2024/07/16 by François Clément, Carola Doerr, Clément, François +5
Mathematics · #Computational Geometry (cs.CG) #FOS: Computer and information sciences #FOS: Mathematics #Mathematical Approximation and Integration #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2407.11533
openalex publication_date 2024/07/16 · openalex created_date 2024/10/26 · openalex updated_date 2026/07/28
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∞-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∞ 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.