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

TL;DR

This paper introduces multivector-based O(n)-equivariant graph neural networks that combine scalar invariants with multivector representations for improved efficiency and accuracy in geometric tasks.

Contribution

It proposes a novel GNN architecture using Clifford multivectors and the geometric product to enhance equivariance and expressiveness while maintaining computational efficiency.

Findings

Outperforms baseline models on N-Body and protein denoising tasks

Achieves 8% lower error on N-body dataset, reaching 0.0035

Maintains high efficiency with improved accuracy

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.