几何代数变换器

摘要

涉及几何数据的问题出现在各个领域，包括计算机视觉、机器人技术、化学和物理学。这类数据可以采用多种形式，如点、方向向量、平面或变换，但迄今为止还没有一种单一的架构可以应用于如此广泛的几何类型，并尊重它们的对称性。在本文中，我们介绍了几何代数变换器（GATr），这是一种通用的用于几何数据的架构。GATr在射影几何代数中表示输入、输出和隐藏状态，射影几何代数提供了常见几何对象的高效16维向量空间表示，以及作用于它们的运算符。GATr对于3D欧几里得空间的对称群E(3)是等变的。作为一个变换器，GATr具有可扩展性、表现力和多功能性。在n体建模和机器人规划的实验中，GATr相对于非几何基线表现出明显的改进。

English

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.

几何代数变换器

Geometric Algebra Transformers

摘要

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