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。

Clifford群等变神经网络