All Questions
Tagged with convex-geometry pr.probability
69 questions
0
votes
0
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19
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A question on Ibragimov's theorem on strong unimodality
I am not a mathematics student and unfortunately have some confusion about a (well-known) theorem about strong unimodality of distributions. First of all let me clarify some terminologies and then ask ...
- 51
1
vote
0
answers
143
views
Integer points inside the high-dimensional ball (asymptotics)
Let $N(\alpha, n)$ denote the number of integer points inside the origin-centered ball of radius $\alpha \sqrt n$ in $n$ dimensions, where $\alpha \in (0,\infty)$ is some fixed constant. Precisely:
$$...
- 435
1
vote
0
answers
45
views
Inequality Involving Concave Monotonic Function
Assume that $ f: \mathbb{R} \to \mathbb{R}_+ $ is a concave, non-decreasing and positive function. Let $\mathbb{X}$ be a finite set consisting of $ 0\leq x_1 \leq x_2 \leq x_3 \leq \ldots \leq x_n$. ...
0
votes
0
answers
72
views
Probability of being inside a convex n-dimensional polytop
I am currently conducting some post-grad research about wireless transmissions with uncertain transmission delays.
As part of the research, each individual transmission is modelled using a probability ...
1
vote
0
answers
50
views
Interpolation in convex hull
I'm reading a paper, Learning in High Dimension Always Amounts to Extrapolation, that provides a result I don't understand.
It provides this theorem which I do understand:
Theorem 1: (Bárány and ...
0
votes
0
answers
22
views
Directions of differentiability of log-concave measures with infinite-dimensional support
I recently came across the very nice review "Differentiable Measures and the Malliavin Calculus" by Bogachev (1997) which begins by discussing measures $\mu$ on locally convex spaces $X$ ...
- 651
1
vote
0
answers
245
views
A sort of dual to nondegenerate random variables
I was motivated by this classical puzzle/1992 Putnam problem.
Suppose 4 points are independently and uniformly distributed on a sphere in 3d. What is the probability the tetrahedron they form contains ...
- 646
0
votes
0
answers
43
views
For a convex body $K\subset R^n$, does the quantity $\min_{t>0} E[t^{-1} \|tg - \pi_K( t g)\|_2]$ have a name? Where has it been studied?
Consider a convex compact set $K\subset R^n$ (with non-empty interior if that helps).
Let $\Pi_K:R^n\to K$ be the projection onto $K$, defined as
$$
\Pi_K(x) = \operatorname{argmin}_{k\in K} \|k-x\|
$$...
- 1,724
1
vote
1
answer
195
views
Reference request: Inequalities involving convex sets and Gaussian variables stated in a paper by Talagrand
I'm looking for references for two facts that are stated without proof in the paper:
Talagrand, M., Are all sets of
positive measure essentially convex?, Lindenstrauss, J. (ed.) et al.,
Geometric ...
- 484
2
votes
0
answers
164
views
Log Sobolev inequality for log concave perturbations of uniform measure
Suppose $\Omega$ is a convex bounded open set of $\mathbb{R}^n$ (I would be happy with just $\Omega$ as the $n$-dimensional cube). Let $\mu$ be the uniform measure on $\Omega$ and consider the ...
- 873
5
votes
0
answers
159
views
Log Sobolev inequality uniform in parameters
Fix a positive integer $N$. For $\theta \in [0,2\pi]$, set $\sigma_k(\theta) :=(\cos(k\theta),\sin(k\theta)) \in S^1$ for each integer $1\leq k\leq N$. Now for vectors $x_1,\ldots,x_N\in \mathbb{R}^2$,...
- 873
1
vote
1
answer
176
views
Is an inner product $\langle X, \epsilon\rangle$ between log-concave $X$ and $\epsilon\gets \{0,1\}^n$ log concave?
Let $X$ be a random variable with a density $p(x)$ with respect to the Lebesgue measure. We say that $X$ is log concave if $p(x) = \exp(-V(x))dx$ for $V(x)$ a convex function.
Let $X$ be log-concave ...
- 2,183
1
vote
0
answers
83
views
Closed form volumes for intersecting modified cylinders
This question is somewhat related to the question Intersecting cylinders, but where the cylinders are now modified to orbifolds in the hypercube with singularities occurring at the vertices of the ...
- 383
1
vote
0
answers
61
views
$\psi_2$ marginals of the permutahedron?
Let $K$ be a convex body.
I in particular care about the permutahedron.
I will view this as being the convex hull of all coordinate-wise permutations of the vector
$$v = \frac{1}{2n+2}(-n, -n+2,\dots, ...
- 2,183
7
votes
4
answers
476
views
What does $\mathbb E_V \max_{x \in V,\,\|x\|=1} x^T Ax$ evaluate to when $V$ is random $k$-dim suspace of $\mathbb R^n$ and $A$ is fixed psd matrix?
Let $G_{k,n}$ be the grassmannian of $k$-dimensional vector spaces of $\mathbb R^n$. By the Courant–Fisher characterization, the $k$th largest eigenvalue of an $n \times n$ psd matrix $A$ is given by
$...
- 6,853
1
vote
0
answers
81
views
Expectation of dual norm induced by probability measure
Let $\mu$ be a probability measure supported on the unit sphere $\mathbb{S}^{d-1}$. Assuming that $\mu$ is even and not supported on any great subsphere, its cosine transform $w \in \mathbb{R}^d \...
- 103
0
votes
0
answers
140
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
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
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
1
vote
1
answer
176
views
Tight upper-bounds for the Gaussian width of intersection of intersection of hyper-ellipsoid and unit-ball
Let $\Lambda$ be a positive-definite matrix of size $n$ and let $R \ge 0$, which may depend on $n$. Consider the set $S := \{x \in \mathbb R^n \mid \|x\|_2 \le R,\,\|x\|_{\Lambda^{-1}} \le 1\}$ where $...
- 6,853
2
votes
2
answers
308
views
Expected value of Tukey’s half-space depth for log-concave measures
Let ${\mathbb P}$ be a probability measure in ${\mathbb R}^n$. Let $x\in{\mathbb R}^n$ be an arbitrary point. Let ${\mathbb H}_x$ be the set of halfspaces of ${\mathbb R}^n$ containing $x$. Let
\begin{...
- 781
0
votes
2
answers
534
views
Wasserstein distance between $N(0,1/d)$ and the marginal distribution of $x_1$ when $x=(x_1,\ldots,x_d)$ is uniform on the unit-sphere in $R^d$
Let $x=(x_1,\ldots,x_d)$ be uniformly distributed on the unit-sphere in $\mathbb R^d$.
Question.
What is a good upper-bound for Wasserstein distance between $N(0,1/d)$ and the marginal distribution ...
- 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
2
votes
1
answer
113
views
Existence of fine approximate of a convex body in $\mathbb R^d$ with convex hull of $\mathcal O(d)$ points
Let $K$ be a convex body in $\mathbb R^d$ which contains the origin and let $\theta \in (0,1)$.
Question. Is it always possible to find $n$ points $x_1,\dotsc,x_n \in \mathbb R^d$ such that
$$
\theta ...
- 6,853
1
vote
0
answers
153
views
Is there a polynomial expression for the volume of the following set?
Denote the unit $\ell_2$ ball in $\mathbb{R}^n$ as $\mathcal{B}_n$. It is widely kown that for a convex body $\mathcal{K}\subseteq \mathbb{R}^n$, the $n$-dimensional volume of the parallel body $\...
- 550
2
votes
1
answer
133
views
Discrete random walk on polytope via involutions
Let $P$ be a convex polytope (or more generally convex body, I suppose) in $V=\mathbb{R}^n$. For each $v\in \mathbb{P}V$, we define an involution $\tau_v\colon P\to P$ by setting $\tau_v(p)$ to be the ...
- 24.2k
1
vote
1
answer
67
views
Lower-bound for $\mathbb E[e^{-b(v^\top X - c)^2}]$, when $X$ is log-concave in high-dimensions
Let $d$ be a large positive integer. Fix a unit-vector $v \in \mathbb R^d$, and scalars $b,c \in \mathbb R$ with $b > 0$. Let $X$ be a log-concave random vector in $\mathbb R^d$ normalized so that ...
- 6,853
0
votes
0
answers
118
views
Weak derivative of projection onto probabilist's simplex
Let $\Delta_n:=\{x\in [0,1]^n:\boldsymbol{1}^{\top}x=1\}$ denote the probabilist's $n$-simplex and let $P:\mathbb{R}^n\rightarrow\Delta_n$ denote the (Euclidean) metric projection onto this simplex ...
- 5,405
4
votes
0
answers
116
views
Improving log-Sobolev inequalities via quadratic regularisation
Suppose that $\mu(dx) = \exp(-\psi(x)) \, \mathrm{dx}$ is a probability measure on $\mathbf{R}^d$.
For suitable functions $g \geqslant 0$, define
$$\text{Ent}(g) = \int \mu(dx) g(x) \log \left( \frac{...
- 801
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
2
votes
0
answers
106
views
How to judge whether the following convex set contains a given point?
Let the set $\mathcal{S}=\left\{ \sum_{i=1}^n x_i\mathbf{h}_i：x_i\in[0,1] \text{ for all }i\right\}\subset\mathbb{R}^r$, i.e., a zonotope generated by $n$ column vector $\mathbf{h_1},\cdots,\mathbf{h}...
- 550
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
2
votes
0
answers
104
views
Weak convergence rates for integral operators
Suppose $q=\sum_{i=1}^m\pi_i\delta_{x_i}$ is a discrete measure on $\mathbb{R}^n$ and let $q\ast \varphi_\epsilon$ denote the convolution of $q$ with some mollifier $\varphi_\epsilon$, so that $q\ast\...
- 75
1
vote
0
answers
45
views
Analytic lower-bound for minimal value of $\|x\|^2$ such that $\|Cx-b\|^2 \le c^2$ (a hyperellipsoid)
Let $C$ be an $n \times p$ matrix and $b$ be a column vector of length $n$, and $c>0$. Let $E := \{x \in \mathbb R^p \mid \|Cx-b\| \le c\}$, a hyperellipsoid in nonstandard position.
Question 1. ...
- 6,853
8
votes
1
answer
355
views
Lower Bound of KL-Divergence Between Two Gibbs Measures
Suppose we have two Gibbs measures with densities
$$
p_f(x) \propto \exp(f(x)),\quad q_g(x)\propto \exp(g(x)).
$$
Consider the KL-divergence between $p_f$ and $q_g$, as a functional of $f$ and $g$, ...
- 1,127
9
votes
2
answers
988
views
Average size of extreme points of convex hull of $N$ points
Fix $n$ a (small) integer.
Let $N$ be a (big) integer. Consider $N$ random points in the $n$-dimensional unit cube $[0, 1]^n$. The $N$ points are independently uniformly distributed.
Define $V(N)$ ...
- 3,432
20
votes
2
answers
922
views
A functional inequality about log-concave functions
Let $f,g$ be smooth even log-concave functions on $\mathbb{R}^{n}$, i.e.,$f=e^{-F(x)}, g=e^{-G(x)}$ for some even convex functions $F(x),G(x)$. Is it true that:
$$
\int_{\mathbb{R}^{n}} \langle \...
- 3,946
1
vote
0
answers
83
views
Given a large random matrix, how to prove that every large submatrix whose range contains a large ball?
Context. Studing a problem in machine-learning, I'm led to consider the following problem in RMT...
Definition. Given positive integers $m$ and $n$ and positive real numbers $c_1$ and $c_2$, let's ...
- 6,853
1
vote
1
answer
59
views
Characterization of random variables whose tensor powers have subexponential "small-ball" probabilities
Is there a succinct characterization of all random variables $\zeta$ on $\mathbb R$ with the following properties
1. Symmetry: $\zeta \overset{d}{=} - \zeta$.
2. Small-ball probability: there exists ...
- 6,853
12
votes
3
answers
3k
views
Area Enclosed by the Convex Hull of a Set of Random Points
Consider $n$ points generated randomly and uniformly on a unit square. What is the expected value of the area (as a function of $n$) enclosed by the convex hull of the set of points?
- 123
5
votes
4
answers
906
views
Probability that convex hull of multivariate Gaussian sample contains a given point
I am generating random vectors $X_1, \dots, X_N$ from a $d$-dimensional multivariate normal $\text N(\mu, \Sigma)$. I would like to know what is the probability that a given point $y \in R^d$ falls ...
- 101
14
votes
1
answer
449
views
References for reasoning about the spectrum of a convex body?
By "spectrum of a convex body", I mean: start with a convex body $B$ in $\mathbb{R}^d$, then consider the corresponding $d \times d$ covariance matrix resulting from a uniform distribution over $B$ -- ...
- 143
1
vote
0
answers
248
views
Gaussian mean width of normal random cones
Suppose $1 \leq n < m < \infty$ are integers. For $g \sim \mathcal N(0, I_n)$ define the gaussian mean width of a non-empty set $T \subseteq \mathbb R^n$ by
$$
w(T) := \mathbb E \sup_{x \in T} \...
- 119
4
votes
0
answers
116
views
Log-Sobolev Inequalities for convex bodies
For a measure $\mu$ supported on a convex body $K$, what are the conditions on $\mu$ and $K$ to satisfy a Log-Sobolev inequality of the form:
$$\int f^{2} \log f^{2}\,d\mu -\int f^{2}\,d\mu \log\left(\...
- 519
4
votes
1
answer
290
views
On the 1/2 assumption on concentration of measure for continuous cube
The concentration of measure on $ [0, 1]^n $ equipped with uniform probability measure $\mu_{\infty}$,
states that for any $A \subset [0, 1]^n $ with $ \mu_{\infty}(A) \geq \frac{1}{2} $,
we have:
$$...
- 365
3
votes
3
answers
439
views
Inner radius of a random convex hull
Let $\sigma_1,\ldots,\sigma_M$ i.i.d. random vectors in $\mathbb{R}^d$, and for notational convenience, let $\Sigma=(\sigma_1,\ldots,\sigma_M)$. I am interested in understanding
$$ \gamma(\Sigma) = \...
- 980
11
votes
1
answer
609
views
Tighter Caratheodory on the moment curve?
The moment curve is the set of points of the form
$$(t,t^2,t^3,...,t^n) \in R^n$$
Let $M$ be the portion of the moment curve where $t\in [0,1]$, and let $\overline{M}$ be the convex hull of $M$.
...
- 3,979
15
votes
2
answers
2k
views
Intuitive explanation of Dvoretzky's theorem
I am wondering if anyone has an enlightening explanation of why Dvoretzky's theorem (which says that a high-dimensional convex body has an almost round central section) is true -- there are a number ...
- 96.4k
4
votes
0
answers
156
views
Geometric meaning of the chi-square "measure of association"
In Statistics, there's a standard test of independence of two random variables taking values in finite sets $I,J$. It relies on the computation of $\chi$-square statistics,
$$
\chi^2:=\sum_{(i,j)\in ...
- 8,992
0
votes
0
answers
165
views
Probability that the perturbed convex hull is larger than the original one
I am wondering if any convex geometers/probabilists have looked at the following question:
Given $n$ randomly distributed (not sure what assumption to put there) points in $\mathbb{R}^d$, for each ...
- 133