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 whose definition we 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, 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.

翻译：我们提出克利福德群等变神经网络：一种构建$\mathrm{O}(n)$和$\mathrm{E}(n)$等变模型的新方法。我们识别并研究了$\textit{克利福德群}$，这是克利福德代数中的一个子群，我们调整其定义以获得若干有利性质。主要而言，该群的作用构成一个正交自同构，从典型向量空间延伸到整个克利福德代数，同时尊重多重向量分级。这产生了对应于多重向量分解的若干非等价子表示。此外，我们证明该作用不仅尊重克利福德代数的向量空间结构，还尊重其乘法结构，即几何积。这些发现意味着每个多重向量多项式，只要层使用该群的等变映射，必然满足等变性。我们在两组权重约束下推导出所有这样的映射，并展示如何高效实现。所得到的架构具有几个理想特性：内部表示保留代数结构，同时赋予等变性。一个值得提及的优势是，我们获得了具有表达力的层，能够优雅地推广到任意维数的内积空间。我们证明，仅凭单一核心实现，就在多个不同任务上取得了最先进性能，包括三维$n$体实验、四维洛伦兹等变高能物理实验和五维凸包实验。