将构造低差异点集的挑战转化为排列选择问题
Transforming the Challenge of Constructing Low-Discrepancy Point Sets into a Permutation Selection Problem
AI总结：
将低差异点集构造转化为排列选择问题，提出简单方法获得$L_{\infty}$优化点放置，在2维和3维中分别最多400和50个点，差异比当前最佳小25%，比斐波那契集好50%。
摘要：
低差异点集已被广泛用作数值过程中用离散对象近似连续对象的工具，例如在数值积分中。经过一个世纪的研究，点集的差异能低到何种程度仍不清楚；换言之，在给定空间中点能如何规则地分布。近期利用优化和机器学习技术的见解在低差异点集的构造上带来了实质性改进，产生了比先前已知差异值低得多的配置。基于最优构造，我们提出了一种简单的方法来获得$L_{\infty}$优化的点放置，这些点遵循与（任意）输入集相同的相对顺序。将此方法应用于维度2和3中分别最多400和50个点的点集，我们获得的点集的$L_{\infty}$星形差异比当前最佳集合小高达25%，并且比经典构造（如斐波那契集）好约50%。
英文摘要：
Low discrepancy point sets have been widely used as a tool to approximate continuous objects by discrete ones in numerical processes, for example in numerical integration. Following a century of research on the topic, it is still unclear how low the discrepancy of point sets can go; in other words, how regularly distributed can points be in a given space. Recent insights using optimization and machine learning techniques have led to substantial improvements in the construction of low-discrepancy point sets, resulting in configurations of much lower discrepancy values than previously known. Building on the optimal constructions, we present a simple way to obtain $L_{\infty}$-optimized placement of points that follow the same relative order as an (arbitrary) input set. Applying this approach to point sets in dimensions 2 and 3 for up to 400 and 50 points, respectively, we obtain point sets whose $L_{\infty}$ star discrepancies are up to 25% smaller than those of the current-best sets, and around 50% better than classical constructions such as the Fibonacci set.
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