Most current deep learning models equivariant to $O(n)$ or $SO(n)$ either
consider mostly scalar information such as distances and angles or have a very
high computational complexity. In this work, we test a few novel message
passing graph neural networks (GNNs) based on Clifford multivectors, structured
similarly to other prevalent equivariant models in geometric deep learning. Our
approach leverages efficient invariant scalar features while simultaneously
performing expressive learning on multivector representations, particularly
through the use of the equivariant geometric product operator. By integrating
these elements, our methods outperform established efficient baseline models on
an N-Body simulation task and protein denoising task while maintaining a high
efficiency. In particular, we push the state-of-the-art error on the N-body
dataset to 0.0035 (averaged over 3 runs); an 8% improvement over recent
methods. Our implementation is available on Github.

本研究通过测试几种基于 Clifford 多矢量的新颖消息传递图神经网络（GNNs），旨在解决当前大多数对 $O (n)$ 或 $SO (n)$ 等等变换具有等变性的深度学习模型在处理标量信息时效果较好，但计算复杂度较高的问题。我们的方法利用了高效的不变标量特征，并同时在多矢量表示上执行表达性学习，特别是通过使用等变的几何乘积算子。通过整合这些要素，我们的方法在 N-Body 模拟任务和蛋白质去噪任务上优于已建立的高效基准模型，同时保持高效性。特别是，在 N-body 数据集上，我们的方法的误差推动到 0.0035（平均 3 次运行）；比最近的方法提升了 8%。我们的实现在 Github 上提供。

多维向量神经元：更好更快的 O (n) 等变 Clifford 图神经网络

Multivector Neurons: Better and Faster O(n)-Equivariant Clifford Graph  Neural Networks

As is well known, the common elementary functions defined over the real
numbers can be generalized to act not only over the complex number field but
also over the skew (non-commuting) field of the quaternions. In this paper, we
detail a number of elementary functions extended to act over the skew field of
Clifford multivectors, in both two and three dimensions. Complex numbers,
quaternions and Cartesian vectors can be described by the various components
within a Clifford multivector and from our results we are able to demonstrate
new inter-relationships between these algebraic systems. One key relationship
that we discover is that a complex number raised to a vector power produces a
quaternion thus combining these systems within a single equation. We also find
a single formula that produces the square root, amplitude and inverse of a
multivector over one, two and three dimensions. Finally, comparing the
functions over different dimension we observe that $ C\ell \left (\Re^3 \right)
$ provides a particularly versatile algebraic framework.

介绍定义在 Clifford 多向量上的一些初等函数，探寻其中向量、复数、四元数之间的相互关系，并为多向量提供平方根、振幅和反函数的公式。