All Questions
Tagged with mg.metric-geometry pr.probability
223 questions
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
7
votes
2
answers
460
views
Gaussian Surface Area of Positive Semidefinite Cone
Let $\mathbb{R}^n$ be the Euclidean space and $A \subseteq \mathbb{R}^n$ be a sufficiently regular set, e.g., one that has smooth boundary or is convex. We define the $\epsilon$-neighbor of $A$ in the ...
- 1,127
19
votes
3
answers
931
views
Is the circle in the square best at avoiding random lines?
This question is inspired by a recent one (and takes a great deal from the answers there). Given a convex subset $\Delta$ of the unit square, let $p(\Delta)$ be the probability that a random line does ...
- 30.1k
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 ...
20
votes
5
answers
1k
views
Probability that biggest area stays greater than 1/2 in a unit square cut by random lines
The square $[0,1]^2$ is cut into some number of regions by $n$ random lines. We can chose these random lines by randomly picking a point on one of the four sides, picking another point randomly from ...
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
4
votes
1
answer
263
views
Knotted TSP tours in 3D?
In the plane, the Euclidean TSP tour never crosses itself—it is always a simple polygon.
I am wondering if there is a similar constraint for the Euclidean TSP tour
of points in $\mathbb{R}^3$.
...
- 151k
5
votes
2
answers
474
views
Another graph characteristic
This question concerns a method of drawing graphs and a graph characteristic about which I want to learn more.
Consider a connected directed graph with at least one node with in-degree 0 and one node ...
- 9,720
4
votes
1
answer
355
views
Random spherical caps cover a spherical cap
Let $S^{n-1}$ be the unit sphere in $n$ dimensional Euclidean space. Define the spherical cap at $x \in S^{n-1}$ with angle $\theta$ to be $C(x,\theta) = \{z \in S^{n-1} \mid z^\top x \geq \cos(\theta)...
- 83
4
votes
1
answer
133
views
Union of random half spaces cover a ray
Let $x, y \in \mathbb{R}^{n}$ be two fixed unit vectors with angle $\alpha \in (\frac{\pi}{2}, \frac{3\pi}{4})$. Define the positive half space associated with a vector $z$ to be $\mathcal{H}(z) = \{h ...
- 83
32
votes
1
answer
1k
views
Does projection of 3D points reduce distances by exactly 1/3?
Let $P$ be a set of $n$ random points uniformly distributed inside
a unit-radius sphere centered on the origin.
Orthogonally project $P$ to a random plane through the origin;
call the projected points ...
- 151k
10
votes
1
answer
494
views
Ping-pong progress through a quincunx
A quincunx or
Galton board consists of
staggered pegs from which ping-pong balls bounce and eventually display
a binomial / normal distribution in catch-bins. I am wondering if the
downward progress ...
- 151k
3
votes
1
answer
105
views
Asymptotic radius of the smallest enclosing ball
Let $X_1,..., X_n$ be i.i.d. $d$-dimensional standard normal random variables, and let $R_n$ be the radius of the smallest ball containing $X_1,...,X_n$. What is known about the distribution of $R_n$ ...
- 724
6
votes
0
answers
281
views
Covariance operator analogue for manifolds and respective measure manifolds
Assume $E$ is a connected riemannian manifold with geodesic metric space structure given by $d$ and $P$ is a probability measure over $E$ with Borel sigma-algebra given by this metric structure. Also ...
- 519
7
votes
1
answer
318
views
Finding a short path using $(0.99n)!$ permutations
Suppose I have $n$ points $x_1,\dots,x_n$ that are all independent uniform samples in the unit square, and I'd like to find a short path (in terms of Euclidean length) that touches all of them (a ...
- 4,049
10
votes
2
answers
846
views
Minimum separation among $m$ random points on an $n$-dimensional unit sphere
Consider $m$ points $v_1, \ldots, v_m \in R^{n}$, which are uniformly distributed on the $n$-dimensional unit sphere $S^{n-1} = \{v:\|v\|_2 = 1\}$. Let the minimum separation be
$$
\rho = \min_{i,j\in{...
- 1,127
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
5
votes
0
answers
266
views
Throwing darts at a barn and putting a bullseye around them in higher dimensions
Let $X \in \mathbb R^d$ be a large domain (a ball of radius $r$ for $r$ large should suffice)
Let $B$ be a ball of radius $1$.
Consider the ratio
$$ \frac{ \left| \left\{ x_1,\dots,x_n \in X \mid ...
- 148k
19
votes
1
answer
448
views
Precise estimate for probability an $n$-point set has diameter smaller than $1$
This question was inspired by an earlier question that I answered but would like a more precise bound for.
Consider random points $x_1, \dots, x_n$ in the unit ball in $\mathbb R^d$, uniformly and ...
- 148k
14
votes
2
answers
319
views
Shortest path through $n^{1/3}$ points out of $n$
Say I sample $n$ points uniformly at random in the unit cube in $\mathbb{R}^3$, and then I look for the shortest path through $n^{1/3}$ of those points (rounding up, say). What happens to the length ...
- 141
3
votes
1
answer
187
views
Moment matching on the standard simplex
Let $\vec{\mu}_1, \vec{\mu}_2,\ldots, \vec{\mu}_k \in \Delta^{d-1}$ be $k\ (k\geq 2)$ distinct vectors on the standard simplex, where
$$\Delta^{d-1} = \{\vec{\mu}\in R^{d}:\| \vec{\mu}\|_1 = 1,\mu_j \...
- 1,127
10
votes
1
answer
484
views
Stochastic Covering Number of a Convex Set
Consider a convex set, say $S = [0,1]^d$. Let $X_1, X_2,\ldots,X_n, \ldots$ be i.i.d. random variables that are uniformly distributed on $S$. Denote the Euclidean ball centered at $x \in \mathbb{R}^d$ ...
- 1,127
4
votes
0
answers
94
views
Finding closest set of K disjoint hyperspheres to a point in $\mathbb{R}^n$ with uniform radius
I am interested in the following problem: in $\mathbb{R}^n$, we have $N$ overlapping hyperspheres all with the same radius. Given a point $p$ in $\mathbb{R}^n$, the objective is to find the $K$ non ...
- 161
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
7
votes
1
answer
757
views
Length of nearest neighbor path in travel salesman problem
Given $n$ nodes uniformly distributed in $[0,1]^2$, consider the nearest neighbor algorithm to solve traveling salesman problem, i.e., each time I select the nearest neighbor not visited so far as the ...
- 367
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
9
votes
4
answers
371
views
Diameter of random segment intersection graph?
I have an even number of points $n$ randomly distributed (uniformly) in a disk.
Then the points are randomly connected to form $n/2$ segments, a perfect
matching.
Finally, I form the intersection ...
- 151k
0
votes
0
answers
320
views
Gromov-Hausdorff distance measure between minimum spanning trees
I am trying to compare minimum spanning trees through time. I have two questions:
1-Is it possible to measure the similarity between two minimum spanning trees with Gromov-Hausdorff distance measure ...
- 1
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 ...
21
votes
2
answers
3k
views
A measure on the space of probability measures
This question was originaly posted in the stackexchange https://math.stackexchange.com/questions/1226701/a-measure-on-the-space-of-probability-measures but since it only got a comment I decided to ...
- 361
8
votes
0
answers
183
views
Can the GUE be thought of as a uniform point in a high-dimensional polytope
I have thought about this question for a long time and could only find partial answers.
The Gaussian Unitary Ensemble (or GUE) is the eigenvalues of a random Hermitian matrix with complex Gaussian ...
- 22.8k
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
12
votes
1
answer
694
views
History of the Jaccard distance $d(A,B) = \mathbb P(\overline A\cup\overline B\mid A\cup B)$
I'm wondering where the relative probabilistic distance or Jaccard distance was first studied:
$$d(A,B) =\mathbb P(\overline A\cup\overline B\mid A\cup B)$$
where $\overline A$ is the complement of $A$...
- 24.8k
3
votes
1
answer
197
views
Three-dimensional Apollonian spirals
Given mutually (externally) tangent spheres $S_1$, $S_2$, $S_3$, $S_4$, let $S_n$ be the unique sphere externally tangent to $S_{n-1}$, $S_{n-2}$, $S_{n-3}$, and $S_{n-4}$ for $n \geq 5$.
Let $P_{\...
- 19.7k
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
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
3
votes
1
answer
492
views
Random non-intersecting circles in the plane
If I give a finite region of $\mathbb{R}^{2}$ and place $k$ circles of radius $r(k)$ uniformly at random inside, are there any known results for the probability that the circles do not overlap? ...
- 903
10
votes
4
answers
904
views
The distribution of the shortest path through $n$ points
In the big picture, I'd like to know: if I sample $n$ points uniformly at random in the unit square, what is the probability that the shortest path that visits each one of them is very small?
More ...
- 121
5
votes
0
answers
275
views
Is there a connection between |roots| $\rightarrow$ 1 and Gromov's waist theorem?
Recent questions showed that roots of a random polynomial tend to lie on the
unit circle ("Why do roots of polynomials tend to have absolute value close to 1?"; "Distribution of roots of complex ...
- 151k
10
votes
1
answer
673
views
A random variation on Pólya's orchard problem
Pólya's orchard problem is as follows:
"How thick must the
trunks of the trees in a regularly spaced circular orchard grow if they are
to block completely the view from the center?"
See, e....
- 151k
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
0
answers
211
views
Quadrilaterals from a Unit Stick
This question could be seen as a coordinate-free variant of Sylvester's Four Point Problem (cf e.g. http://mathworld.wolfram.com/SylvestersFour-PointProblem.html):
Suppose one are given an ...
- 13.2k
18
votes
0
answers
667
views
The lonely molecule
Suppose $n$ air molecules (infinitesimal points) are bouncing around in
a unit $d$-dimensional cube, with perfectly elastic wall collisions.
Let $k=n^{\frac{1}{d}}$.
For example, in 3D, $d=3$, with $n=...
- 151k
30
votes
2
answers
1k
views
Shortest path through $\sqrt{n}$ points out of $n$
Say I sample $n$ points uniformly at random in the unit square, and then I look for the shortest path through $\sqrt{n}$ of those points (rounding up, say). What happens to the length of this path as ...
- 335
3
votes
2
answers
207
views
Connectivity of points sampled in a grid
Suppose that I partition an $n\times n$ square into $n^2$ squares $S_1 ,\dots, S_{n^2}$ each of area $1$, and then I sample a point $X_i$ uniformly at random in each $S_{i}$. Now fix a radius $r$ and ...
- 31
40
votes
5
answers
5k
views
"Entropy" proof of Brunn-Minkowski Inequality?
I read in an information theory textbook the Brunn-Minkowski inequality follows from the Entropy Power inequality.
The first one says that if $A,B$ are convex polygons in $\mathbb{R}^d$, then
$$ m(...
- 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
3
votes
1
answer
202
views
Diameter of $n$-unit-vector closed scribble
Suppose one creates a random, closed, likely self-crossing polygon
from $n$ unit-length vectors arranged head-to-tail,
randomly oriented except for the requirement
that their sum is zero (so the ...
- 151k
12
votes
1
answer
2k
views
Limit of distance between two random points in a unit-radius $n$-sphere
This is a companion contrast to the earlier analogous question for unit $n$-cubes,
where the answer (provided by several respondents) is $\infty$ .
What is the limit, as $n \to \infty$, of the ...
- 151k
11
votes
2
answers
3k
views
Limit of distance between two random points in a unit $n$-cube
What is the limit, as $n \to \infty$, of the expected distance between two
points chosen uniformly at random within a unit edge-length hypercube
in $\mathbb{R}^n$?
For $n=1$, the average distance ...
- 151k