All Questions
Tagged with pr.probability measure-concentration
324 questions
1
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118
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What is the order of the left tail of a mixture of non-central chi-square?
Let $\mu\sim N(0,1)$, $Z\sim N(\mu,1)$. Then $Z$ can be viewed as a mixture of Gaussians. It can also be viewed as a Gaussian but there is a prior for the mean.
Let $X\sim\exp(\lambda)$ where the ...
- 1,495
1
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1
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122
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Variance bound of a functional
$X_1,\ldots,X_n$ are i.i.d standard normal random variables.
$a_1,\ldots, a_n$ are constants with $a_i \in [\kappa_1, \kappa_2]$ for all $i$ and $\kappa_1>0$.
$\hat c_n$ is given as the solution ...
1
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1
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239
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Concentration bound for a martingale-like setting (the expected difference decreases as the sequence increases)
I went through several martingales concentration bounds, but none of them fit the settings I am interested in, which is the following. Suppose I have a sequence of nonnegative random variables $0=Y_{0}...
- 225
1
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42
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Sub-Gaussian analysis via bounded decomposition?
Let $\psi_\alpha(x) := \exp(x^\alpha)-1$.
The Sub-Gaussian Norm $\lVert X \rVert_{\psi_2}$ of a random variable $X$ is defined as
$$
\lVert X\rVert_{\psi_2} = \inf\{c>0\mid \mathbb{E}[\varphi_2(|X|/...
- 2,183
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34
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Discrepancy between probability measures, tested against bounded functions of bounded variance
When studying some concentration inequalities, it became relevant to consider the following discrepancy between two probability measures $\pi$ and $\nu$ (treating $\sigma \in \left( 0, \frac{1}{2} \...
- 801
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0
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57
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Limiting value of expectation of trace of truncated Gram matrix
Let $n$ and $d$ be large positive integers such that $d/n = a \in (0,1)$, fixed. Let $x_1,\ldots,x_n$ be iid random vectors from $N(0,I_d)$. Fix $b \in (0,1]$ and a unit-vector $v \in \mathbb R^d$, ...
- 6,853
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131
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Large-deviation inequalities for a class of simple random multivariate polynomials
Let $N$ be a large positive integer and let $[N] := \{1,2,\ldots,N\}$. For any $k$, let $K_{N,k}$ denote the collection of $k$-element subsets of $[N]$. Let $x=(x_1,\ldots,x_N)$ be a uniformly random ...
- 6,853
1
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1
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285
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Rate of convergence to uniform distribution
Let $p=(p(1),\ldots,p(N))$ be a discrete distribution on $[N]:=\{1,2,\ldots,N\}$ with full support (i.e all the $p(i)$'s are strictly positive and sum to $1$). Let $i_1,i_2,\ldots,i_T$ be an iid ...
- 6,853
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121
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Composing an Orlicz norm related to Bernstein's inequality?
This is related to my previous question, but is hopefully more precise.
I would like to reason about tail-bounds for polynomial products of concentrated random variables in $R:=\mathbb{R}[x]/(x^n-1)$.
...
- 2,183
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129
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Concentration of a combinatorial sum
Let $X=(x_1,\ldots,x_p)$ be an $p \times n$ random matrix with iid entries from $\{\pm 1\}$, distributed so that $\mathbb P(x_{ij} = 1) \equiv 1/2$, where $x_i=(x_{i1},\ldots,x_{in})$. Let $y$ be a ...
- 6,853
1
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1
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153
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Minimax estimation rate of sparse vector $w_\star$, w.r.t to mixed norm $\|\hat w_n-w_\star\| := \|\hat w_n - w_\star\|_2 + \|\hat w_n-w_\star\|_q$
Let $n,d,s$ be positive integers with $s \le d$, and let $B_0(d,s)$ be the set of all (real) $d$-dimensional vectors with at most $s$ nonzero components. Given an $n \times d$ matrix $X$ with rows $...
- 6,853
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370
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Lower bound on the sum of the product of random variables
Let $X_i$ be the $i$-th element of the vector $X=(X_1, ..., X_m)$ of i.i.d. random variables.
I am looking for a lower bound for the expression
$\mathbb{P}((\sum^n_{i=1}\prod^{m_i}_{j=1}(X_j))^2 \geq ...
- 129
1
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1
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230
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VC dimension of a certain derived class of binary functions
Let $X$ be a measurable space and let $P$ be a probability distribution on $X \times \{\pm 1\}$. Let $F$ be a function class on $X$, i.e., a collection of (measurable) functions from $X$ to $\mathbb R$...
- 6,853
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143
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$\newcommand\v{\operatorname{vol}_d(C}$Compact subsets of $ℝ^d$ which maximize $\inf_{|v|\le1}\dfrac{\v\cap(𝜀v+C))}{\v)}$ for fixed $\v)$ and $𝜀>0$
Let $\operatorname{vol}_d$ be the volume measure on $\mathbb R^d$ and let $B_d$ be the unit-ball. For $\varepsilon \ge 0$ and a compact subset $C$ of $\mathbb R^d$ with $\operatorname{vol}_d(C)>0$, ...
- 6,853
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160
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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).
$$
Let $w_1,\dotsc,w_N$ be iid from $N(0,(1/d)I_d)$ and let $f:\mathbb R \...
- 6,853
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349
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Tail bounds for random Gaussian chaos?
Let $g = (g_1, \dots, g_d)$ be a sequence of independent standard Normal random variables, and suppose $\Sigma$ is a $d \times d$ (deterministic), real, symmetric, positive definite matrix. The Hanson-...
- 420
1
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415
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Approximate the singular values of a certain random dot-product kernel matrix (in the sense of El Karoui, Cheng-Singer, etc.)
Let $g:\mathbb R \to \mathbb R $ be a continuous function which is
"sufficiently smooth" (e.g $\mathcal C^3$) around $0$, and
"sufficiently integrable" (e.g integrable w.r.t $N(0,...
- 6,853
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0
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57
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Concentration inequality for matrix martingale with dynamic upper bounds
Consider a sequence of stochastic PSD matrices $X_1, X_2, \dots, X_n \in \mathbb{R}^{d\times d}$. Let $\mathcal{F}_k = \sigma(X_1, X_2, \dots, X_{k-1})$ be the natural filtration and $Y_k = \mathbb{E}[...
- 11
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155
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Relation between the class $\mathcal{M}(m,\sigma)$ and subgaussianity
In this paper, Adamczak defines, for $m>0$ and $\sigma\geq 0$, the class of probability distributions $\mathcal{M}(m,\sigma)$ over $\mathbb{R}$ as those $\mu$ satisfying the tail conditions
$$\nu^+(...
- 1,372
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
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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
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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
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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
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334
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Strong data-processing inequality ? Upper bound on a certain modified total-variation metric
Let $\mathcal X=(\mathcal X,d)$ be a Polish space equipped with the Borel sigma-algebra. Let $p\ge 1$ and $P_1,P_2$ be probability distributions on $\mathcal X$ such that $\max_{k=1,2}\int d(x,x_0)^...
- 6,853
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176
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Rate of convergence of empirical distribution with respect to Wasserstein distance induced by binary cost function
Let $\mathcal X=(\mathcal X, d)$ be a Polish space (i.e complete metric space), and let $\Omega$ be a non-empty subset. Consider the binary cost function $c_\Omega$ on $\mathcal X^2$ defined by $c_\...
- 6,853
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136
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Linearly independent functions evaluated at random points create full rank matrices
Assume $f_1, f_2,...,f_n: \mathbb{R}^d\mapsto\mathbb{R}^d$ are linearly independent functions. Now let $w_1,w_2,..,w_k\in\mathbb{R}^d$ be i.i.d. Gaussian random vectors distributed as $\mathcal{N}(0,\...
- 859
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79
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Showing that additive Gaussian noise never increases sparsity
Let $\mathbf{1}\in\mathbb{R}^d$ be the $d$-dimensional all-ones vector and let $n\sim\mathcal{N}(0, \sigma^2 I_{d\times d})$, show that
$$ \frac{\| \mathbf{1} + n \|_1}{\|\mathbf{1} + n \|_2} \ge c \...
- 392
1
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0
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109
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Concentration inequality for Lipschitz functions with orthogonal gradients
Let $f_j:\mathbb{R}^n\to\mathbb{R}$ be a set of 1-Lipschitz functions for $1\leq j\leq M$. From Gaussian isoperimetry or a log-Sobolev inequality, it can be shown that
$$
\mathbf{Pr}(|f_j(X)-\mathbf{...
- 452
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0
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123
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Sanov-type finite-sample bound on $KL(P\|\hat{P}_n)$
Let $P$ be a distribution on an alphabet of size $k$ and let $\hat{P}_n$ be an empirical version of $P$ via $n$ i.i.d samples $a_1,\ldots,a_n \sim P$, i.e $\hat{P}_n := (1/n)\sum_{i=1}^n\delta_{a_i}$.
...
- 6,853
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105
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Gaussian isoperimetry for $\ell_p$ norms
Let $\gamma_n$ be the standard Gaussian measure on $\mathbb R^n$. It is well-known (e.g see Proposition 1) that for a given Gaussian volume content, half-spaces $H=\{x \in \mathbb R^n | a^Tx \le b\}$ ...
- 6,853
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686
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Almost orthogonality of independent random vectors [closed]
If $X_1$ and $X_2$ are two independent isotropic random vectors in $\mathbb{R}^n$,
then $\mathbb{E}\|X_i\|_{2}^{2}=n$, $\mathbb{E}\langle X_1,X_2\rangle^{2}=n$.
How can I show from the above result ...
- 137
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175
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Matrix Bernstein for spherical random variables
Theorem 4.1 in Tropp's Matrix Concentration Inequalities provides an exponential concentration inequality for the spectral norm of a matrix $Z = \sum_i \gamma_i B_i $, where $\gamma_i$ are an i.i.d. ...
- 111
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34
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Limiting law of quadratic functions of sample averages
Let $X_1,\cdots,X_n$ be independent centered univariate random variables. Let also $\{w_{ij}\}_{i,j=1}^{k,n}$ be a set of deterministic scalar weights, where $k\ll n$. Define sample averages
$$
\...
- 373
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676
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Are Outer Products of Sub-Gaussian Vectors Sub-Exponential?
$\newcommand\xx{\mathbf{x}}\newcommand\yy{\mathbf{y}}\newcommand\A{\mathbf{A}}\newcommand\aalpha{\boldsymbol{\alpha}}\newcommand\bbeta{\boldsymbol{\beta}}\newcommand\E{\mathbb{E}}\newcommand\inner[1]{\...
- 153
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0
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376
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Anti-concentration bounds for folded normal and inverse of gaussian variables
Are there any easy to use bounds on sums of the following kind :
$$
\sum_{i = 1}^{i = N} |a_i| \geq P \\
a_i \sim \mathcal{N}(0, 1) \\
$$
and also for sums of the form :
$$
\sum_{i = 1}^{i = M} \...
- 111
1
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110
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Tail bound without independence
Suppose $X_i , X_j\in \mathbb{R}^d$ are gaussian vectors and $A$ is an $n\times n$ symmetric PSD matrix where $A_{ij} = f(\|X_i-X_j\|_2), \quad i,j\in 1,\ldots,n\;$ for some non-negative Lipschitz ...
- 195
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0
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146
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minimum eigenvalue of Katri-Rao product of two Gaussian matrices
Let $\mathbf{A}\in\mathbb{R}^{k\times n}$ and $\mathbf{B}\in\mathbb{R}^{d\times n}$ be independent matrices with i.i.d. $\mathcal{N}(0,1)$ entries. I'm interested in lower bounding the minimum ...
- 363
1
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0
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295
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One-sided Talagrand concentration inequality for empirical processes
Let $\mathcal{F}$ denote a function class. A classic result by Talagrand states that
\begin{align*}
\mathbb{P}\bigg\{\sup_{f\in\mathcal{F}}\big|\sum_{i=1}^nf(X_i)-\mathbb{E}\big[\sum_{i=1}^nf(X_i)\...
- 859
1
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0
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360
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concentration inequalities for quadratic forms of correlated random vectors
Let $\mathbf{n}$ is a Gaussian random vector with mean $\mathbf{0}$ and co-variance matrix $\mathbf{H}$. Let $\mathbf{r} = Sign(\mathbf{n})$, where $Sign(n_i) = 1$ if $n_i>0$ and $Sign(n_i) = -1$ ...
- 39
1
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0
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98
views
Small ball probabilities for functions of correlated normals
Let $f : \mathbb{R}^k \rightarrow \mathbb{R}$ and let $X$ be distributed k-dimensional normal with mean $0$ (with "arbitrary" covariance matrix). I am looking for references with bounds of the form: ...
- 111
1
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0
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610
views
Upper bound on expectations of the sum of product of a martingale difference sequence with a predictable sequence, weighted by certain random weights
Let $(\mathcal{F}_i)_{i\geq 1}$ be a filtration. Let $0\leq p_i\leq 1$, be a random variable measurable w.r.t. $\mathcal{F}_i$. Consider two sequences of random vectors $v_i\in\mathbb{R}^M,w_i\in\...
- 323
0
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1
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808
views
Concentration of $\ell_2$ norm of a vector sampled from a distribution
Let $X=(X_1,\ldots,X_n)$, where $X_i \sim P_{p_i}(0,\frac{1}{\lambda})$ are iid, $P_{p_i}$ is sub gaussian distribution for $i^\text{th}$ element, and 0 and $1/\lambda$ are mean and variance.
I'm ...
- 61
0
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1
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254
views
Does log-concave approximable distribution satisfy transportation-cost inequality?
Definition: Recall that a distribution $\mu$ on $\mathbb R^d$ is said to be log-convave with constant $c > 0$, if density $d\nu \propto e^{-V}dvol$ satisfying the curvature condition
$$
\...
- 6,853
0
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1
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231
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Concentration inequalities for random sampling without replacement
Let a population $C$ consist of $N$ values $c_1, c_2, \cdots, c_N$, with $c_i\in \{0,1\}$. Let $X_1, X_2, \cdots, X_n$ denote a random sample without replacement from $C$ and let $Y_1, Y_2, \cdots, ...
- 105
0
votes
2
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280
views
Bounds tighter than the additive Chernoff
Additive Chernoff
Suppose $X_1, \ldots, X_n$ are i.i.d. random variables, taking values in $\{0,1\}$. Let $p=\mathrm{E}\left[X_i\right]$ and $\varepsilon>0$.
\begin{gather*}
\operatorname{Pr}\left(\...
- 105
0
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1
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182
views
Deducing norm concentration from MGF bounds
Suppose that $X$ is a centered, $\mathbf{R}^d$-valued random variable such that for all $t \in \mathbf{R}^d$, there holds the bound $$\log \mathbf{E} \left[ \exp \langle t, X \rangle \right] \leqslant ...
- 801
0
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1
answer
110
views
Positivity of linear combination of gaussian variables
Consider a collection of independent standard Gaussian variables $w_i$ for $i = 1, 2, \ldots, N$. Define its linear combination $f:=\sum_{i=1}^Na_iw_i+b_i$, where $a_i=pb_i$ ($p$ is a fixed parameter),...
- 49
0
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2
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204
views
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
0
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1
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378
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Concentration of norm of linearly transformed normal random vector as dimension go to infinity
Earlier asked on MSE, but didn't get an answer, so posting here:
Let $X=(X_1 \dots X_n) \in \mathbb{R}^n, X_i\sim N(0,1), iid.$ Let $B: \mathbb{R}^n \to \mathbb{R}^n $ be the diagonal linear map: $...
- 1,467
0
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1
answer
115
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Compute lower bound on $\min_{E} \mathcal N(0,\sigma^2 I_n)(E)$ subject to $vol(E \cap H_n(r)) / vol(H_n(r)) \ge p$ where $H_n(r)$ is $n$-hemisphere
Let $n \ge 2$ be an integer, which may be assumed to be very large. For $r > 0$, consider the hemi-sphere $H_n(r) := S_n(r) \cap (\mathbb R^+ \times \mathbb R^{n-1})$, where
$$
S_n(r):= \{x \in \...
- 6,853