We introduce Clifford Group Equivariant Neural Networks: a novel approach for
constructing $\mathrm{E}(n)$-equivariant networks. We identify and study the
$\textit{Clifford group}$, a subgroup inside the Clifford algebra, whose
definition we slightly adjust 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, including
their grade projections, constitutes an equivariant map with respect to the
Clifford group, allowing us to parameterize equivariant neural network layers.
Notable advantages are that these layers operate directly on a vector basis and
elegantly generalize to 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.

介绍了一种新的方法：Clifford Group Equivariant Neural Networks。从 Clifford 代数中识别和研究 Clifford group 子群，形成了一个直接在向量基础上运算、高效推广到任意维数的 Equivariant Neural Network 层。从三维 N 体实验、四维 Lorentz-equivariant 高能物理实验到五维凸包实验都获得了最新的技术成果。