Efficient Graph Field Integrators Meet Point Clouds
This work addresses computational bottlenecks in point cloud processing for applications such as mesh dynamics and distance metrics, representing an incremental advancement by adapting existing methods to new graph-based contexts.
The paper tackles the problem of efficient field integration on graphs representing point clouds by introducing two new algorithm classes, SeparatorFactorization and RFDiffusion, which extend Fast Multipole Methods to non-Euclidean spaces. The result includes extensive theoretical analysis and empirical evaluation on tasks like on-surface interpolation and Wasserstein distance computations.
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.