We present two new classes of algorithms for efficient field integration on graphs encoding point clouds. The first class, SeparatorFactorization(SF), leverages the bounded genus of point cloud mesh graphs, while the second class, RFDiffusion(RFD), uses popular epsilon-nearest-neighbor graph representations for point clouds. Both can be viewed as providing the functionality of Fast Multipole Methods (FMMs), which have had a tremendous impact on efficient integration, but for non-Euclidean spaces. We focus on geometries induced by distributions of walk lengths between points (e.g., shortest-path distance). We provide an extensive theoretical analysis of our algorithms, obtaining new results in structural graph theory as a byproduct. We also perform exhaustive empirical evaluation, including on-surface interpolation for rigid and deformable objects (particularly for mesh-dynamics modeling), Wasserstein distance computations for point clouds, and the Gromov-Wasserstein variant.

翻译：我们提出了两类用于编码点云图上的高效场积分新算法。第一类——分隔子分解（SeparatorFactorization，SF）利用了点云网格图的有界亏格特性，而第二类——射频扩散（RFDiffusion，RFD）则采用常用的点云ε近邻图表示。这两类算法均可视为提供快速多极子方法（FMMs）的功能——该方法对高效积分产生了巨大影响——但适用于非欧几里得空间。我们聚焦于由点间游走长度分布（如最短路径距离）诱导的几何结构。我们对算法进行了全面的理论分析，并作为副产品获得了结构图论的新成果。我们还开展了详尽的实证评估，包括刚性与可变形物体的表面插值（特别针对网格动力学建模）、点云的Wasserstein距离计算以及Gromov-Wasserstein变体。