Heuristic Approaches to Obtain Low-Discrepancy Point Sets via Subset Selection

TL;DR

This paper presents a heuristic method for selecting low-discrepancy point sets that improves upon existing sequences, achieving significant discrepancy reductions across multiple dimensions, with promising results demonstrated through extensive testing.

Contribution

The paper introduces a new heuristic for star discrepancy subset selection that outperforms recent energy functional methods and is applicable across all dimensions.

Findings

Achieves up to 35% better discrepancy than Sobol' sequence in dimension 6.

Works effectively across all tested dimensions despite limitations in discrepancy calculation.

Outperforms Steinerberger's energy functional method on all tested instances.

Abstract

Building upon the exact methods presented in our earlier work [J. Complexity, 2022], we introduce a heuristic approach for the star discrepancy subset selection problem. The heuristic gradually improves the current-best subset by replacing one of its elements at a time. While we prove that the heuristic does not necessarily return an optimal solution, we obtain very promising results for all tested dimensions. For example, for moderate point set sizes $30 \leq n \leq 240$ in dimension 6, we obtain point sets with $L_{\infty}$ star discrepancy up to 35% better than that of the first $n$ points of the Sobol' sequence. Our heuristic works in all dimensions, the main limitation being the precision of the discrepancy calculation algorithms. We also provide a comparison with a recent energy functional introduced by Steinerberger [J. Complexity, 2019], showing that our heuristic performs better on all tested instances.