  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.
