All Questions
Tagged with pr.probability mg.metric-geometry
58 questions with no upvoted or accepted answers
18
votes
0
answers
667
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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
15
votes
0
answers
398
views
Will a unit disk be completely covered by randomly placed disks of area $\pi,\frac{\pi}{2},\frac{\pi}{3},\dots$ with probability $1$?
On a "bottom" disk of area $\pi$, we place "top" disks of area $\pi,\frac{\pi}{2},\frac{\pi}{3},\dots$ such that the centre of each top disk is an independent uniformly random ...
- 3,547
11
votes
0
answers
307
views
Entropy, magnitude, diversity of finite metric spaces in number theory
I was reading the article by Tom Leinster, (Maximizing
diversity in biology and beyond, arXiv link), and find it very interesting.
Since I was searching for entropies of finite metric spaces I found
...
user6671
11
votes
0
answers
601
views
High-dimensional geometry: Top-down Vs. Bottom-up
There are several ways to leverage one's intuition from low-dimensional geometry to understand high-dimensional phenomena. For example, one can get a clearer picture of the behaviour of high-...
- 1,666
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
8
votes
0
answers
257
views
Variations on Gauss' trick
Cross-posted from MSE. This question is inspired by these two:
Non-trivial values of error function erf(x)?
Where is the mass of a hypercube?
Upon reading these two, I realized there might be a ...
- 956
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
7
votes
0
answers
162
views
Approximating any convex shape in $\mathbb{R}^d$ with a polytope having $\mathrm{poly}(d)$ facets
We denote by $V(A)$ the $d$-volume of any convex set $A$. Furthermore, given any two convex sets $A,B\in\mathbb{R}^d$, we denote by $V_{A,B}$ the $d$-volume of the symmetric difference $V\left(A \...
- 1,677
7
votes
0
answers
122
views
Discrepancy of the finite approximation of the Lebesgue measure
Let $\mu$ be a probabilistic measure on the unit square $Q$ which is the average of $N$ delta-measures in some points in this square; let $\lambda$ denote the Lebesgue measure on $Q$. What is the rate ...
- 109k
7
votes
0
answers
209
views
Stabbing disks in space, or: Galactic alignment
I have a collection of $n$ unit-radius disks in $\mathbb{R}^3$, whose centers are
random within a sphere of radius $R>1$, and which are each oriented randomly.
I'd like to find a line $L$ that ...
- 151k
6
votes
0
answers
197
views
What are compact manifolds such that GROWTH (of spheres volumes) is well approximated by the Gaussian normal distribution?
Consider some compact Riemannian manifold $M$. Fix some point $p$.
Consider a "sub-sphere of radius $r$" - i.e. set of points on distance $r$ from $p$.
Consider growth function $g(r)$ to be ...
- 24.9k
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
5
votes
0
answers
184
views
Question about $n$ random points in a regular polygon, and a limiting probability
Suppose we choose $n$ uniformly random points in a disk, then draw the smallest circle that encloses all of those points. There is evidence suggesting that the probability that the enclosing circle is ...
- 3,547
5
votes
0
answers
74
views
Concentration bound on additive functions with constraints
Given a family of sets $F \subseteq P(\{1,\ldots,n\})$. I define the function $f_F:[0,1]^n \rightarrow R$ to be $f_F(x_1,\ldots,x_n)= \max_{S \in F} \sum_{j \in S} x_j$.
Given a series of independent ...
- 107
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
$$ \frac{ \left| \left\{ x_1,\dots,x_n \in X \mid ...
- 148k
5
votes
0
answers
275
views
Is there a connection between |roots| $\rightarrow$ 1 and Gromov's waist theorem?
Recent questions showed that roots of a random polynomial tend to lie on the
unit circle ("Why do roots of polynomials tend to have absolute value close to 1?"; "Distribution of roots of complex ...
- 151k
4
votes
0
answers
52
views
Isomorphism of Wasserstein space implies isomorphism of base spaces?
Assume $(X_i,d_i)$ are polish spaces (or compact metric spaces) for $i=1,2$.
Further assume that the 1- Wasserstein spaces $(P_1(X_1),W_1)$ and $(P_1(X_2),W_1)$ are isometrically isomorphic. Does that ...
4
votes
0
answers
87
views
Statistics of random Voronoi S-tessellations
Given a locally finite set of points $\{x_1,x_2,\dots\}\subset\mathbb{R}^d$, the Voronoi cell of a point $x_{i}$, denoted by $C(x_{i})$, consists of all the points in $\mathbb{R}^d$ that are closer to ...
- 41
4
votes
0
answers
144
views
Approximation of a convex shape in the $d$-dimensional Euclidean space for $d\gg 1$
We are given a convex shape $C$ lying inside the hypercube $[0,1]^d$ in the $d$-dimensional Euclidean space. Let the volume of $C$ be $\tfrac12$ (I guess nothing changes for any other fixed constant ...
- 1,677
4
votes
0
answers
156
views
Geometric meaning of the chi-square "measure of association"
In Statistics, there's a standard test of independence of two random variables taking values in finite sets $I,J$. It relies on the computation of $\chi$-square statistics,
$$
\chi^2:=\sum_{(i,j)\in ...
- 8,992
4
votes
0
answers
756
views
Tangent space and gradient on subspace of Wasserstein space given by finitely supported measures
Let $\mathcal{P}_2(M)$ be the 2-Wasserstein space over some Riemannian manifold $(M,g)$ (connected, complete, and without boundary). Let $\mathcal{FP}_2(M,n)$ be the subspace of probability measures ...
- 1,675
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
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
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
4
votes
0
answers
173
views
On understanding Discrete-Valued Stochastic Processes( time series, panel data )
It seems to me that a significant proportion of work in probability theory, statistics and machine learning are on understanding continuous-valued, relatively weakly dependent, or linear dependent ...
- 41
4
votes
0
answers
128
views
Metrized categories
Motivation: Let $\Gamma = (V,E)$ be a directed graph. To each edge $e \in E$, choose a value $\kappa^e \in \mathbb R$, representing the cost of transporting one unit of "stuff" through the edge. Let $\...
- 8,512
3
votes
0
answers
228
views
Are 1-Wasserstein and 2-Wasserstein distances between multivariate normal distributions equivalent?
The $p$-Wasserstein between two measures $\nu_1$ and $\nu_2$ on $X$ is given by
$$W^p_p(\nu_{1},\nu_{2})=\underset{\pi\in\Gamma(\nu_{1},\nu_{2})}{\inf}\int_{\mathbf{\mathcal{X}}^{2}}d(x,y)^p\pi(dx,dy)$...
- 1,058
3
votes
0
answers
187
views
Approximating any $d$-dimensional convex shape that occupies a constant fraction of its bounding box with a polytope having $\mathrm{poly}(d)$ facets
Given any convex set $A\in\mathbb{R}^d$, we denote by $V(A)$ its $d$-volume. Furthermore, given any two convex sets $A_1,A_2\in\mathbb{R}^d$, we denote by $V_{A_1,A_2}$ the $d$-volume of the symmetric ...
- 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
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
3
votes
0
answers
234
views
Are random convex polygons on a sphere themselves sphere-like?
Say $\mathbb{R}^n$ is divided by $k>n$ randomly chosen hyperplanes. Each connected region away from the hyperplanes is the intersection of $k$ half-spaces, so it is a convex cone. It is known that ...
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
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
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
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
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
1
vote
0
answers
184
views
Bounding the total variation metric between Gaussian mixtures
Let $\mathcal{P}(\mathbb{R}^d)$ the space of probability measures on $\mathbb{R}^d$ with total variation metric $\delta$, fix $k \in \mathbb{N}$, and let $\mathcal{P}'\subset \mathcal{P}(\mathbb{R}^d)$...
- 5,405
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
1
vote
0
answers
50
views
Volume estimates of rooted embedded tree containing certain subtrees.
Consider a rooted embedded tree of $n+1$ vertices. It is known that around the root for small $r$, volume of the ball of radius $r$ grows like $r^2$. Now suppose we are given that a certain subtree is ...
- 141