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

François Clément; Carola Doerr; Luís Paquete

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arXiv · 2306.15276

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

François Clément·Carola Doerr·Luís Paquete

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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.

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François Clément, Carola Doerr, Luís Paquete. 2023-06-27. Heuristic Approaches to Obtain Low-Discrepancy Point Sets via Subset Selection. https://arxiv.org/abs/2306.15276

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