hyperuniform-samplers

Exclusion Sets and incomplete orbits:

The sampling process to build the hyperuniform database proceeded as follows:

Initialize with random i.i.d. points.
Select a spectral exclusion set $\Omega$ based on a chosen geometry (L2 ball or WD ball), with a budget of $M = \lfloor \chi \cdot D \cdot (N-1) \rfloor$ wave vectors (note that we count the pair $(k, -k)$ as a single wave vector due to spectral symmetry), where $\chi = 0.43$, to compute the Fourier loss:

\[\text{loss} = \sum_{k \in \Omega} \frac{1}{\lvert k \rvert^2} \left\lvert \sum_{\text{point} \in \text{points}} e^{2i\pi \langle k, \text{point}\rangle} \right\rvert^2\]

This is what a typical exclusion set looks like for 2D wave vectors (the geometrical constructions also generalise to $D > 2$) and balls L2 and WD:

The set $\Omega$ is built by accumulating complete orbits ($weight(k) = cst$, where weight is the squared L2 norm of the wave vector $k$ for L2, or the wrap-around discrepancy coefficient for WD) until the budget $M$ is reached. The last orbit, which would be incomplete, is truncated (discarded), so that $\Omega$ only contains complete orbits.

The spectral bound $S(k) \le \varepsilon$ is only guaranteed on complete orbits, which is why the incomplete one is discarded. As a consequence, the number of wave vectors actually used is $M_{\text{eff}} \le M$, which results in an effective

\[\chi_{\text{eff}} = \frac{M_{\text{eff}}}{D\,(N-1)} \le \chi = 0.43.\]

This is especially critical in high dimensions, where a single orbit can contain thousands of wave vectors: discarding the last one can leave $\chi_{\text{eff}}$ at only a small fraction of the nominal $\chi$.

1. Orbit Definitions

Ball	Exclusion Region	Weight $`w(k)`$	Orbit Definition (Modes of Equal Weight)
`L2`	$`L_2`$ ball	$`\Vert k\Vert_2^2`$	Shells where $`\Vert k\Vert_2^2 = \text{const}`$
`WD`	Star ball	$`r(k) = \prod_i \hat{K}(k_i)`$ (Fourier coefficient of the wrap-around discrepancy)	Shells where $`r(k) = \text{const}`$

2. Worst-Case $`\chi_{\text{eff}} / \chi`$ by Dimension

Analysis for $N = 2^{10}$ to $2^{17}$, with $D = 2$ to $16$, and $D = 20, 24, 28, 32$.

For the remaining dimensions ($17 \le D \le 19$, $21 \le D \le 23$, $25 \le D \le 27$, and $29 \le D \le 31$), the incomplete-orbit effect combines with an additional projection effect. The exact $\chi_{\text{eff}} / \chi$ table, including both effects, is provided in PROJECTIONS.md.

The following table reports the worst-case $\chi_{\text{eff}} / \chi$, in percent (higher is better). The worst case is the minimum across $N = 2^{10}, \ldots, 2^{17}$.

Dimension ($`D`$)	`L2`	`WD`
2–3	98%	94%
4–6	79%	96%
7–9	67%	87–96%
10–12	35%	76%
13–16	18%	63%
20	14.08%	49.10%
24	10.26%	57.38%
28	6.37%	65.26%
32	7.27%	38.17%

3. Black, Gray and Light Gray Lists

These are the point clouds with the lowest $\chi_{\text{eff}} / \chi$, and thus the least effective use of the spectral exclusion budget. All of them are L2 point clouds: WD never falls below 38% in the dimensions considered.

Black: 5–10% (3 point clouds)

Ball	D	N
`L2`	24	$`2^{13}`$ (10.26%)
`L2`	28	$`2^{10}`$ (6.37%)
`L2`	32	$`2^{10}`$ (7.27%)

Gray: 10–20% (9 point clouds)

Ball	D	N
`L2`	16	$`2^{11}`$ (17.7%)
`L2`	20	$`2^{12}`$ (14.08%)
`L2`	20	$`2^{15}`$ (15.52%)
`L2`	24	$`2^{12}`$ (20.52%)
`L2`	24	$`2^{15}`$ (13.85%)
`L2`	24	$`2^{16}`$ (13.85%)
`L2`	28	$`2^{12}`$ (14.08%)
`L2`	28	$`2^{17}`$ (11.26%)
`L2`	32	$`2^{13}`$ (18.51%), $`2^{17}`$ (17.11%)

Light Gray: 20–30% (7 point clouds)

Ball	D	N
`L2`	13	$`2^{10}`$ (23.0%)
`L2`	14	$`2^{10}`$ (26.8%)
`L2`	15	$`2^{13}`$ (24.6%), $`2^{15}`$ (29.1%)
`L2`	20	$`2^{11}`$ (28.18%)
`L2`	20	$`2^{17}`$ (25.95%)
`L2`	24	$`2^{14}`$ (27.71%)
`L2`	28	$`2^{14}`$ (28.17%)

Conclusion

Low-dimensional regime ($D \le 3$): $\chi_{\text{eff}} \approx \chi$ for both balls, with negligible loss.
High-dimensional regime ($D \ge 20$): Orbits become very wide, causing $\chi_{\text{eff}} / \chi$ to drop to as low as 6.37% for the L2 ball at $D = 28$.
Robustness of the WD ball: Thanks to a much larger number of distinct orbits, truncating its last shell has a smaller impact. For the supplied dimensions $D = 20, 24, 28, 32$, the worst-case WD ratios are 49.10%, 57.38%, 65.26%, and 38.17%, respectively.

This site is open source. Improve this page.