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

TL;DR

A heuristic approach for the star discrepancy subset selection problem, which gradually improves the current-best subset by replacing one of its elements at a time, and it is proved that the heuristic does not necessarily return an optimal solution.

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.

Figures (14)

• Figure 1: Local discrepancy values for a random point set of 15 points (red dots) in dimension 2
• Figure 2: Two subsets for $k=8$ taken from the first $n=10$ points of the Sobol' sequence in dimension 2. The ten initial points are shown, with those present in both subsets shown as blue circles. The points shown as red squares and the blue points form a local optimum for 1-swap with discrepancy 0.234. The black triangles plus the blue points correspond to the global optimum of discrepancy 0.203. Neither of the two sets can be improved via 1-swaps. Intuitively, replacing the lower red square by the lower black triangle would create an overfilled box at the bottom, whereas replacing it by the upper triangle would create an overfilled box on the left.
• Figure 3: Performance of the different instantiations in dimension 3, from left to right: TA_BF, TA_NBF, DEM_BF and DEM_NBF. Different colors indicate a change of the initial set size (red for $n=100$, blue for $n=150$, green for $n=200$ and yellow for $n=250$), and the black curve corresponds to the Sobol' sequence (it is the same in all four plots). The plot includes the $k=n$ case for all four different $n$, the rightmost point in this color.
• Figure 4: Performance of the different instantiations in dimension 6, from left to right: TA_BF, TA_NBF, DEM_BF and DEM_NBF. Different colors indicate a change of the initial set size (red for $n=100$, blue for $n=150$, green for $n=200$ and yellow for $n=250$), and the black curve corresponds to the Sobol' sequence (it is the same in all four plots). The plot includes the $k=n$ case for all four different $n$, the rightmost point in each color.
• Figure 5: Best discrepancy obtained for different values of $n$, $k$ fixed to 90, and $d=6$, with a cutoff time of 1 hour.

Theorems & Definitions (2)

• Proposition A.1
• proof