Geometric Algebra TransformersDownload PDF

Published: 24 Jun 2023, Last Modified: 30 Jun 2023RSS 2023 Workshop SymmetryReaders: Everyone
Keywords: Geometry, geometric algebra, equivariance, transformer
TL;DR: GATr: geometric algebra + E(3) equivariance + transformer = general-purpose architecture for 3D data
Abstract: 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 and 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.
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