Geometric Algebra Transformer
It addresses the lack of a single architecture for geometric data in fields like physics and robotics, offering a versatile solution, though it appears incremental as it builds on Transformer and geometric algebra concepts.
The paper tackles the problem of handling diverse geometric data types while respecting symmetries by introducing the Geometric Algebra Transformer (GATr), a general-purpose architecture that outperforms baselines in error, data efficiency, and scalability across applications like n-body modeling and robotic motion planning.
Problems involving geometric data arise in physics, chemistry, robotics, computer vision, and many other fields. Such data can take numerous forms, for instance 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.