# Efficient Graph Field Integrators Meet Point Clouds

arXiv: https://arxiv.org/abs/2302.00942 · HTML: https://mlsanity.com/paper/2302.00942/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 69 references that are in the map.
- First posted on arXiv 2 Feb 2023.

## Builds on (most cited first, 20 of 20)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Rethinking Attention with Performers](https://mlsanity.com/paper/2009.14794/) (2020)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [PCT: Point cloud transformer](https://mlsanity.com/paper/2012.09688/) (2020)
- [TUDataset: A collection of benchmark datasets for learning with graphs](https://mlsanity.com/paper/2007.08663/) (2020)
- [Thingi10K: A Dataset of 10,000 3D-Printing Models](https://mlsanity.com/paper/1605.04797/) (2016)
- [Optimal Transport for structured data with application on graphs](https://mlsanity.com/paper/1805.09114/) (2018)
- [Gromov-Wasserstein Learning for Graph Matching and Node Embedding](https://mlsanity.com/paper/1901.06003/) (2019)
- [Asymmetric LSH (ALSH) for Sublinear Time Maximum Inner Product Search (MIPS)](https://mlsanity.com/paper/1405.5869/) (2014)
- [Fused Gromov-Wasserstein distance for structured objects: theoretical foundations and mathematical properties](https://mlsanity.com/paper/1811.02834/) (2018)
- [Predicting Physics in Mesh-reduced Space with Temporal Attention](https://mlsanity.com/paper/2201.09113/) (2022)
- [Random Fourier Features for Kernel Ridge Regression: Approximation Bounds and Statistical Guarantees](https://mlsanity.com/paper/1804.09893/) (2018)
- [Linear-Time Gromov Wasserstein Distances using Low Rank Couplings and Costs](https://mlsanity.com/paper/2106.01128/) (2021)
- [Graph Kernels: A Survey](https://mlsanity.com/paper/1904.12218/) (2019)
- [From block-Toeplitz matrices to differential equations on graphs: towards a general theory for scalable masked Transformers](https://mlsanity.com/paper/2107.07999/) (2021)
- [Debiased Sinkhorn barycenters](https://mlsanity.com/paper/2006.02575/) (2020)
- [Bridging the Gap Between Spectral and Spatial Domains in Graph Neural Networks](https://mlsanity.com/paper/2003.11702/) (2020)
- [Stability of the Lanczos Method for Matrix Function Approximation](https://mlsanity.com/paper/1708.07788/) (2017)
- [A Simple Baseline Algorithm for Graph Classification](https://mlsanity.com/paper/1810.09155/) (2018)
- [Covering Metric Spaces by Few Trees](https://mlsanity.com/paper/1905.07559/) (2019)

## Built on by (most cited first, 7 of 7)

- [Towards Foundation-model-based Multiagent System to Accelerate AI for Social Impact](https://mlsanity.com/paper/2412.07880/) (2024)
- [The Bandit Whisperer: Communication Learning for Restless Bandits](https://mlsanity.com/paper/2408.05686/) (2024)
- [Fast Tree-Field Integrators: From Low Displacement Rank to Topological Transformers](https://mlsanity.com/paper/2406.15881/) (2024)
- [Manifold Random Features](https://mlsanity.com/paper/2602.03797/) (2026)
- [SWING: Unlocking Implicit Graph Representations for Graph Random Features](https://mlsanity.com/paper/2602.12703/) (2026)
- [The Evidence Ladder for Reinforcement Learning in Healthcare: From Retrospective Policies to Trusted Interventions](https://mlsanity.com/paper/2609.23374/) (2026)
- [Unlocking Geodesic Gromov-Wasserstein Distances for 3D Modeling](https://mlsanity.com/paper/2609.32824/) (2026)

## Abstract (arXiv)

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.