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Transforming the Challenge of Constructing Low-Discrepancy Point Sets into a Permutation Selection Problem

François Clément, Carola Doerr, Kathrin Klamroth, Luís Paquete

TL;DR

The paper addresses constructing low-discrepancy point sets in $[0,1]^d$ with small $d^*_{\infty}$ by transforming the task into permutation selection. It fixes a permutation (via an assignment formulation) and optimizes the coordinate lists to minimize the discrepancy bound $f$, enabling scalable exploration of many permutations. In 2D, shifting Fibonacci-based permutations and using Kronecker-derived permutations yield substantial improvements, achieving $d^*_{\infty}$ values significantly below traditional constructions; in 3D, lifted sequences (Kronecker/Sobol) outperform nonlifted counterparts and baselines. The work reveals structure in good permutations (notably Kronecker) and demonstrates that reducing the problem to optimizing $d-1$ permutations can yield high-quality point sets, with potential implications for tighter discrepancy bounds and ML-inspired methodologies.

Abstract

Low discrepancy point sets have been widely used as a tool to approximate continuous objects by discrete ones in numerical processes, for example in numerical integration. Following a century of research on the topic, it is still unclear how low the discrepancy of point sets can go; in other words, how regularly distributed can points be in a given space. Recent insights using optimization and machine learning techniques have led to substantial improvements in the construction of low-discrepancy point sets, resulting in configurations of much lower discrepancy values than previously known. Building on the optimal constructions, we present a simple way to obtain $L_{\infty}$-optimized placement of points that follow the same relative order as an (arbitrary) input set. Applying this approach to point sets in dimensions 2 and 3 for up to 400 and 50 points, respectively, we obtain point sets whose $L_{\infty}$ star discrepancies are up to 25% smaller than those of the current-best sets, and around 50% better than classical constructions such as the Fibonacci set.

Transforming the Challenge of Constructing Low-Discrepancy Point Sets into a Permutation Selection Problem

TL;DR

The paper addresses constructing low-discrepancy point sets in with small by transforming the task into permutation selection. It fixes a permutation (via an assignment formulation) and optimizes the coordinate lists to minimize the discrepancy bound , enabling scalable exploration of many permutations. In 2D, shifting Fibonacci-based permutations and using Kronecker-derived permutations yield substantial improvements, achieving values significantly below traditional constructions; in 3D, lifted sequences (Kronecker/Sobol) outperform nonlifted counterparts and baselines. The work reveals structure in good permutations (notably Kronecker) and demonstrates that reducing the problem to optimizing permutations can yield high-quality point sets, with potential implications for tighter discrepancy bounds and ML-inspired methodologies.

Abstract

Low discrepancy point sets have been widely used as a tool to approximate continuous objects by discrete ones in numerical processes, for example in numerical integration. Following a century of research on the topic, it is still unclear how low the discrepancy of point sets can go; in other words, how regularly distributed can points be in a given space. Recent insights using optimization and machine learning techniques have led to substantial improvements in the construction of low-discrepancy point sets, resulting in configurations of much lower discrepancy values than previously known. Building on the optimal constructions, we present a simple way to obtain -optimized placement of points that follow the same relative order as an (arbitrary) input set. Applying this approach to point sets in dimensions 2 and 3 for up to 400 and 50 points, respectively, we obtain point sets whose star discrepancies are up to 25% smaller than those of the current-best sets, and around 50% better than classical constructions such as the Fibonacci set.
Paper Structure (9 sections, 3 theorems, 2 equations, 7 figures, 2 tables)

This paper contains 9 sections, 3 theorems, 2 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Optimizing Point Sets for a Given Permutation
  3. The Global Assignment Formulation
  4. L-infty Star Point Sets from Random Permutations and Classical Sequences
  5. L-infty Star Point Sets from Shifted Sequences
  6. L-infty Star Point Sets in Three Dimensions
  7. Investigating the Structure of Good Permutations
  8. Kronecker Permutations
  9. Conclusion

Key Result

Theorem 3.1

Given an initial point in $(0,0)$, there exist only $O(n^2)$ different Kronecker permutations generated by the associated lattices.

Figures (7)

  • Figure 1: Best discrepancy values obtained with our permutation approach $\pi$(Fib+1), compared to the traditional Fibonacci set shifted by one point (Fib+1, see Sections \ref{['sec:classic']} and \ref{['sec:shift']}) and the MPMC results from MPMC. The black crosses labelled "Best found" correspond to the best values obtained via a more extensive search of the different shifts at the start of the Fibonacci sequence (see Figure \ref{['fig:Shifts']}). The current best theoretical upper bound for the asymptotic discrepancy order, obtained by Ostromoukhov Ostro for a far larger number of points (between $60^7$ and $60^8$), is given by the green line $0.2223\log(n)/n$. The $0.3\log(n)/n$ curve is added as a comparison point.
  • Figure 2: Optimized discrepancy values $f(\pi)$ obtained by taking 2000 random permutations, $n=100$. We recall that the discrepancy for the (unoptimized but shifted by 1) Fibonacci set at 100 points is 0.0261, represented by the black vertical line in the plot.
  • Figure 3: Evolution of the optimized $L_\infty$ star discrepancy $f(\pi)$ obtained with model \ref{['eq:M5_2dcont']} for permutations $\pi$ obtained from shifted Fibonacci sequences, depending on the starting point of the sequence, for $n=$50, 100, 150, 200, from left to right. The best values obtained are 0.027088 for $n=50$, 0.014444 for $n=100$, 0.01 for $n=150$ and 0.007718 for $n=200$.
  • Figure 4: Discrepancies obtained by optimizing the Fibonacci permutation starting in 0 (No shift), 1 (+1), or three random integers (which we then split into a minimum, median and maximum value curves). Starting in 1 seems to be a reliably good choice, consistently outperforming the start in 0 and generally also better than the random choices.
  • Figure 5: Point sets obtained by taking the permutation for the Fibonacci set starting at the 75th point of the Fibonacci sequence. From first to last: local discrepancies of $F^*$, local discrepancies of open boxes for $F^*$, local discrepancies of closed boxes for $F^*$, the new optimized 100 point set $F^*$, and a comparison between the offset between the Fibonacci set $F_{74+100}$ (black) and $F^*$ (red).
  • ...and 2 more figures

Theorems & Definitions (4)

  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Proposition 3.3: Sos