrix $\Sigma$, an $n$-by-$n$ positive definite matrix with ...
- 6,853
0
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1
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971
views
Bound the norm of sum of random vector that generated from standard basis
I have a question like this:
Consider $N$ samples $X_1, X_2, ..., X_N$ that uniformly random generated from standard basis $\{e_i, i=1,2,...,d\}$, i.e. $(1,0,0,\cdots,0),(0,1,0,\cdots,0),(0,0,1,0,\...
- 25
3
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1
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177
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Gaussian concentration/isoperimetric inequality with correlated Gaussian measure
Famous Gaussian concentration inequality states that:
If $\mathrm{F}$ is 1 -Lip, and $\mathbb{E} F(X)=0,$ and $X=(X_1,...,X_n) \sim N\left(0, I_{n}\right),$ then we have for some absolute constant $C&...
- 519
2
votes
2
answers
690
views
Concentration and anti-concentration of gap between largest and second largest value in Gaussian iid sample
Let $n \ge 3$ be an integer and let $X=(X_1,\ldots,X_n)$ be random vector with iid coordinates from $N(0,1)$. For $1 \le k \le n$, let $X_{(k)}$ be the value of the $k$th largest coordinate of $X$.
...
- 6,853
1
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1
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261
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Concentration inequality for a function whose parameter depends on input samples
Concentration inequalities can be used to establish results such as sample mean cannot be too far from the actual population mean, and so on. For example, let $X_1 \ldots X_n$ be i.i.d instances of a ...
- 615
3
votes
1
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115
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Lower-bound for $\underset{p \le \gamma_d(A) \le q}{\inf} \gamma(A^\epsilon)$, where $\gamma_d$ is the standard gaussian distribution on $\mathbb R^d$
Let $\gamma_d = \gamma_1 \otimes \ldots \otimes \gamma_1$ be the standard Gaussian distribution on $\mathbb R^d$, where $d$ is a large positive integer. Given $\epsilon \ge 0$ and a measurable $A \...
- 6,853
9
votes
1
answer
351
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Concentration inequalities for very rare events on a multiplicative scale
Let $E_1, \dots, E_N$ be independent events, each of probability $p$, where $p$ is very close to $0$. Let $A_N = \frac{1}{N} ( 1_{E_1} + \dots + 1_{E_N} )$ be the proportion of the events $E_i$ that ...
- 323
1
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0
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57
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Good lower-bound for $\inf_{x \in \Delta_n} \|Gx\|$ where $G$ is an $N \times n$ random matrix with iid entries from $\mathcal N(0,1/\sqrt{N})$
Let $G$ be an $N \times n$ random matrix with independent entries distributed according to a centered Gaussian with variance $1/\sqrt{N}$ and let $n/N = \lambda \in (0, 1)$. Let $\Delta_n$ be the $(n-...
- 6,853
3
votes
2
answers
637
views
Exponential inequality for the sum of martingale differences $X_1, \dots, X_n$ when $\sum_{i=1}^{n} \operatorname{Var}(X_i) \leq B^2$
Let $X_1, X_2, \dots, X_n$ be a martingale difference sequence such that
$$
X_i \leq y \quad \text{and} \quad \sum_{i=1}^{n} \operatorname{Var}(X_i) \leq B^2.
$$
Question 1: Does the following hold?
$$...
- 33
1
vote
1
answer
141
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Central limit theorem for chi-squared random field on $\mathbb R^p$
Let $X:x \mapsto X(x)$ be a centered stationary Gaussian process on the $\Omega:=\mathbb R^p$, such that $X(x) \overset{d}{=}X(x')$ for all $x,x' \in \Omega$. Set $\sigma^2 := \mbox{Var}(X(0)) = \...
- 6,853
1
vote
1
answer
343
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Concentration inequality for the supremum of $L_2$ norm of a vector-valued Gaussian process with iid components
Let $\Omega$ be a compact subset of $\mathbb R^p$ and let $f_1,\ldots,f_k$ be zero mean identically distrubuted Gaussian processes on $\Omega$ such that $f_1(x),\ldots,f_k(x)$ are independent $x \in \...
- 6,853