Efficient Graph Field Integrators Meet Point Clouds

TL;DR

This paper introduces two novel algorithms, SF and RFD, for efficient field integration on graphs representing point clouds, extending FMM-like methods to non-Euclidean spaces with theoretical and empirical validation.

Contribution

The paper presents two new classes of algorithms, SF and RFD, that enable efficient field integration on graph-based point cloud representations, with theoretical analysis and extensive empirical evaluation.

Findings

Algorithms achieve efficient integration on complex graphs

Theoretical results in structural graph theory

Effective on-surface interpolation and Wasserstein computations

Abstract

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.