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 low dimension and small number of points, i.e., for which the optimal discrepancy can be determined. Code for generating MPMC points can be found at https://github.com/tk-rusch/MPMC.

翻译：差异度是衡量点集分布不规则性的一个经典指标。具有较小差异度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领域的诸多问题中发挥着核心作用，包括数值积分、计算机视觉、机器感知、计算机图形学、机器学习和仿真模拟。在本研究中，我们首次提出了一种机器学习方法，用于生成名为消息传递蒙特卡洛点集的新型低差异点集。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之上。我们进一步将框架扩展至高维空间，该框架能灵活生成定制化点集，可针对特定问题重点优化关键维度的均匀性。最后，我们通过实验证明，所提出的模型以显著优势超越了现有方法的性能极限。实证表明，在低维且点数较少的情况下，MPMC点集的差异度达到最优或接近最优水平。MPMC点集生成代码已发布于https://github.com/tk-rusch/MPMC。