Multivector Neurons: Better and Faster O(n)-Equivariant Clifford Graph Neural Networks

TL;DR

The paper tackles the challenge of building $O(n)$- and $SO(n)$-equivariant graph neural networks that are both expressive and scalable for geometric data. It introduces three Clifford-based architectures—Clifford-EGNN, MVN-GNN, and MVP-GNN—that blend invariant scalar networks with multivector updates enabled by the geometric product in $Cl(R^3)$ to preserve equivariance while controlling parameter growth. The authors demonstrate state-of-the-art or competitive performance on two tasks: an N-body dynamics benchmark achieving a mean squared error of $0.0035$ (averaged over runs) and a protein denoising task, with favorable memory and speed profiles relative to prior CGENN baselines. These results indicate that decoupling scalar invariants from multivector features can yield fast, scalable, $O(n)$-equivariant GNNs suitable for large-scale geometric learning and potential diffusion-model applications.

Abstract

Most current deep learning models equivariant to $O(n)$ or $SO(n)$ either consider mostly scalar information such as distances and angles or have a very high computational complexity. In this work, we test a few novel message passing graph neural networks (GNNs) based on Clifford multivectors, structured similarly to other prevalent equivariant models in geometric deep learning. Our approach leverages efficient invariant scalar features while simultaneously performing expressive learning on multivector representations, particularly through the use of the equivariant geometric product operator. By integrating these elements, our methods outperform established efficient baseline models on an N-Body simulation task and protein denoising task while maintaining a high efficiency. In particular, we push the state-of-the-art error on the N-body dataset to 0.0035 (averaged over 3 runs); an 8% improvement over recent methods. Our implementation is available on Github.