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

TL;DR

This paper introduces a permutation-based method to optimize low-discrepancy point sets, achieving significantly lower discrepancy values in numerical integration tasks, especially in low dimensions.

Contribution

It presents a novel permutation selection approach that enhances the construction of low-discrepancy point sets, surpassing classical and existing optimized methods.

Findings

Achieved up to 25% lower $L_{ extinfty}$ star discrepancy in 2D and 3D point sets.

Improved discrepancy by around 50% over classical constructions like Fibonacci sets.

Demonstrated effectiveness of the method in dimensions 2 and 3 with up to 400 points.

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.