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Outperforming the Best 1D Low-Discrepancy Constructions with a Greedy Algorithm

François Clément

TL;DR

The paper tackles the problem of constructing uniformly distributed sequences with small star discrepancy, comparing classic 1D constructions (Kronecker with the golden ratio and Ostromoukhov permutation) to a greedy approach introduced by Kritzinger. It introduces a fast algorithm to generate the Kritzinger sequence and demonstrates, through extensive numerical experiments, that it outperforms existing 1D constructions for large $n$, while also showing robustness to starting conditions. The work extends the method to dimensions 2 and 3 using an $L_2$ framework and various optimization strategies (NLP and heuristics), finding that the exact Kritzinger sequence remains competitive with or superior to Sobol' in these dimensions. The results support the conjecture that the discrepancy of the Kritzinger sequence scales as $O( ext{log}^d(n)/n)$ in dimension $d$, highlighting a practical and robust greedy approach for high-quality low-discrepancy sequences.

Abstract

The design of uniformly spread sequences on $[0,1)$ has been extensively studied since the work of Weyl and van der Corput in the early $20^{\text{th}}$ century. The current best sequences are based on the Kronecker sequence with golden ratio and a permutation of the van der Corput sequence by Ostromoukhov. Despite extensive efforts, it is still unclear if it is possible to improve these constructions further. We show, using numerical experiments, that a radically different approach introduced by Kritzinger in seems to perform better than the existing methods. In particular, this construction is based on a \emph{greedy} approach, and yet outperforms very delicate number-theoretic constructions. Furthermore, we are also able to provide the first numerical results in dimensions 2 and 3, and show that the sequence remains highly regular in this new setting.

Outperforming the Best 1D Low-Discrepancy Constructions with a Greedy Algorithm

TL;DR

The paper tackles the problem of constructing uniformly distributed sequences with small star discrepancy, comparing classic 1D constructions (Kronecker with the golden ratio and Ostromoukhov permutation) to a greedy approach introduced by Kritzinger. It introduces a fast algorithm to generate the Kritzinger sequence and demonstrates, through extensive numerical experiments, that it outperforms existing 1D constructions for large , while also showing robustness to starting conditions. The work extends the method to dimensions 2 and 3 using an framework and various optimization strategies (NLP and heuristics), finding that the exact Kritzinger sequence remains competitive with or superior to Sobol' in these dimensions. The results support the conjecture that the discrepancy of the Kritzinger sequence scales as in dimension , highlighting a practical and robust greedy approach for high-quality low-discrepancy sequences.

Abstract

The design of uniformly spread sequences on has been extensively studied since the work of Weyl and van der Corput in the early century. The current best sequences are based on the Kronecker sequence with golden ratio and a permutation of the van der Corput sequence by Ostromoukhov. Despite extensive efforts, it is still unclear if it is possible to improve these constructions further. We show, using numerical experiments, that a radically different approach introduced by Kritzinger in seems to perform better than the existing methods. In particular, this construction is based on a \emph{greedy} approach, and yet outperforms very delicate number-theoretic constructions. Furthermore, we are also able to provide the first numerical results in dimensions 2 and 3, and show that the sequence remains highly regular in this new setting.
Paper Structure (9 sections, 1 theorem, 10 equations, 12 figures)

This paper contains 9 sections, 1 theorem, 10 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Competitiveness of the Kritzinger Sequence in Dimension One
  3. Generating More Points: from Thousands to Millions
  4. Robustness of the Kritzinger sequence
  5. Extending the Sequence to Dimensions 2 and 3
  6. The L_2 Star Discrepancy
  7. Construction methods
  8. Results in dimensions 2 and 3
  9. Non-Linear Programming Formulation

Key Result

Theorem 2.1

Kritz Let $P^*:=(x_n)_{n \in \mathbb{N}}$ be the sequence generated by the above method. Let $\Gamma_{n}:=\{(2i+1)/(2(n+1)):i\in \{0,\ldots,n\}\}$. Then, for all $n \in \mathbb{N}$, we have $x_n \in \Gamma_{n}$ and $x_n$ is different from all previous $x_k$, $k<n$.

Figures (12)

  • Figure 1: The Kronecker sequence with golden ratio for the first 5, 8, 15 and 30 elements, from left to right. We represent $[0,1)$ and the fractional part as a torus to better illustrate how the points are gradually more and more uniformly distributed. As shown in the two leftmost images by the numbering, the elements are added one by one starting from the first in (1,0), each time with a rotation of $2\phi\pi$ to add the next point.
  • Figure 2: For both images, the segment corresponds to $[0,1]$. On the left, the first three elements of the van der Corput sequence. They correspond to the numbers that are written 01, 10 and 11 in binary. 01 gives $0.5$, 10 is $0.25$ while 11 corresponds to $0.75$. The next four elements to be added are exactly in the middle of the segments between the previous points. They are also added regularly: one element is added to each half of $[0,1]$ at first, then one to each quarter and so on.
  • Figure 3: Comparison of the first million points of the Kritzinger sequence with the Fibonacci sequence (left) and the Ostromoukhov sequence (right). Values are calculated every 1 000 points and scaled by $n/\log(n)$. The black line corresponds to the best theoretical upper bound on the asymptotic discrepancy constant by Ostromoukhov. We clearly notice that this asymptotic constant is much better than the discrepancy of the Ostromoukhov sequence for a million points. While the asymptotic discrepancy order of the Ostromoukhov sequence is the best known to this day, it is outperformed by the Kritzinger sequence for the first million points.
  • Figure 4: Localized plots in regions where the Kritzinger sequence was not performing well. Discrepancy values are calculated for all $n$ and scaled by $n/\log(n)$.
  • Figure 5: Proportion of $n$ for which the Fibonacci sequence is better than the Kritzinger one, initialized in $x=0.5$, counting only one instance per 1000 points.
  • ...and 7 more figures

Theorems & Definitions (3)

  • Theorem 2.1
  • Conjecture 2.2
  • Conjecture 3.1