All Questions
Tagged with measure-concentration convex-geometry
14 questions
19
votes
0
answers
554
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{...
- 484
0
votes
0
answers
142
views
Anti-concentration of the $\ell_2$ norm of log-concave measures
This question is regarding a special case of this question, for which it is plausible the details are known.
The Carbery-Wright inequality is an "anti-concentration inequality" that states ...
- 2,183
1
vote
1
answer
231
views
For $x_1,...,x_n$ iid random on sphere of radius $\sqrt{d}$ in $R^d$, what is a good upper-bound on min distance of $x_{n}$ from the other $x_i$'s?
Let $n$ and $d$ be large positive integers with $n \le d^\gamma$, for some absolute constant $\gamma>0$; i.e., $n$ is at most polynomial in $d$. Let $x_1,\ldots,x_n,x_{n+1}$ be drawn iid from the ...
- 6,853
2
votes
1
answer
165
views
Existence of preferred direction for a random vector with arbitrary distribution on sphere, under a condition on its covariance matrix
Let $X$ be random vector on the unit-sphere $S_{n-1}$ in $\mathbb R^n$. We don't assume that the distribution of $X$ is uniform on $S_{n-1}$
I'm interested in proving the existence of a (...
- 6,853
1
vote
1
answer
67
views
Lower-bound for $\mathbb E[e^{-b(v^\top X - c)^2}]$, when $X$ is log-concave in high-dimensions
Let $d$ be a large positive integer. Fix a unit-vector $v \in \mathbb R^d$, and scalars $b,c \in \mathbb R$ with $b > 0$. Let $X$ be a log-concave random vector in $\mathbb R^d$ normalized so that ...
- 6,853
6
votes
3
answers
447
views
Isoperimetric inequality for $\epsilon$-expansion of a set only along a certain subspace
Let $\gamma_n$ be the standard gaussian distribution on $\mathbb R^n$. Let $V$ be a $k$-dimensional subspace of $\mathbb R^n$. Finally let $A$ be any (nonempty) Borel subset of $A$ with $\gamma_n(A) = ...
- 6,853
2
votes
1
answer
114
views
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
3
votes
0
answers
83
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 ...
- 941
1
vote
1
answer
59
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 ...
- 6,853
4
votes
0
answers
107
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 ...
- 6,853
1
vote
0
answers
34
views
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
$$
\...
- 373
4
votes
0
answers
93
views
On symmetry and measure concentration rate for convex bodies
The concentration of measure on the cube $ [0, 1]^n $ equipped with uniform probability measure $\mu_{\infty}$,
states that for any $A \subset [0, 1]^n $ with $ \mu_{\infty}(A) \geq \frac{1}{2} $,
we ...
- 365
4
votes
1
answer
290
views
On the 1/2 assumption on concentration of measure for continuous cube
The concentration of measure on $ [0, 1]^n $ equipped with uniform probability measure $\mu_{\infty}$,
states that for any $A \subset [0, 1]^n $ with $ \mu_{\infty}(A) \geq \frac{1}{2} $,
we have:
$$...
- 365
4
votes
1
answer
503
views
An elementary probability question
Let $X$ be a $d$-dimensional random vector distributed according to probability measure $D$. At least the second moment of the coordinates of $X$ is finite.
Consider $n+1$ samples $X_0, \ldots, X_n ...
- 461