RBF-GNN: Rational Basis Functions for Pseudo-Coordinate based Graph Convolutions

2026-09-29

AI Summary

This study addresses the curse of dimensionality in SplineCNN, where the number of B-spline basis functions grows exponentially in high-dimensional spaces. To overcome this limitation, we propose RBF-GNN, an architecture that replaces B-splines with rational Padé basis functions. By integrating pseudo-coordinates—such as Euclidean, spherical, or angular representations—the method establishes strong spatial inductive biases. Furthermore, subspace initialization and variance-preserving weight rescaling techniques are introduced to ensure training stability. When substituted for SplineCNN across semantic keypoint matching, shape correspondence, and event-based vision tasks, RBF-GNN consistently yields significant performance improvements. These results demonstrate that the proposed approach effectively resolves the efficiency bottlenecks of high-dimensional graph convolutions while maintaining robust representational capacity.

Abstract

We propose RBF-GNN, a new pseudo-coordinate based graph neural network architecture that takes into account Euclidean, spherical or angular coordinates and uses them to induce a powerful spatial inductive bias. Similar in architecture to SplineCNN, we improve upon the latter by replacing the less efficient sparse-activation based B-splines whose number grows exponentially with dimension by rational Padé basis functions. For effective training we propose a spline-subspace initialization and a variance-preserving weight rescaling. Experimentally, we evaluate on a number of popular neural network architectures that use SplineCNNs. We replace only the SplineCNNs with RBF-GNN. We achieve improved results, including on semantic keypoint matching, shape matching, event based camera computer vision tasks. We will make our implementation publicly available upon acceptance of the paper.

Problem

• Graph Neural Networks
• Pseudo-coordinate
• B-splines
• Spatial Inductive Bias
• Computational Efficiency

Innovation

• Graph Neural Networks
• Pseudo-Coordinate
• Rational Basis Functions
• Padé Approximation
• Spatial Inductive Bias