Problems involving geometric data arise in physics, chemistry, robotics, computer vision, and many other fields. Such data can take numerous forms, such as points, direction vectors, translations, or rotations, 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 (or Clifford) 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 versatile, efficient, and scalable. We demonstrate GATr in problems from n-body modeling to wall-shear-stress estimation on large arterial meshes to robotic motion planning. GATr consistently outperforms both non-geometric and equivariant baselines in terms of error, data efficiency, and scalability.

翻译：涉及几何数据的问题出现在物理学、化学、机器人学、计算机视觉以及诸多其他领域。此类数据可采用多种形式，如点、方向向量、平移或旋转。然而，迄今为止，尚无一种通用架构能够在尊重其对称性的同时，适用于如此广泛的几何类型。本文提出几何代数变换器（GATr），一种面向几何数据的通用架构。GATr将输入、输出及隐藏状态表示在射影几何（或克利福德）代数中，该代数提供了一种高效的16维向量空间表示，以描述常见几何对象及其作用算子。GATr关于三维欧氏空间对称群E(3)具有等变性。作为变换器，GATr兼具通用性、高效性与可扩展性。我们在从n体建模、大型动脉网格壁面剪切应力估计到机器人运动规划等各类问题中验证了GATr。在误差、数据效率与可扩展性方面，GATr始终优于非几何基准方法与等变基准方法。