Related papers: Sharp estimates on random hyperplane tessellations
A tessellation of a graph is a partition of its vertices into vertex disjoint cliques. A tessellation cover of a graph is a set of tessellations that covers all of its edges. The $t$-tessellability problem aims to decide whether there is a…
We study the sublinear multivariate mean estimation problem in $d$-dimensional Euclidean space. Specifically, we aim to find the mean $\mu$ of a ground point set $A$, which minimizes the sum of squared Euclidean distances of the points in…
We consider homogeneous STIT tessellations in the $\ell$-dimensional Euclidean space ${\mathbb R}^\ell$. Based on results for the spatial $\beta$-mixing coefficient an upper bound for the variance of additive functionals of tessellations is…
We establish higher-order nonasymptotic expansions for a difference between probability distributions of sums of i.i.d. random vectors in a Euclidean space. The derived bounds are uniform over two classes of sets: the set of all Euclidean…
Based on general relativity, it can be argued that deviations from a uniform Hubble flow should be thought of as variations in the Universe's expansion velocity field, rather than being thought of as peculiar velocities with respect to a…
The Erd\H os unit distance conjecture in the plane says that the number of pairs of points from a point set of size $n$ separated by a fixed (Euclidean) distance is $\leq C_{\epsilon} n^{1+\epsilon}$ for any $\epsilon>0$. The best known…
We solve the problem of finding a sharp upper bound on the minimum angle formed by $N$ points in the Euclidean and Hyperbolic planes.
Statistical divergences are ubiquitous in machine learning as tools for measuring discrepancy between probability distributions. As these applications inherently rely on approximating distributions from samples, we consider empirical…
We consider two independent symmetric Markov random flights $\bold Z_1(t)$ and $\bold Z_2(t)$ performed by the particles that simultaneously start from the origin of the Euclidean plane $\Bbb R^2$ in random directions distributed uniformly…
This paper presents a number of new findings about the canonical change point estimation problem. The first part studies the estimation of a change point on the real line in a simple stump model using the robust Huber estimating function…
A set of n segments in the plane may form a Euclidean TSP tour, a tree, or a matching, among others. Optimal TSP tours as well as minimum spanning trees and perfect matchings have no crossing segments, but several heuristics and…
We introduce two versions of a new sketch for approximately embedding the Gaussian kernel into Euclidean inner product space. These work by truncating infinite expansions of the Gaussian kernel, and carefully invoking the…
Recently a new class of planar tessellations, named T-tessellations, was introduced. Splits, merges and a third local modification named flip where shown to be sufficient for exploring the space of T-tessellations. Based on these local…
In this report, the explicit probability density functions of the random Euclidean distances associated with regular hexagons are given, when the two endpoints of a link are randomly distributed in the same hexagon, and two adjacent…
We investigate how the complexity of Euclidean TSP for point sets $P$ inside the strip $(-\infty,+\infty)\times [0,\delta]$ depends on the strip width $\delta$. We obtain two main results. First, for the case where the points have distinct…
We observe a realization of a stationary generalized weighted Voronoi tessellation of the d-dimensional Euclidean space within a bounded observation window. Given a geometric characteristic of the typical cell, we use the minus-sampling…
Let $\{p_1, \ldots , p_n \} \subset {\Bbb{R}}^2$ be a separated point set, i.e., any two points have a distance at least $1$. Let $k \ge 1$ be an integer, and $1 \le t_1 < \ldots < t_k$ be real numbers. Let $\delta > 0$. Suppose for all $1…
A general theory is provided delivering convergence of maximal cyclically monotone mappings containing the supports of coupling measures of sequences of pairs of possibly random probability measures on Euclidean space. The theory is based…
This paper addresses the question of when projections of a high-dimensional random vector are approximately Gaussian. This problem has been studied previously in the context of high-dimensional data analysis, where the focus is on…
In this article, we consider the Euclidean dispersion problems. Let $P=\{p_{1}, p_{2}, \ldots, p_{n}\}$ be a set of $n$ points in $\mathbb{R}^2$. For each point $p \in P$ and $S \subseteq P$, we define $cost_{\gamma}(p,S)$ as the sum of…