Cause of 4D Kronecker underperformance

Determine whether the inability of 4-dimensional Kronecker point sets, whose parameters are optimized specifically for each sample size by Covariance Matrix Adaptation Evolution Strategy (CMA-ES) and Reactive Tabu Search (RTS), to outperform truncated Sobol' sequences arises from intrinsic limitations of Kronecker sequences in dimension 4 or from insufficient effectiveness of the employed optimization algorithms in finding high-quality Kronecker parameters.

Background

The paper evaluates Kronecker sequences for generating low-star-discrepancy point sets, focusing primarily on 3 dimensions, where optimized parameters via CMA-ES and Irace achieve state-of-the-art performance for medium to large sample sizes.

In contrast, in 4 dimensions, even when parameters are tuned specifically for each n using CMA-ES and RTS, the resulting Kronecker point sets do not surpass the discrepancy of truncated Sobol' sequences. This raises an unresolved issue: whether the underperformance reflects an inherent limitation of Kronecker sequences in dimension 4, or whether better optimization could yield competitive parameters.

Resolving this question would clarify whether further algorithmic improvements in parameter search are warranted, or whether Kronecker sequences are fundamentally ill-suited for low-discrepancy construction in higher dimensions.

References

As shown in Table~\ref{tab:kronecker_4d}, even the CMA-ES and RTS algorithms that tune the parameters specifically for one given n cannot outperform the discrepancy values of the truncated Sobol' sequence in 4 dimensions. One question that remains is whether this is due to the nature of Kronecker sequences or to the ability of the algorithms to find very good parameters. The former would mean that Kronecker sequences would be unable to provide good point sets for dimension~$D=4$, and possibly beyond.

— Finding Low Star Discrepancy 3D Kronecker Point Sets Using Algorithm Configuration Techniques  (2604.00786 - Abderrahim et al., 1 Apr 2026) in Subsection "Assessing the Kronecker method for dimension D = 4"