Finding Low Star Discrepancy 3D Kronecker Point Sets Using Algorithm Configuration Techniques

A team of researchers has published a paper demonstrating that classical point set constructions used for uniform sampling can be significantly outperformed through size-specific optimization. The work, led by Imène Ait Abderrahim, Carola Doerr, and Martin Durand, focuses on the Kronecker construction in 3D. They show that by optimizing its two configurable parameters using the algorithm configuration technique called irace, they can generate point sets with lower star discrepancy—a key measure of uniformity—than established sequences like Sobol', Halton, and Hammersley for sets containing at least 500 points.

This optimization yields new state-of-the-art discrepancy values across entire ranges of set sizes, not just isolated examples. The findings are highly relevant because low-discrepancy point sets are critical infrastructure for several computational techniques. They serve as designs for experiments, initial designs for Bayesian optimization algorithms, and the foundation for quasi-Monte Carlo integration methods used in rendering and simulation.

The research underscores a shift in thinking: instead of relying solely on sequences known for their asymptotic convergence properties, practitioners can now seek optimized, fixed-size point sets tailored for specific applications. The use of irace, an automated algorithm configuration tool, was key to efficiently searching the parameter space and discovering these superior configurations. This work provides a practical method and specific parameters for generating better sampling patterns, directly impacting the efficiency and accuracy of downstream AI and simulation tasks.