Discrepancy is a well-known measure for the irregularity of the distribution of a point set. Point sets with small discrepancy are called low-discrepancy and are known to efficiently fill the space in a uniform manner. Low-discrepancy points play a central role in many problems in science and engineering, including numerical integration, computer vision, machine perception, computer graphics, machine learning, and simulation. In this work, we present the first machine learning approach to generate a new class of low-discrepancy point sets named Message-Passing Monte Carlo (MPMC) points. Motivated by the geometric nature of generating low-discrepancy point sets, we leverage tools from Geometric Deep Learning and base our model on Graph Neural Networks. We further provide an extension of our framework to higher dimensions, which flexibly allows the generation of custom-made points that emphasize the uniformity in specific dimensions that are primarily important for the particular problem at hand. Finally, we demonstrate that our proposed model achieves state-of-the-art performance superior to previous methods by a significant margin. In fact, MPMC points are empirically shown to be either optimal or near-optimal with respect to the discrepancy for every dimension and the number of points for which the optimal discrepancy can be determined.

翻译：差异性是衡量点集分布不规则性的经典度量。具有小差异性的点集被称为低差异点集，已知其能以均匀方式高效填充空间。低差异点在科学与工程诸多问题中扮演核心角色，包括数值积分、计算机视觉、机器感知、计算机图形学、机器学习与仿真。本研究首次提出基于机器学习的方法来生成新型低差异点集——消息传递蒙特卡洛（MPMC）点集。受生成低差异点集的几何特性启发，我们借助几何深度学习工具，将模型建立在图神经网络基础上。我们进一步将框架扩展至高维空间，该框架可灵活生成定制化点集，这些点集能针对特定问题重点强化关键维度上的均匀性。最后，我们证明所提模型以显著优势实现了超越现有方法的性能。实验表明，对于所有可确定最优差异性的维度与点数，MPMC点集在差异性度量上均达到最优或接近最优水平。