We introduce Clifford Group Equivariant Simplicial Message Passing Networks,
a method for steerable E(n)-equivariant message passing on simplicial
complexes. Our method integrates the expressivity of Clifford group-equivariant
layers with simplicial message passing, which is topologically more intricate
than regular graph message passing. Clifford algebras include higher-order
objects such as bivectors and trivectors, which express geometric features
(e.g., areas, volumes) derived from vectors. Using this knowledge, we represent
simplex features through geometric products of their vertices. To achieve
efficient simplicial message passing, we share the parameters of the message
network across different dimensions. Additionally, we restrict the final
message to an aggregation of the incoming messages from different dimensions,
leading to what we term shared simplicial message passing. Experimental results
show that our method is able to outperform both equivariant and simplicial
graph neural networks on a variety of geometric tasks.

我们介绍了 Clifford Group 等变成分维消息传递网络，这是一种用于 simplicial 复合体上的可操控 E (n)- 等变消息传递的方法。实验结果表明，我们的方法能够在各种几何任务中优于等变和 simplicial 图神经网络。

克利福德群等变单纯消息传递网络

Clifford Group Equivariant Simplicial Message Passing Networks

Clifford algebras are a natural generalization of the real numbers, the
complex numbers, and the quaternions. So far, solely Clifford algebras of the
form $Cl_{p,q}$ (i.e., algebras without nilpotent base vectors) have been
studied in the context of knowledge graph embeddings. We propose to consider
nilpotent base vectors with a nilpotency index of two. In these spaces, denoted
$Cl_{p,q,r}$, allows generalizing over approaches based on dual numbers (which
cannot be modelled using $Cl_{p,q}$) and capturing patterns that emanate from
the absence of higher-order interactions between real and complex parts of
entity embeddings. We design two new models for the discovery of the parameters
$p$, $q$, and $r$. The first model uses a greedy search to optimize $p$, $q$,
and $r$. The second predicts $(p, q,r)$ based on an embedding of the input
knowledge graph computed using neural networks. The results of our evaluation
on seven benchmark datasets suggest that nilpotent vectors can help capture
embeddings better. Our comparison against the state of the art suggests that
our approach generalizes better than other approaches on all datasets w.r.t.
the MRR it achieves on validation data. We also show that a greedy search
suffices to discover values of $p$, $q$ and $r$ that are close to optimal.

在知识图嵌入中，提出了使用二次零指数的 nilpotent base 向量的 Clifford 代数 $Cl_{p,q,r}$ 来捕捉实体嵌入的无高阶相互作用的模式，设计了两种新的模型来发现参数 $p$、$q$ 和 $r$，评估结果表明，nilpotent 向量能够更好地捕捉嵌入，与其他方法相比在所有基准数据集上，我们的方法在验证数据上达到的最大排名 (MRR) 更好，同时还展示了贪婪搜索足以发现接近最优值的 $p$、$q$ 和 $r$。

将知识图嵌入退化的 Clifford 代数

Embedding Knowledge Graphs in Degenerate Clifford Algebras

The octonions are the largest of the four normed division algebras. While
somewhat neglected due to their nonassociativity, they stand at the crossroads
of many interesting fields of mathematics. Here we describe them and their
relation to Clifford algebras and spinors, Bott periodicity, projective and
Lorentzian geometry, Jordan algebras, and the exceptional Lie groups. We also
touch upon their applications in quantum logic, special relativity and
supersymmetry.

本文介绍了八元数及其与多个数学领域的相关性，包括 Clifford 代数、自旋、Bott 周期性、投影几何、洛伦兹几何、Jordan 代数和例外李代数，并涉及其在量子逻辑、特殊相对论和超对称性等方面的应用。