All Questions
Tagged with pr.probability mg.metric-geometry
223 questions
3
votes
0
answers
144
views
What is the probability that these four random areas can yield a tetrahedron?
This is inspired by this problem about randomly broken sticks that can form a triangle. It goes in a different direction than this generalization about randomly broken sticks that can form a ...
- 13.4k
3
votes
0
answers
119
views
Probability that a random projection doesn't reduce the distance of a point from a subspace too much
Consider the natural uniform measure (is it called the Haar measure?) on the set of $(n-k)$-dimensional subspaces of $R^n$. We are given a $d$-dimensional affine subspace $U$ (think of $d, k \ll n$; ...
- 111
3
votes
0
answers
146
views
The mean number of vertices in small connected components of random geometric graphs
I place $N$ points on a circular plane of radius $R$, and draw edges to connect points that are less than or equal to some distance $D$ to form a set of graphs or cliques $G_i$. As a function of $N$, ...
2
votes
3
answers
290
views
Geometric probabilistic problem on triangles on a plane
We are given a triangle $T$ on a plane $P$, with sidelengths $a$, $b$ and $c$, where $c \ge b \ge a > 0$. A straight line $L$ on $P$ is selected uniformly at random from the set of all the ...
- 1,677
2
votes
1
answer
1k
views
Sampling a uniformly distributed point INSIDE a hypersphere?
There is a simple algorithm to pick a random point ON an $n$-dimensional hypersphere.
Is there one to sample a point from inside it? (Sampling points from a hypercube and rejecting them if they are ...
- 1,667
2
votes
1
answer
185
views
$\zeta(2n)$ and Levy processes
I am missing some steps in the final derivation of a probabilistic computation of the even values of $\zeta$. They show the Cauchy distribution is relate to a certian Levy process:
$$ |\mathbb{C}_1| \...
- 22.8k
2
votes
2
answers
146
views
Volume of specials sets on sphere $S^N$
Suppose I'm given $m$ points $\{q_i\}$ on the sphere $S^N$. I want to get a lower/upper bound for the volume of the following sets with respect to uniform probability measure $\mathbb{P}$ on the ...
- 41
2
votes
2
answers
1k
views
Distance measure for noisy $SE(3)$ transforms
I have a transformation $T \in SE(3)$ parameterized by a mean quaternion $q$ with covariance matrix $\Sigma_q \in R^{4\times4}$ and a mean translation $t \in R^3$ with covariance matrix $\Sigma_t \in ...
- 21
2
votes
2
answers
248
views
On an angle distribution of a random linear subspace of a given dimension
$\newcommand\R{\mathbb R}$ Let $u$ be a fixed unit vector in $\R^n$, and let $\Pi_u$ be the hyperplane in $\R^n$ with normal vector $u$. Let $B$ be the (say open) unit ball in $\R^n$ centered at the ...
- 128k
2
votes
1
answer
245
views
Union bound over infinitely many events
Let $p_1, ... ,p_n$ be chosen independently from the uniform distribution on the unit torus $[0,1]^2$.
I want to prove a theorem of the form: "With high probability, every circle of radius $r$ ...
- 1,643
2
votes
1
answer
151
views
Given an input point in $\mathbb{R}^n$, select (one of) the closest point(s) from a fixed large set of points given in advance
We are given a set $S$ of $m\gg 1$ points in $\mathbb{R}^n$.
In the problem I am trying to solve, in a sequential fashion, we obtain a new point $p_r\not\in S$ at each round $r\ge 1$ and the goal is ...
- 1,677
2
votes
1
answer
350
views
The equilibrium position for the body of a spider-like spring system after randomly perturbing the anchor positions of its legs
Take $N$ springs, $(s_1, ..., s_N) \in S$ of length $(l_1, ..., l_N)$, and for each spring, label one end "A" and one end "B". Connect the "A" ends of the $N$ springs to a point-like particle on a ...
- 55
2
votes
1
answer
157
views
Inferring the location of a reflecting boundary in a toroidal cage with a Brownian particle
Let's say I have a Brownian particle of some radius $r_b$ and coefficient of diffusion $D$, freely moving about in a toroidal/doughnut-shaped chamber with inner and outer radius $R_{inner}$ and $R_{...
- 599
2
votes
1
answer
190
views
Estimating the volume of a convex shape in higher dimensions based only on normal sections
We are given a $d$-dimensional convex shape $S$ inscribed in the hypercube $[-1,1]^d$. We want find an approximation of its volume based only on a set of curves given by the intersection of the $S$ ...
- 1,677
2
votes
1
answer
404
views
Euclidean distance bound with geometric constraints
Let $S_n$ be a set of $n$ points belonging to $\mathcal{B}_d:=\{\mathbf{x}\in\mathbb{R}^d:\|\mathbf{x}\|_2\le 1\}$, where $d\ll \log(n)$.
Let $s_n$ and $\ell_n$ be respectively defined as follows:
$$...
- 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
2
votes
1
answer
715
views
The largest circle that encloses no points on a plane with points placed at $N$ random coordinates
I randomly scatter $N$ points on a bounded rectangular plane $P$ with dimensions $A \times B$. To be more specific, for $N$ iterations, I choose a real number $x \in [0, A]$ and a real number $y \in [...
- 75
2
votes
1
answer
319
views
Minimum distance to a sampled point with given pdf
Let $f(x)>0$ be a probability density function defined on the unit square $[0,1]^2$ in $\mathbb{R}^2$. Suppose that we take $N$ independent samples, $X_1,\dots,X_N$, of $f$. Now, sample a point $...
- 645
2
votes
2
answers
810
views
Terrain Generation: Infinite 2D space filled with Diffusion-limited aggregation clusters?
Disclaimer: I don't have a deep understanding of fractals or any higher math, I'm just personally interested in it, so please excuse me if I'm using wrong terms or if I'm being inaccurate. Making ...
- 121
2
votes
1
answer
81
views
Rate of convergence of random samples wrt Hausdorff distance
Let $X$ be a compact metric space with a probability measure $\mu$. We can draw random samples $X_n = \{x_1,\cdots, x_n\}$ from $X$ using $\mu$, and I am interested in the rate of convergence of $X_n$ ...
- 305
2
votes
1
answer
110
views
A questions concerning Laguerre/Voronoi tessellations
Fix $n>1$ distinguished points $x_1,\ldots, x_n\in \mathbb R^d$, the Voronoi tessellations are the subsets $V_1,\ldots V_n\subset\mathbb R^d$ defined by
$$V_k~~ := ~~ \big\{x\in\mathbb R^d:\quad |...
user128095
2
votes
2
answers
331
views
what's the best way to characterise the distribution of prime elements in simple perfect squared squares
DEFINITIONS: A squared rectangle is a rectangle dissected into a finite number, two or more, of squares, called the elements of the dissection. If no two of these squares have the same size the ...
2
votes
1
answer
235
views
The distance between the centroid of $P$ points and the centroid of a subset of the points
Imagine I have an $(p_1, ..., p_N) \in P$ points, on a two-dimensional plane, patterned in a rectangular or hexagonal lattice arrangement in a circle of radius $R_c$, with a spacing between the points ...
- 55
2
votes
0
answers
100
views
Distributions of random walks on boundaries of balls in hyperbolic metric spaces
Suppose $G$ is a finitely-generated non-elementary hyperbolic group and consider a symmetric random walk on the Cayley graph $\text{Cay}(G,S)$ with generating set $S$. Denote the set of points $B_{\...
- 21
2
votes
0
answers
124
views
Generalization of the triangle inequality to complex exponents: What is $P\left(\left| x^{a+bi} + y^{a+bi} \right| \ge \left|z^{a+bi}\right|\right)$?
Let $x \le y \le z$ be the length of the sides of a triangle whose vertices are uniformly random on the circumference of a circle. In this question, it was proved that if $a \ge 1$, then the ...
- 5,470
2
votes
0
answers
64
views
Limiting distribution of separated points in a unit square
Let $n$ and $r$ be fixed, and consider the following process, with $S=\emptyset$ to start:
For $i\in\{1,\dots,n\}$:
Sample a random point $X$ in the unit square.
If $X$ is a distance at least $r$ ...
- 4,049
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
2
votes
0
answers
159
views
Maximizing $\iiint|(x-z)\times(y-z)|d\mu d\mu d\mu$ over probability measures on the unit sphere
This is a follow-up question to the one asked here (the unit circle case). What probability measure(s) maximize the quantity $\iiint_{\mathbb{S}^2}|(x-z)\times(y-z)|d\mu(x)d\mu(y)d\mu(z)$?
The ...
- 3,209
2
votes
0
answers
83
views
Wasserstein distance to the set of Gaussians, relation to Boltzman dissipation rate
$\newcommand{\RR}{\mathbb{R}}$
I am interested in the 2-Wasserstein distance for probabilities over $\RR^n$,
$W_2(μ,ν)=\left(\inf\int_{\RR^n×\RR^n}|w−v|^2dπ(v,w)\right)^1/2$
where the infimum is taken ...
2
votes
0
answers
60
views
Mean width of intersection of two elipsoid
My question is regarding mean widths. For a set $\mathcal{T}$ define the mean width
\begin{align*}
\omega(T)=\mathbb{E}_{\mathbf{g}\sim\mathcal{N}(0,\mathbf{I})}\bigg[\underset{\mathbf{u}\in\mathcal{...
- 363
2
votes
0
answers
59
views
Totally distance non-preserving transformations
JL lemma (https://en.wikipedia.org/wiki/Johnson%E2%80%93Lindenstrauss_lemma)
guarantees if you have a set of $K$ points in $\Bbb R^N$ a random transformation guarantees that the set can be projected ...
- 13.9k
2
votes
0
answers
104
views
Expected Area of Triangle Formed form Region [1,n]
We take three pieces of random lengths from the interval $[1,n]$, and then guarantee that they can form a triangle (ie that the triangle inequality is satisfied). That is to say we say that the sum of ...
2
votes
0
answers
157
views
Better Sobolev inequality holds in this case when assuming doubling and Poincare inequality?
Let $X$ be a Polish space and let $m$ be a locally finite Borel measure on $X$.
Let $\epsilon$ be a strongly local, regular Dirichlet form on $L^2(X,m)$ with Domain $V :=\{f\in L^2(X,m):\epsilon(f)&...
- 471
2
votes
1
answer
121
views
Expected length of a certain kind of nearest-neighbor graph
Suppose I have sets of points $Z_1,\dots,Z_N$, such that $|Z_i|=m$ for all $i$, and where all $m\times N$ points are independently distributed uniformly at random in the unit square. Can someone give ...
1
vote
1
answer
555
views
The probability a self-avoiding random walk (SAW) on a rectangular or hexagonal lattice takes more than $N$ steps before trapping itself
What is the probability that a self-avoiding random walk (SAW) on a rectangular or hexagonal lattice is able to take more than $N$ steps, i.e. able to take more than $N$ steps before trapping itself ...
- 55
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
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
1k
views
Pairwise distance distribution for point clouds (normal distribution) [closed]
I have a point cloud (2D for now) of $N$ normally distributed points (with a certain $\sigma$).
My first question would be how the pairwise distance distribution looks (just by chance I discovered a ...
user76454
1
vote
1
answer
1k
views
Closed-form upper-bounds for Wasserstein distance between finite measures
Let $x_1,\dots,x_n,y_1,\dots,y_n\in \mathbb{R}$ and such that $x_i\neq x_j$ and $y_i\neq y_j$ if $i\neq j$. Let $a,b$ be elements of the probability n-simplex. Define the measures $\mu\triangleq \...
- 5,405
1
vote
1
answer
301
views
Distances between and among points in a region
Let $X = \{x_1, \dots, x_n\}$ denote a finite set of $n$ points in the unit square $S$, and let's center $S$ at the origin. Let $F(X) = \sum_{i=1}^n \| x_i \| $ and let $G(X) = \iint_S \min_i \|x - ...
- 195
1
vote
1
answer
115
views
Approximating the probability of a half-space using random Voronoi diagrams
Fix a half-space $H = \{x_1 \geq 0: ~ (x_1,\dots,x_n) \in \mathbb{R}^n\}$. Let $p$ be a distribution with support in $\mathbb{R}^n$. I am interested in the following way of estimating the weight $p(H) ...
- 33
1
vote
1
answer
100
views
$L^p$-barycenters via continuous selectors
Let $\mathbb{P}$ be a Borel probability measure on $\mathbb{R}^n$ with $\int_{x \in \mathbb{R}^n} \|x\|^p\mathbb{P}(dx)<\infty$. For which $p\in [1,\infty)$ (other than $p=2$) is the following ...
- 5,405
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
1
vote
1
answer
242
views
Barycenter Map on Wasserstein Space
Let $(X,d)$ be a complete separable metric space, $P_1(X,d)$ be the set of Radon probability measures on $X$ satisfying
$$
P_1(X,d)\triangleq \left\{
\nu:\,(\exists x_0\in X)\, \int_{x\in X} d(x,x_0)d\...
- 5,405
1
vote
1
answer
138
views
Least square assignment and hyperplanes
Let $S$ be a finite set of points in $\mathbb{R}^{d}$, $c(s) \in [0,1]$ such that $\sum_{s \in S} c(s) = 1$, $\rho$ continuous and non-vanishing probability distribution on $[0,1]^{d}$ and $\mu $ ...
user178738
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 ...
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
1
vote
1
answer
337
views
Estimating the volume of a union of balls
Let $\{ B_i \}_{i=1}^n$ be a set of $n$ ball in the unit cube $C$ of dimension $d$.
If I want to estimate
$$
\frac{ \lambda \left( \cup B_i \right) }{\lambda\left( C \right) }, \tag{1}
$$
where $\...
user24451
1
vote
1
answer
516
views
Properties of the minimum circumscribing circle for points sequentially placed within a circle of radius $r$ of previously placed points on a 2D plane
Imagine I perform the following procedure:
[1] At time point $t_1$, I place a single point on a two-dimensional plane at the coordinate $(x, y) = (0, 0)$.
[2] At time point $t_2$, I center a ...
- 197
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