title: Geometric Algebra Transformers
summary: 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.
paper_authors._id: 6476daf5592bb80eba2bf95a
paper_authors.name: Johann Brehmer
paper_authors._id: 6476daf5592bb80eba2bf95b
paper_authors.name: Pim de Haan
paper_authors._id: 6476daf5592bb80eba2bf95c
paper_authors.name: Sönke Behrends
paper_authors._id: 6476daf5592bb80eba2bf95d
paper_authors.name: Taco Cohen
paper_title: Geometric Algebra Transformers
paper_summary: 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.
paper_ai_keywords: Geometric Algebra Transformer (GATr) projective geometric algebra vector space representation common geometric objects operators E(3) 3D Euclidean space equivariant transformer scalable expressive versatile n-body modeling robotic planning