All Questions
25 questions
9
votes
0
answers
242
views
Does there exist such a probability distribution?
Does there exist a probability distribution over the set $\{(x,y,z)\in[0,1]^3\colon x+y+z=3/2\}$ whose projection on each of the three coordinate axes is the uniform distribution over the interval $[0,...
- 128k
0
votes
0
answers
24
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Is there a log-concave distribution not spherical symmetric s.t $ \langle X, \theta \rangle$ is almost normal for all directions $\theta$?
Klartag's results indicate that for a log-concave isotropic random vector, with high probability over $\theta$, $\langle X, \theta \rangle$ is close to a normal distribution.
It is known that for the ...
- 11
7
votes
1
answer
508
views
An order statistics problem with some interesting geometry
Let $a_n$ be a given sequence of positive numbers, and $X_n$ a sequence of independent random variables with each $X_n$ uniformly distributed on $[0, a_n]$.
Question: Let $N \geq 2$ be an arbitrary ...
- 6,215
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
1
answer
288
views
Probability that three vectors of a unit sphere lie on one side of a hyperplane if angle between the vectors are given
As the title says, How to find the probability of vectors a, b, c, on some unit sphere, all lies on same side of some hyperplane passing through the origin. Information present are the angles between ...
- 23
0
votes
1
answer
146
views
Upper bound of Wasserstein distance given by subvariables of codim 1
recently I am considering the upper-bound of Wasserstain distance. Say we have random vectors $X,Y$ of dimension $n$, and let $\tilde{X}_i (\tilde{Y}_i,$ resp.) be the $(n-1)$-dim random vector of $X (...
- 363
3
votes
1
answer
206
views
Random planes separating points in $\mathbb{R}^3$
We are given a unit origin-centered sphere $S$ in $\mathbb{R}^3$, and three points $\mathbf{x},\mathbf{y},\mathbf{z}\in S$. Let $\mathbf{h}$ be a point selected uniformly at random from $S$ and let $H$...
- 1,677
1
vote
1
answer
117
views
Modulus of continuity of parameterizing Wasserstein
Let $x_1,\dots,x_n\in X$ some Polish space $X$ and let $\Delta$ be the probability simplex in $\mathbb{R}^n$. Consider the map sending every $(w_1,\dots,w_n)\in\Delta$ to the finitely supported ...
0
votes
0
answers
113
views
How much a probability distribution is non-uniform in a convex subspace of $\mathbb{R}^d$?
I know a number of (standard and well known) ways to measure the distance between two probability distributions and, more in general, to quantify how much one is far from another.
Could you please ...
- 1,677
2
votes
0
answers
174
views
Random sets of points and hyperplanes in high dimensions
We are given a set $X$ of $n$ points $\mathbf{x}_1, \mathbf{x}_2, \ldots, \in\mathbb{R}^d$ selected uniformly at random from the unit origin-centered ball $\mathcal{B}^{d}$.
Consider the random ...
- 1,677
1
vote
1
answer
92
views
Geometric sampling problem in the Euclidean space in high dimensions
Let $T$ be the triangle whose vertices are three given points $\mathbf{x}, \mathbf{y}, \mathbf{z}\in\mathbb{R}^d$.
Question: What computationally efficient strategy can we use to sample a point $\...
- 1,677
1
vote
2
answers
197
views
Sampling method for a specific distribution in high dimensions
We are given a set $X$ of $n$ points $\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n\in\mathbb{R}^d$, where $d\ll n$. Given any point $\mathbf{p}$ on the unit $(d-1)$-sphere $\mathcal{S}$, we define
...
- 1,677
2
votes
2
answers
379
views
Probabilistic combinatorial optimization problem on the distances between pairs of points in $[0,1]$
Let $S$ be a set of $n \gg 1$ points lying on the interval $[0,1]$. Given a point $p\in[0,1]$, let $S_p\subseteq S\times S$ be the set formed by all pairs of points $(x,y)$ with $x,y\in S$, such that ...
- 1,677
3
votes
0
answers
132
views
Probability that a Voronoi cell contains exactly k random points
Consider two independent point processes in the unit square $[0,1]^2$. The two point processes are identically independent and typically binomial/Poisson. One, say $\Phi^*$, is used to generate a ...
- 137
1
vote
0
answers
113
views
Metrics on the space of distributions in terms of p.d.fs
If two probability distributions (on the same measure space) are s.t they have p.d.fs and the $L^1$ distance between the p.d.f.s is large, then is there a choice of a ``nice" metric $d_{\rm ...
- 2,246
3
votes
0
answers
253
views
Metric ($f$-divergence) on space of probability measures that satisfies pythagorean theorem
Let $E$ be a polish space, $\mathcal{P}(E)$ the Borel probability measures on $E$ with the topology of weak convergence and $\mathcal{Q} \subset \mathcal{P}(E)$ a convex and compact set.
First, the ...
- 1,095
1
vote
1
answer
239
views
Probability of two Points being divided by an high-Dimensional Hyperplane
I have two points $x_1,x_2 \in \mathbb S^n $ which are distant $d$ from each other, where $d<<1$.
I also have a vector $v$ sampled uniformly at random from $\mathbb S^n$.
What is the ...
- 899
13
votes
1
answer
10k
views
KL divergence and mixture of Gaussians
Do we have an exact formula to compute the KL divergence between 2 mixtures of Gaussians (i.e convex combinations of a finite number of Gaussian distributions)?
If not exactly known, are there good ...
- 2,246
4
votes
0
answers
100
views
Asymptotics of the joint pdf of two sums of powers of independent $\mathcal U(0,1)$ random variables
As a warm-up in words: The sum of twelve uniform random variables is a classic approximation to a normal distribution. What is the joint pdf for the sum of their cubes and the sum of their fourth ...
- 128k
9
votes
1
answer
2k
views
Uniform sampling from general simplex with a twist
This is part of a question I had asked elsewhere, and then some of the links redirected me to CS stack exchange.
Given $0\leq a_1\leq\dots\leq a_D\leq1$ (all strictly positive), I want to draw points ...
- 221
3
votes
1
answer
187
views
Moment matching on the standard simplex
Let $\vec{\mu}_1, \vec{\mu}_2,\ldots, \vec{\mu}_k \in \Delta^{d-1}$ be $k\ (k\geq 2)$ distinct vectors on the standard simplex, where
$$\Delta^{d-1} = \{\vec{\mu}\in R^{d}:\| \vec{\mu}\|_1 = 1,\mu_j \...
- 1,127
17
votes
2
answers
1k
views
A probability distribution in n dimensional space which its projection on any line is a uniform distribution?
Does there exist, for any natural $n$, a probability distribution in $\mathbb{R}^n$ whose projection on any line is a uniform distribution?
- 761
6
votes
1
answer
718
views
What is the distribution of the maximum nearest-neighbor distance of a point cloud sampled from a solid body like?
Let $\mathcal{B} \subseteq \mathbb{R}^n$ be an $n$-dimensional solid body. Assume that we sample $N$ points, say $S = \{ x_1, ..., x_N \}$, from $\mathcal{B}$ uniformly at random. Consider the ...
1
vote
1
answer
524
views
Convergence in the Wasserstein metric and the square root function
Let $f$ be a smooth probability distribution on the unit square $S$ such that $f(x)>0$ on $S$. Let $\{g_i\}$ be a sequence of smooth probability distributions such that $g_i(x)>0$ on $S$ as ...
- 11
6
votes
1
answer
333
views
Trasportation metric (AKA Earth-Mover's, Wasserstein, etc.) as "natural" / "induced"?
Context: Given a discrete finite metric space $X$ (in my case X={0,1}$^n$ with the Hamming/L$_1$ distance), I need to define the natural or canonical metric on the set of all probability distributions ...
- 111