The Geometric Algebra Transformer (GATr) is a versatile architecture for
geometric deep learning based on projective geometric algebra. We generalize
this architecture into a blueprint that allows one to construct a scalable
transformer architecture given any geometric (or Clifford) algebra. We study
versions of this architecture for Euclidean, projective, and conformal
algebras, all of which are suited to represent 3D data, and evaluate them in
theory and practice. The simplest Euclidean architecture is computationally
cheap, but has a smaller symmetry group and is not as sample-efficient, while
the projective model is not sufficiently expressive. Both the conformal algebra
and an improved version of the projective algebra define powerful, performant
architectures.

基于射影几何代数的几何代数变换器 (GATr) 是一种多功能的几何深度学习架构，我们将其推广为一种蓝图，可用于根据任何几何（或 Clifford）代数构建可扩展的变压器架构。我们研究了欧几里德、射影和共形代数的不同版本的该架构，它们都适用于表示 3D 数据，并在理论和实践中对它们进行评估。最简单的欧几里德架构计算成本低廉，但对称群较小且样本效率较低，而射影模型的表达能力不足。共形代数和改进版本的射影代数都定义了强大而高效的架构。

选择欧几里德、射影、共形几何代数用于等变变换器

Euclidean, Projective, Conformal: Choosing a Geometric Algebra for  Equivariant Transformers

Problems involving geometric data arise in a variety of fields, including
computer vision, robotics, chemistry, and physics. Such data can take numerous
forms, such as points, direction vectors, planes, or transformations, but to
date there is no single architecture that can be applied to such a wide variety
of geometric types while respecting their symmetries. In this paper we
introduce the Geometric Algebra Transformer (GATr), a general-purpose
architecture for geometric data. GATr represents inputs, outputs, and hidden
states in the projective geometric algebra, which offers an efficient
16-dimensional vector space representation of common geometric objects as well
as operators acting on them. GATr is equivariant with respect to E(3), the
symmetry group of 3D Euclidean space. As a transformer, GATr is scalable,
expressive, and versatile. In experiments with n-body modeling and robotic
planning, GATr shows strong improvements over non-geometric baselines.

本文介绍了一种用于处理几何数据的通用框架 ——Geometric Algebra Transformer (GATr)，它能够高效地表示常见几何物体及其运算符，并能够保持几何物体的对称性，经实验表明其在 n-body modeling 和机器人规划方面的表现明显优于无几何基线模型。