# Questions tagged [measure-concentration]

The tag has no usage guidance.

286 questions
Filter by
Sorted by
Tagged with
212 views
+50

### 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 ...
150 views

### 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 ...
56 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 ...
39 views

### 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 ...
57 views

31 views

### Gaussian maxima, graph connectivity, superconcentration and mean outsourcing : reference request

Consider a sequence (indexed by $n$) of centred multivariate Gaussian arrays $\{X_i\}_{i\le n}$ such that $\operatorname{Var}(X_i)=1$ for all $i$ and $\sum_{i\le n}\mathbb{E}(X_iX_j)=c_n$ for all $j$. ...
183 views

### 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 ...
156 views

### 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. ...
112 views

### 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 ...
139 views

34 views

### Spectral norm of product of fixed matrix and random semiorthogonal matrix

Suppose I have a fixed matrix $A \in \mathbb{R}^{a \times b}$ and a random matrix $B \in \mathbb{R}^{b \times c}$ with $c < b$ where $B'B = I_c$. I am hoping to find a concentration inequality for ...
129 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 ...
91 views

69 views

48 views

### Characterization of random variables whose tensor powers have subexponential “small-ball” probabilities

Is there a succinct characterization of all random variables $\zeta$ on $\mathbb R$ with the following properties 1. Symmetry: $\zeta \overset{d}{=} - \zeta$. 2. Small-ball probability: there exists ...
152 views

### Prove / disprove: If $1 \le n < N$ and $A$ is an $N \times n$ matrix with iid from $\mathcal N(0,1)$, then $s_\min(A) \ge c\sqrt{N}$ w.p $1-2e^{-N}$

Let $1 \le n < N$ be integers and $A$ be a random $N\times n$ matrix with iid entries from $\mathcal N(0,1)$. This paper (Rudelson and Vershynin) claims in the paragraph just before formula (3.4) ...
76 views

### Lower-bound on smallest singular-value of rectangular random matrix

Let $X$ be a random $N \times n$ matrix with iid entries from $\mathcal N(0, 1)$ and with $n/N =: \lambda(N,n) \le \lambda_0$, for some $\lambda_0 \in (0, 1)$. That is, $X$ is genuinely rectangular (...
Suppose that I have Bernoulli trials with unknown bias $p$. I need $\Omega(\frac{\log 1/\delta}{\epsilon^2})$ samples for the average of the samples to estimate $p$ within $\epsilon$ error with ...