Geometric Algebra Transformer
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.
Code (2)
Tasks
Motion PlanningMethods 이 논문이 사용한 방법론
Similar Papers 제목 키워드 기반
Euclidean, Projective, Conformal: Choosing a Geometric Algebra for Equivariant Transformers
The Geometric Algebra Transformer (GATr) is a versatile architecture for geometric deep learning based on projective geometric algebra. We generalize this architecture into a blueprint that allows one to construct a scal…
Lorentz-Equivariant Geometric Algebra Transformers for High-Energy Physics
Extracting scientific understanding from particle-physics experiments requires solving diverse learning problems with high precision and good data efficiency. We propose the Lorentz Geometric Algebra Transformer (L-GATr)…
A Lorentz-Equivariant Transformer for All of the LHC
We show that the Lorentz-Equivariant Geometric Algebra Transformer (L-GATr) yields state-of-the-art performance for a wide range of machine learning tasks at the Large Hadron Collider. L-GATr represents data in a geometr…
AllHybrid Diffusion Policies with Projective Geometric Algebra for Efficient Robot Manipulation Learning
Diffusion policies are a powerful paradigm for robot learning, but their training is often inefficient. A key reason is that networks must relearn fundamental spatial concepts, such as translations and rotations, from sc…
Robot ManipulationToward a Functional Geometric Algebra for Natural Language Semantics
Distributional and neural approaches to natural language semantics have been built almost exclusively on conventional linear algebra: vectors, matrices, tensors, and the operations that accompany them. These methods have…