Uniformly distributed point sets of low discrepancy are heavily used in experimental design and across a very wide range of applications such as numerical integration, computer graphics, and finance. Recent methods based on Graph Neural Networks [Rusch et al., PNAS 2024] and solver-based optimization [Clément et al., accepted in Proceedings of the AMS] identified point sets having much lower discrepancy than previously known constructions. We show that further substantial improvements are possible by separating the construction of low-discrepancy point sets into (i) the relative position of the points, and (ii) the optimal placement respecting these relationships. Using tailored permutations, we construct point sets that are of 20\% smaller discrepancy on average than those proposed by Rusch et al. In terms of inverse discrepancy, our sets reduce the number of points in dimension 2 needed to obtain a discrepancy of 0.005 from more than 500 points to less than 350. For applications where the sets are used to query time-consuming models, this is a significant reduction.
The presentation is based on joint work with François Clément (University of Washington, US), Kathrin Klamroth (University of Wuppertal, Germany), and Luís Paquete (University of Coimbra, Portugal).
Zoom link:
https://zoom.us/j/91427615044?pwd=6hIVXIbu84vmbnFCUb0bnvRNAQyWlz.1