Related papers: Constructing Low Star Discrepancy Point Sets with …
The irregularities of a distribution of $N$ points in the unit interval are often measured with various notions of discrepancy. The discrepancy function can be defined with respect to intervals of the form $[0,t)\subset [0,1)$ or arbitrary…
In this paper we discuss various connections between geometric discrepancy measures, such as discrepancy with respect to convex sets (and convex sets with smooth boundary in particular), and applications to numerical analysis and…
Consider a unit interval $[0,1]$ in which $n$ points arrive one-by-one independently and uniformly at random. On arrival of a point, the problem is to immediately and irrevocably color it in $\{+1,-1\}$ while ensuring that every interval…
Approximation of a target probability distribution using a finite set of points is a problem of fundamental importance in numerical integration. Several authors have proposed to select points by minimising a maximum mean discrepancy (MMD),…
The inverse of the star-discrepancy problem asks for point sets $P_{N,s}$ of size $N$ in the $s$-dimensional unit cube $[0,1]^s$ whose star-discrepancy $D^\ast(P_{N,s})$ satisfies $$D^\ast(P_{N,s}) \le C \sqrt{s/N},$$ where $C> 0$ is a…
Low-discrepancy points are designed to efficiently fill the space in a uniform manner. This uniformity is highly advantageous in many problems in science and engineering, including in numerical integration, computer vision, machine…
In this paper we propose an acceptance-rejection sampler using stratified inputs as diver sequence. We estimate the discrepancy of the points generated by this algorithm. First we show an upper bound on the star discrepancy of order…
Points in the unit cube with low discrepancy can be constructed using algebra or, more recently, by direct computational optimization of a criterion. The usual $L_\infty$ star discrepancy is a poor criterion for this because it is…
We study the expected star discrepancy under a newly designed class of non-equal volume partitions. The main contributions are twofold. First, we establish a strong partition principle for the star discrepancy, showing that our newly…
For two decades, reproducing kernels and their associated discrepancies have facilitated elegant theoretical analyses in the setting of quasi Monte Carlo. These same tools are now receiving interest in statistics and related fields, as…
We study some notions of negative dependence of a sampling scheme that can be used to derive variance bounds for the corresponding estimator or discrepancy bounds for the underlying random point set that are at least as good as the…
In this paper, we consider the upper bound of the probabilistic star discrepancy based on Hilbert space filling curve sampling. This problem originates from the multivariate integral approximation, but the main result removes the strict…
We introduce a class of convex equivolume partitions. Expected star discrepancy results are compared for stratified samples under these partitions, including simple random samples. There are four main parts of our results. First, among…
Uniform distribution of the points has been of interest to researchers for a long time and has applications in different areas of Mathematics and Computer Science. One of the well-known measures to evaluate the uniformity of a given…
With the aim of generalizing histogram statistics to higher dimensional cases, density estimation via discrepancy based sequential partition (DSP) has been proposed to learn an adaptive piecewise constant approximation defined on a binary…
We propose a generalized minimum discrepancy, which derives from Legendre's ODE and spherical harmonic theoretics to provide a new criterion of equidistributed pointsets on the sphere. A continuous and derivative kernel in terms of…
Given a set system (V,S), V={1,...,n} and S={S1,...,Sm}, the minimum discrepancy problem is to find a 2-coloring of V, such that each set is colored as evenly as possible. In this paper we give the first polynomial time algorithms for…
Several researchers have proposed minimisation of maximum mean discrepancy (MMD) as a method to quantise probability measures, i.e., to approximate a target distribution by a representative point set. We consider sequential algorithms that…
Discrepancies play an important role in the study of uniformity properties of point sets. Their probability distributions are a help in the analysis of the efficiency of the Quasi Monte Carlo method of numerical integration, which uses…
Artificial Neural Networks (ANN) comprise important symmetry properties, which can influence the performance of Monte Carlo methods in Neuroevolution. The problem of the symmetries is also known as the competing conventions problem or…