Rhomboid Tiling for Geometric Graph Deep Learning

TL;DR

The paper tackles the limitation of conventional GNN pooling in leveraging geometric information from geometric graphs. It introduces Rhomboid Tiling clustering (RT clustering), a geometry-driven hierarchical clustering based on high-order Voronoi tessellations and Delaunay complexes, which is realized as rhomboids. Building on this, the authors develop RTPool, a clustering-based pooling model that uses RT clustering with a weighting N(Q,Q') and supports both Delaunay and generated underlying graphs, accompanied by a theoretical complexity analysis. Empirically, RTPool outperforms 21 competitive baselines on 7 chemistry and biology datasets, with ablations and hyperparameter studies validating the component contributions and showing favorable efficiency.

Abstract

Graph Neural Networks (GNNs) have proven effective for learning from graph-structured data through their neighborhood-based message passing framework. Many hierarchical graph clustering pooling methods modify this framework by introducing clustering-based strategies, enabling the construction of more expressive and powerful models. However, all of these message passing framework heavily rely on the connectivity structure of graphs, limiting their ability to capture the rich geometric features inherent in geometric graphs. To address this, we propose Rhomboid Tiling (RT) clustering, a novel clustering method based on the rhomboid tiling structure, which performs clustering by leveraging the complex geometric information of the data and effectively extracts its higher-order geometric structures. Moreover, we design RTPool, a hierarchical graph clustering pooling model based on RT clustering for graph classification tasks. The proposed model demonstrates superior performance, outperforming 21 state-of-the-art competitors on all the 7 benchmark datasets.

Figures (3)

• Figure 1: Flowchart of the Rhomboid Tiling clustering process. A: An example geometric graph and the corresponding point cloud $X$, obtained by embedding the graph's vertices into $\mathbb{R}^2$. B: 1-, 2-, and 3-order Voronoi tessellations constructed from $X$. C: 1-, 2-, and 3-order Delaunay complexes obtained as the nerves of the corresponding Voronoi tessellations. D: The Rhomboid Tiling constructed based on $X$. E: Illustration of 1-layer Rhomboid Tiling clustering, focusing on a single rhomboid.
• Figure 2: Visualization of the hierarchical clustering process performed by RTpool on the molecular graph of Formaldehyde. The original molecular structure is shown on the left, followed by three successive clustering layers. Each cluster center (colored circle) is positioned based on the geometric realization of rhomboid tiling, and is connected to its members by dashed lines.
• Figure 3: An illustrative example of generating a rhomboid tiling from a point cloud. A: Molecular graph of Formaldehyde. B: Point cloud obtained by embedding the molecular graph into $\mathbb{R}^2$. C: Minimal circumcircles corresponding to each local point cluster in the point cloud. D: Rhomboids associated with the circumcircles. E: Rhomboid tiling structure of Formaldehyde formed by the union of all rhomboids.

Theorems & Definitions (8)

• Theorem 1
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• Theorem 8