Geometric Algebra Transformers
Authors: name: Johann Brehmer, name: Pim de Haan, name: Sönke Behrends, name: Taco Cohen

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.