Questions tagged [measure-concentration]

The tag has no usage guidance.

140 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
17 votes
0 answers
369 views

Talagrand's "Creating convexity" conjecture

We say a subset $A$ of $\mathbb{R}^N$ is balanced if \begin{equation} x \in A, \lambda \in [-1,1] \implies \lambda x \in A. \end{equation} Given a subset $A$ of $\mathbb{R}^N$, we write \begin{...
Samuel Johnston's user avatar
12 votes
0 answers
441 views

Is this extension of Hoeffding's inequality known?

Question Overview: Is it already known that, when using Hoeffding's inequality to lower bound the mean of i.i.d. random variables, you can replace the upper bound on the random variables with the ...
PThomasCS's user avatar
  • 389
9 votes
0 answers
1k views

Balls and bins -- concentration bounds pertaining to the minimal load bin

Consider the standard balls and bins process, where $m$ balls are thrown uniformly at random into $n$ bins. Previous work has been done on estimating the value of the maximum load (i.e., the number of ...
JoelO's user avatar
  • 201
8 votes
0 answers
163 views

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 ...
Sergio Zamora's user avatar
7 votes
0 answers
697 views

Product of two random Gaussian matrices - orthant probability

Let $X \in \mathbb{R}^{m \times n}$ and $Y \in \mathbb{R}^{n \times k} $ be two independent Gaussian random matrices, i.e., with entries independently sampled from $\mathcal{N}(0,1)$ (a normal ...
Daniel Soudry's user avatar
6 votes
1 answer
704 views

Simple proof of sharp constant in DKW inequality

The DKW inequality says that if $F_n$ is the empirical CDF corresponding to real-valued random variables $X_1, \dots, X_n$ distributed identically and independently from a distribution with CDF $F$, ...
Drew Brady's user avatar
6 votes
0 answers
268 views

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 ...
Clement C.'s user avatar
  • 1,342
6 votes
0 answers
525 views

a variation on Hanson-Wright inequality

The classic Hanson-Wright inequality states that for a Gaussian random vector $\mathbf{x}\in\mathbb{R}^n$ distributed as $\mathcal{N}(\mathbf{0},\mathbf{I})$ and $\mathbf{A}\in\mathbb{R}^{n\times n}$ ...
mohi's user avatar
  • 859
6 votes
0 answers
325 views

Chernoff bound in the not-quite-sub-exponential case

In Terry Tao's notes on Concentration of measure, Exercise 7 indicates that the Chernoff bound can be generalized to sub-exponential random variables: http://terrytao.wordpress.com/2010/01/03/254a-...
Dustin G. Mixon's user avatar
5 votes
0 answers
218 views

Concentration inequalities for random measures

For random variables $X_1,\dots,X_n$ with common mean $\mathbb{E}[X_i]=\mu$ and common bounds $a\leq X_i\leq b$, we have the very useful Hoeffding's inequality: $$\mathbb{P}\left(\left|\mu -\frac1n\...
Tyler6's user avatar
  • 91
5 votes
0 answers
535 views

Concentration inequality for max component of a multivariate Gaussian in the general case

I am looking to bound the variance of the maximum component of a vector distributed multivariate Gaussian in the general case where the Gaussian distribution has arbitrary mean and full covariance ...
ted's user avatar
  • 271
5 votes
0 answers
1k views

Asymptotic behavior of row sums in 2-d array of random variables

Set-up. Let $f : \mathbb{N} \to \mathbb{N}$ be increasing. For each $m \in [0,1]$, consider an infinite two-dimensional array of random variables, where row $n$ has $f(n)$ variables: $B^m_{1,1}$ $B^...
cosmo-grant's user avatar
5 votes
0 answers
493 views

Vector martingale concentration

Let $\varepsilon_1, \dots, \varepsilon_N$ be a martingale difference sequence in $R^d$ with $\|\varepsilon_n\| \le B_n, a.s.$ for each $n=1,\dots,N$. Do we have some Azuma-type concentration ...
Nikolayevich's user avatar
5 votes
0 answers
131 views

Is there a concentration inequality depending on dimension for a symmetric function on product space?

I recently read an elegant paper of Bobkov Bobkov, S.G., On concentration of measure on the cube, J. Math. Sci., New York 165, No. 1, 60-70 (2010); translation from Probl. Mat. Anal. 44, 55-64 (2010)....
Jason Cantarella's user avatar
5 votes
0 answers
161 views

McDiarmid-like inequality for subgassian random variables

Let $X_n$ be a set of $N$ subgaussian random variables, not necessarily independent, with $E\exp(\lambda X_n) \le \exp(\lambda^2/2)$. Let $X=(X_1,\ldots, X_N)$ and $f:\mathbb R^N \rightarrow \mathbb R$...
gappy3000's user avatar
  • 461
5 votes
0 answers
134 views

Improving concentration estimates by controlling sums on subsets

Let $X_1, \dots, X_N$ be uniform random variables (r.v.) in $[-1, 1]$, and let $S_N$ be their sum $S_N = \sum_{i=1}^N X_i$. If the r.v. are taken independent, then the CLT suggests that $S_N$ is ...
TLeble's user avatar
  • 121
5 votes
0 answers
317 views

Eigenvalues of Random Regular Bipartite Graphs

I am looking for a way of getting a good estimate of the eigenvalues of random bipartite d-regular graphs. The literature has very precise values the proofs of which are very involved and since I am ...
user1189053's user avatar
5 votes
0 answers
135 views

Large Deviations: Exponential decay in normed spaces

Let $(X_1,X_2,\cdots)$ be a sequence of independent and identically distributed random variables taking values in some general normed space $(V,||\cdot||)$. Denote $\mu=E[X_1]$ and $S_n=\frac{1}{n}[...
Manuel Schmidt's user avatar
5 votes
0 answers
116 views

Concentration of weighted random chirp

I'm interested in seeing whether the following is true. Assume $u$ is uniform on $[0,1]$. For a fixed $x\in\mathbb{C}^n$ with $\|x\|_{2}=1$ we have \begin{align*} \mathbb{P}\Big\{\Big|\sum_{k=0}^{n-1}...
mohi's user avatar
  • 859
5 votes
0 answers
380 views

Concentration of functions of random unitary matrices

Suppose $U$ and $V$ are $n \times n$ random unitary matrices, chosen independently from the Haar measure. Is there any kind of concentration inequality which would be applicable to polynomials $p(U,V)$...
Michal Kotowski's user avatar
4 votes
0 answers
105 views

Upper bound $\tau_C := \int_{\|x\| \le 1}(vol(C \cap (x + C))/vol(C))dx$ for a convex body $C \subseteq \mathbb R^n$, by reducing to a ball

Let $C$ be a convex body in $\mathbb R^n$, i.e a bounded convex subset of $\mathbb R^n$ which has nonempty interior, and which is (A) open, or (B) closed (I'm not sure one makes more sense; choose the ...
dohmatob's user avatar
  • 6,716
4 votes
0 answers
131 views

For a martingale $f_0,f_1,\ldots $ how can we bound $P(\frac{1}{n} \|f_n\| \le 1$ for all $ n \ge N)$?

Suppose $f_0,f_1, \ldots$ is a martingale (or i.i.d sequence) in $\mathbb R^d$ with $f_0=0$ and all $\|f_n - f_{n-1}\| \le L$ say. There are many concentration results for the initial segment of the ...
Daron's user avatar
  • 1,761
4 votes
0 answers
163 views

Regularity of optimal transport of Gaussian mixtures

In one of the problems that I am working on, I came across the topic of smoothness of optimal transport for Gaussian mixtures. In particular, let $P=P_\theta=\sum_{i=1}^k \frac{1}{k}\mathcal{N}(x| \...
pikachuchameleon's user avatar
4 votes
0 answers
115 views

$\ell_p$ geodesic distance on smooth Riemannian manifold and Logarithmic Sobolev Inequalities

Bear with me, I'm not a professional geometer. Recently, I've been studying Logarithmic Sobolev Inequalities (LSI) for probability distributions on manifolds (e.g as done in works of Bobkvo et al. ...
dohmatob's user avatar
  • 6,716
4 votes
0 answers
159 views

Are sums extremal for subgaussian concentration?

Bobkov and Houdre https://projecteuclid.org/euclid.bj/1178291721 showed that among all $f:R^n\to R$ that are $1$-Lipschitz with respect to the $\ell_1$ metric, the variance is maximized by sums. ...
Aryeh Kontorovich's user avatar
4 votes
0 answers
88 views

On symmetry and measure concentration rate for convex bodies

The concentration of measure on the cube $ [0, 1]^n $ equipped with uniform probability measure $\mu_{\infty}$, states that for any $A \subset [0, 1]^n $ with $ \mu_{\infty}(A) \geq \frac{1}{2} $, we ...
random_shape's user avatar
4 votes
0 answers
127 views

Is there an example that both Berry-Essen bound and DKW bound are attained?

The Berry-Essen bound stated that $$\sup _{{x\in {\mathbb R}}}\left|\widehat{F_{n}(x)}-\Phi (x)\right|\leq C_{0}\cdot \psi _{0}$$ where $\psi _{0}(n)={\Big (}{\textstyle \sum \limits _{{i=1}}^{n}\...
Henry.L's user avatar
  • 7,951
4 votes
0 answers
153 views

Concentration Inequality for Score Functions of Exponential Familty

Let $p$ be the density of a continuous one-parameter exponential family distribution on $\mathbb{R}$. We assume that $$p(x) = c(x)\cdot \exp\bigl [ x \cdot \theta - b(\theta ) \bigr ], $$ where $\...
Steve's user avatar
  • 1,117
4 votes
0 answers
76 views

How well does an estimator perform on another dataset?

Suppose $X \sim N(0, \Sigma)$ is a $d$-dimensional Gaussian random vector. And we have $2n$ $i.i.d$ sample $X_1, \ldots, X_{n}, \ldots, X_{2n}$. Let $\hat{\Sigma}_1 = \frac{1}{n}\sum_{i=1}^nX_i X_i^\...
Wuchen's user avatar
  • 505
4 votes
0 answers
387 views

concentration of functions of Gaussian processes

Let $\mathcal{C}\in\mathbb{R}^n$ be a subset of the unit ball. Also let $\mathbf{a}_1,\mathbf{a}_2,\ldots,\mathbf{a}_m\in\mathbb{R}^n$ be i.i.d. random Gaussian vectors $\mathcal{N}(\mathbf{0},\mathbf{...
mohi's user avatar
  • 859
4 votes
0 answers
1k views

Concentration of sum of independent random variables

Let $X_1, ..., X_n$ be i.i.d. sub-Gaussian random variables with mean $0$ and variance $1$. That is, we have $\Pr[|X_i| > t] \leq \exp(1-t^2/K^2)$ for all $t>0$ and a parameter $K$. Then we can ...
MCH's user avatar
  • 1,304
3 votes
0 answers
56 views

Concentration for Hamming balls

It is well known that Lipschitz functions on the Boolean $n$-cube endowed with the Hamming metric satisfy concentration properties. Specifically, most of their values lie in a range of width $O(\sqrt ...
alesia's user avatar
  • 2,582
3 votes
0 answers
278 views

Tail bound on trace norm / nuclear norm / Schatten-1 norm of Rademacher matrix

Let $0 < r \leq d$ integers. Let $X$, $Y$ be $d \times r$ matrices of independent Rademacher variables, that is, $X,Y \in \mathbb{R}^{d \times r}$ with entries $\pm1$ with probability $1/2$. I am ...
arriopolis's user avatar
3 votes
0 answers
91 views

Explaning why the spectrum of a setting simple structure random matrix is always spiked ($d-1$ eigenvalues close to zero, and $1$ away from zero)

For concreteness, let $m=500$, $d=600$, $N=1000$. Let $W$ be and $d \times m$ matrix with unit-norm rows and let $u$ be a uni-norm vector of length $m$. Given a binary vector $b$ of length $m$, length ...
dohmatob's user avatar
  • 6,716
3 votes
1 answer
302 views

Concentration of very dependent Markov chains

Consider the following simple Markov chain $ X_1\to X_2\to\cdots\to X_n $ where each $X_i$ is $\{-1,1\}$-valued and $X_1\sim\mathrm{Unif}(\{-1,1\})$ (such that the chain is stationary). The flip ...
Yihan Zhang's user avatar
3 votes
0 answers
80 views

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 ...
arjun's user avatar
  • 921
3 votes
0 answers
187 views

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....
M-Brust's user avatar
  • 31
3 votes
1 answer
325 views

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 ...
dohmatob's user avatar
  • 6,716
3 votes
1 answer
940 views

Explicit constant for Carbery–Wright inequality

The Carbery–Wright inequality is a seminal result about the anti-concentration of polynomials of Gaussian random variables. See e.g. Meka, Nguyen, and Vu - Anti-concentration for polynomials of ...
user134977's user avatar
3 votes
0 answers
291 views

Upper-bound for eigenvalues of $E [UU^T]$, where $U$ is uniformly distributed on the unit $n$-sphere

Let $X$ be a $\sigma$-subGaussian random vector on $\mathbb R^n$ (for large $n \ge 3$), meaning that the random variable $X^Tv$ is $\sigma$-subGaussian for every unit vector $v \in \mathbb R^n$. ...
dohmatob's user avatar
  • 6,716
3 votes
0 answers
169 views

Concentration Inequalities for the Exponential of Weighted Bernoulli Sums

I want a concentration inequality for the exponential of a weighted sum of independent Bernoulli random variables around its mean, for one of my research works. I was wondering if there is a well ...
Somabha's user avatar
  • 123
3 votes
0 answers
147 views

Does there exist a compactly supported integrable function with infinite Coulomb energy?

The title of the question pretty much says it all. I am looking for a function $f\in L^1(\Omega)$, where $\Omega \subset \mathbb{R}^3$ is a bounded domain, such that $$ E[f] = \iint\limits_{\Omega\...
Ben Ciotti's user avatar
3 votes
0 answers
152 views

Matrix Chernoff sampling with out replacement

I am interested to know if the matrix Chernoff bound (see Theorem 5.1.1 in https://arxiv.org/pdf/1501.01571.pdf) holds if one samples without replacement. For example, the Bernstein inequality is ...
felasfaw's user avatar
  • 221
3 votes
0 answers
76 views

A concentration problem of product of matrices

Let $A$ be an $n \times m$ matrix with non-negative entries and $B \in \mathbb{R^{n\times n}_{\geq 0}}, C \in \mathbb{R^{m\times m}_{\geq 0}}$ be random matrices where B and C are both symmetric and ...
ie86's user avatar
  • 195
3 votes
0 answers
176 views

Anti-concentration for sum of t-wise independent uniform variables

Let $X_{1},\ldots,X_{n}$ be i.i.d. random variables, each variable is uniform over the set of integers $\{ 0,\ldots,D-1 \}$. Let $S = \sum_{i=1}^{n}X_{i}$. By ``small ball probability'', we have that ...
Daniel86's user avatar
  • 215
3 votes
0 answers
116 views

Concentration of sums of random matrices around the mean, in the Loewner order

Recently, I have found myself interested in concentration properties of random matrices. Specifically I would like to answer questions of the following sort Let $\{X_i\}_{i=1}^n$ be i.i.d. copies ...
Cain's user avatar
  • 393
3 votes
0 answers
367 views

On the precise concentration of permanent of $\pm1$ matrices

Obtain $M\in\{-1,+1\}^{n\times n}$ by unbiased coin flipping. What is known about the distribution of permanent $\mathsf{Perm}(M)$? It seems to be bimodal. Given a function $g(n)$ what is the ...
user avatar
3 votes
0 answers
423 views

concentration bounds on weighted multinomial sum

Consider i.i.d random vectors $Y_{1},..,Y_{n}$ and they are chosen uniformly at random from $\{e_{1},..,e_{L}\}$ where $e_{i}$ is a $L\times 1$ vector with $i$th component be 1 and the others be 0. ...
Cuize Han's user avatar
3 votes
0 answers
187 views

Concentration of the quotient of random variables

Let $X_1, X_2, \cdots, X_n$ be n i.i.d. standard Gaussian random variables. It is clear that we can describe the concentration of $\sum_{i=1}^n \alpha_i X_i$, and $\sum_{i=1}^n \alpha_i X_i^2$ (sub-...
liuliuforph's user avatar
3 votes
0 answers
258 views

A generalization of coupon collector problem - $\geq1$ pick per experiment

Mix $T\geq1$ coupons numbered $1$ to $T$ with a set of $S\geq0$ number of dummy coupons with no numbers. Select $N\geq1$ coupons at each trial at random and put them back. $N=1$ is standard coupon ...
user avatar