Constructing Optimal $L_{\infty}$ Star Discrepancy Sets

TL;DR

The paper addresses the hard problem of constructing exact $L_{ Infty}$ star discrepancy minimal point sets by introducing two nonlinear programming formulations that compute optimal 2D sets (and extensions to 3D). In 2D, the approach achieves exact solutions up to $n=21$, outperforming Fibonacci and other low-discrepancy sequences by about 50% in many cases and revealing structurally different distributions of points. The authors further extend the framework to 3D, lattices, and several discrepancy notions beyond the star discrepancy, including extreme, periodic, and multiple-corner discrepancies, demonstrating the method's versatility. The work provides both exact constructions and practical heuristics, offering new directions for discrepancy-based design and enabling more efficient quasi-Monte Carlo point sets with potential broad applications.

Abstract

The $L_{\infty}$ star discrepancy is a very well-studied measure used to quantify the uniformity of a point set distribution. Constructing optimal point sets for this measure is seen as a very hard problem in the discrepancy community. Indeed, optimal point sets are, up to now, known only for $n\leq 6$ in dimension 2 and $n \leq 2$ for higher dimensions. We introduce in this paper mathematical programming formulations to construct point sets with as low $L_{\infty}$ star discrepancy as possible. Firstly, we present two models to construct optimal sets and show that there always exist optimal sets with the property that no two points share a coordinate. Then, we provide possible extensions of our models to other measures, such as the extreme and periodic discrepancies. For the $L_{\infty}$ star discrepancy, we are able to compute optimal point sets for up to 21 points in dimension 2 and for up to 8 points in dimension 3. For $d=2$ and $n\ge 7$ points, these point sets have around a 50% lower discrepancy than the current best point sets, and show a very different structure.

Figures (11)

• Figure 1: Fibonacci, Sobol' and optimal sets' local discrepancies for $n=21$.
• Figure 2: $L_{\infty}$ star discrepancies for the $L_{\infty}$ star optimal sets (line "optimal") and our multiple-corner optimal sets (line "multiple"), compared to the Fibonacci set. The dashed lines are lower and upper bounds described in \ref{['tab:cont_heuristic']}
• Figure 3: Left: An optimal $10$-point set $P^*$ with $f^*=d^*_{\infty}(P^*)=0.1111$. The red arrow indicates the up-shift $\mathop{\mathrm{up}}\nolimits(P^*,x^{(8)},0.09447)$, which is admissible with $\delta=\frac{1}{n}-(x^{(8)}_2-x^{(4)}_2)$. Note that $x^{(8)}$ is slightly higher than $x^{(4)}$, i.e., the two points are not on the same horizontal grid line. Right: Alternative optimal $10$-point set after implementing all admissible up-shifts and all admissible right-shifts.
• Figure 4: Comparison of the bounds at 40 000s obtained in Table \ref{['tab:cont_heuristic']} with the values of the Fibonacci set.
• Figure 5: Fibonacci, Sobol' and optimal sets' local discrepancies for $n=6$.

Theorems & Definitions (12)

• Theorem 2.1
• proof
• Definition 3.1: up-shift, right-shift
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Definition 3.4: down-shift, left-shift
• Lemma 3.5
• proof