Comparison of the l2 and wd point clouds with standard QMC and random samplers, based on scipy.stats.qmc.discrepancy (lower = better).
Run the benchmark with:
pip install scipy
python src/benchmark_data.py
For each sampler and each discrepancy, we compute the geometric mean over all configurations of
discrepancy(sampler) / discrepancy(Sobol)
A value below 1 means better than Sobol. Sobol is 1 by construction.
| Method | CD | WD | MD |
|---|---|---|---|
| Sobol (SciPy QMC) | 1.00 | 1.00 | 1.00 |
| Halton (SciPy QMC) | 1.72 | 1.47 | 1.62 |
| wd (ours) | 2.72 | 1.28 | 1.88 |
| LHS random-cd (SciPy QMC) | 3.45 | 3.51 | 3.70 |
| l2 (ours) | 17.03 | 9.17 | 15.92 |
| LHS (SciPy QMC) | 25.66 | 12.45 | 18.72 |
| i.i.d. uniform (baseline) | 87.64 | 53.94 | 87.46 |
Setup. Averages are taken over three sample sizes (N ∈ {1024, 2048, 4096}) and seven dimensions (D ∈ {2, …, 8}), i.e. 21 configurations per discrepancy (CD: centered, WD: wrap-around, MD: mixture).
wd is the closest to the QMC sequences: it is better than Halton on the wrap-around discrepancy (1.28 vs 1.47) and better than LHS random-cd on all three measures. It remains behind Sobol, which is the best method on this benchmark.l2 is not designed to minimise discrepancy: its exclusion region is an isotropic ball, which targets blue-noise quality rather than axis-aligned uniformity. It still improves over plain LHS and by a factor of 3 to 6 over i.i.d. sampling.wd for QMC integration when the integrand has favourable low-frequency spectral properties or low effective dimension, and l2 when isotropy matters.