Clifford Group Equivariant Neural Networks
This provides a novel method for building equivariant models that can generalize to inner-product spaces of any dimension, addressing a foundational challenge in geometric deep learning.
The paper tackles the problem of constructing O(n)- and E(n)-equivariant neural networks by introducing Clifford Group Equivariant Neural Networks, which achieve state-of-the-art performance on tasks like 3D n-body experiments, 4D Lorentz-equivariant high-energy physics, and 5D convex hull experiments.
We introduce Clifford Group Equivariant Neural Networks: a novel approach for constructing $\mathrm{O}(n)$- and $\mathrm{E}(n)$-equivariant models. We identify and study the $\textit{Clifford group}$, a subgroup inside the Clifford algebra tailored to achieve several favorable properties. Primarily, the group's action forms an orthogonal automorphism that extends beyond the typical vector space to the entire Clifford algebra while respecting the multivector grading. This leads to several non-equivalent subrepresentations corresponding to the multivector decomposition. Furthermore, we prove that the action respects not just the vector space structure of the Clifford algebra but also its multiplicative structure, i.e., the geometric product. These findings imply that every polynomial in multivectors, An advantage worth mentioning is that we obtain expressive layers that can elegantly generalize to inner-product spaces of any dimension. We demonstrate, notably from a single core implementation, state-of-the-art performance on several distinct tasks, including a three-dimensional $n$-body experiment, a four-dimensional Lorentz-equivariant high-energy physics experiment, and a five-dimensional convex hull experiment.