Greedy energy minimization can count in binary: point charges and the van der Corput sequence

This paper establishes a connection between a problem in Potential Theory and Mathematical Physics, arranging points so as to minimize an energy functional, and a problem in Combinatorics and Number Theory, constructing 'well-distributed' sequences of points on $[0,1)$. Let $f:[0,1] \rightarrow \mathbb{R}$ be (i) symmetric $f(x) = f(1-x)$, (ii) twice differentiable on $(0,1)$, and (iii) such that $f''(x)>0$ for all $x \in (0,1)$. We study the greedy dynamical system, where, given an initial set $\{x_0, \ldots, x_{N-1}\} \subset [0,1)$, the point $x_N$ is obtained as $$x_{N} = \arg\min_{x \in [0,1)} \sum_{k=0}^{N-1}{f(|x-x_k|)}.$$ We prove that if we start this construction with the single element $x_0=0$, then all arising constructions are permutations of the van der Corput sequence (counting in binary and reflected about the comma): \textit{greedy energy minimization recovers the way we count in binary.} This gives a new construction of the classical van der Corput sequence. The special case $f(x) = 1-\log(2 \sin(\pi x))$ answers a question of Steinerberger. Interestingly, the point sets we derive are also known in a different context as Leja sequences on the unit disk. Moreover, we give a general bound on the discrepancy of any sequence constructed in this way for functions $f$ satisfying an additional assumption.Comment: 18 pages, 7 figures, discrepancy bound adde

On the Discrepancy of Jittered Sampling

We study the discrepancy of jittered sampling sets: such a set $\mathcal{P} \subset [0,1]^d$ is generated for fixed $m \in \mathbb{N}$ by partitioning $[0,1]^d$ into $m^d$ axis aligned cubes of equal measure and placing a random point inside each of the $N = m^d$ cubes. We prove that, for $N$ sufficiently large, $$\frac{1}{10}\frac{d}{N^{\frac{1}{2} + \frac{1}{2d}}} \leq \mathbb{E} D_N^*(\mathcal{P}) \leq \frac{\sqrt{d} (\log{N})^{\frac{1}{2}}}{N^{\frac{1}{2} + \frac{1}{2d}}},$$ where the upper bound with an unspecified constant $C_d$ was proven earlier by Beck. Our proof makes crucial use of the sharp Dvoretzky-Kiefer-Wolfowitz inequality and a suitably taylored Bernstein inequality; we have reasons to believe that the upper bound has the sharp scaling in $N$. Additional heuristics suggest that jittered sampling should be able to improve known bounds on the inverse of the star-discrepancy in the regime $N \gtrsim d^d$. We also prove a partition principle showing that every partition of $[0,1]^d$ combined with a jittered sampling construction gives rise to a set whose expected squared $L^2-$discrepancy is smaller than that of purely random points

Weak multipliers for generalized van der Corput sequences

Generalized van der Corput sequences are onedimensional, infinite sequences in the unit interval. They are generated from permutations in integer base b and are the building blocks of the multi-dimensional Halton sequences. Motivated by recent progress of Atanassov on the uniform distribution behavior of Halton sequences, we study, among others, permutations of the form P(i) = ai (mod b) for coprime integers a and b. We show that multipliers a that either divide b - 1 or b + 1 generate van der Corput sequences with weak distribution properties. We give explicit lower bounds for the asymptotic distribution behavior of these sequences and relate them to sequences generated from the identity permutation in smaller bases, which are, due to Faure, the weakest distributed generalized van der Corput sequences.Les suites de Van der Corput généralisées sont dessuites unidimensionnelles et infinies dans l’intervalle de l’unité.Elles sont générées par permutations des entiers de la basebetsont les éléments constitutifs des suites multi-dimensionnelles deHalton. Suites aux progrès récents d’Atanassov concernant le com-portement de distribution uniforme des suites de Halton nous nousintéressons aux permutations de la formuleP(i) =ai(modb)pour les entiers premiers entre euxaetb. Dans cet article nousidentifions des multiplicateursagénérant des suites de Van derCorput ayant une mauvaise distribution. Nous donnons les bornesinférieures explicites pour cette distribution asymptotique asso-ciée à ces suites et relions ces dernières aux suites générées parpermutation d’identité, qui sont, selon Faure, les moins bien dis-tribuées des suites généralisées de Van der Corput dans une basedonnée

Optimal Jittered Sampling for two Points in the Unit Square

Jittered Sampling is a refinement of the classical Monte Carlo sampling method. Instead of picking $n$ points randomly from $[0,1]^2$, one partitions the unit square into $n$ regions of equal measure and then chooses a point randomly from each partition. Currently, no good rules for how to partition the space are available. In this paper, we present a solution for the special case of subdividing the unit square by a decreasing function into two regions so as to minimize the expected squared $\mathcal{L}_2-$discrepancy. The optimal partitions are given by a \textit{highly} nonlinear integral equation for which we determine an approximate solution. In particular, there is a break of symmetry and the optimal partition is not into two sets of equal measure. We hope this stimulates further interest in the construction of good partitions

On the construction of equivolume partitions of the d−d-dimensional unit cube

The aim of this note is to construct equivolume partitions of the $d$-dimensional unit cube with hyperplanes that are orthogonal to the main diagonal of the cube. Each such hyperplane $H_r \subset \mathbb{R}^d$ is defined via the equation $x_1 + \ldots + x_d + r=0$ for a parameter $r$ with $-d\leq r \leq 0$. For a given $N$, we characterise those real numbers $r_i$ with $1\leq i \leq N-1$ for which the corresponding hyperplanes $H_{r_i}$ partition the unit cube into $N$ sets of volume $1/N$. As a main result we derive an algebraic and a probabilistic characterisation of the $r_i$ for arbitrary $d\geq 2$ and $N \in \mathbb{N}$. Importantly, our results do not only work for equivolume partitions but also for arbitrary predefined distributions of volume among the $N$ sets of the partition generated from hyperplanes $H_r$.Comment: 10 pages, 4 figure