Geometric Algebra Transformer (2305.18415v3)

Abstract: 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.

• The paper introduces a novel integration of projective geometric algebra within the Transformer framework to process complex geometric data while preserving 3D symmetries.
• The method achieves E-equivariance through custom-designed attention mechanisms, linear maps, and MLP layers, excelling in tasks such as n-body dynamics.
• Empirical results show enhanced accuracy and scalability in applications like wall-shear stress estimation and robotic motion planning, enabling robust geometric modeling.

Geometric Algebra Transformer

The paper "Geometric Algebra Transformer" introduces a novel architecture, the Geometric Algebra Transformer (GATr), designed to process geometric data found in a variety of disciplines such as physics, chemistry, robotics, and computer vision. The core innovation of GATr is its integration of geometric algebra (GA), specifically the projective geometric algebra $3,0,1$, into the Transformer architecture, providing a versatile, efficient, and scalable method for handling a broad spectrum of geometric data while maintaining the symmetries of 3D Euclidean space.

Key Contributions

1. Projective Geometric Algebra: GATr leverages geometric algebra, particularly the algebra $3,0,1$, enabling it to represent common geometric objects such as points, lines, and planes, as well as their transformations within a unified 16-dimensional vector space. This approach not only provides a robust mathematical framework but also supports operations on data that are not invariant to translations, thereby enhancing the expressiveness of the model.
2. Equivariance with respect to $E$: GATr exhibits equivariance to the symmetry group of 3D Euclidean space, denoted as $E$. This symmetry group encompasses translations, rotations, and reflections, making GATr suitable for tasks where geometric transformations are intrinsic. To achieve this, the authors developed $E$-equivariant primitives including linear maps, an attention mechanism, and MLP layers that respect these symmetries.
3. Integration with Transformer Architecture: By embedding the GA-based representation within the Transformer framework, GATr inherits the scalability and versatility of transformers. This design choice allows GATr to manage the complexity of geometric interactions through attention mechanisms, which facilitates efficient learning from complex geometric data.

Empirical Evaluation

The authors demonstrate the efficacy of GATr across several domains:

• $n$-body dynamics: In modeling gravitational interactions, GATr outperformed both non-geometric and existing equivariant architectures in terms of error reduction and data efficiency. This demonstrates its capability to generalize well and handle dynamic systems involving multiple interacting bodies.
• Wall-shear-stress estimation: GATr improved state-of-the-art accuracy in predicting wall shear stress on arterial meshes, even when meshes were randomly oriented. This task showcases GATr's ability to handle large-scale geometric input, specifically in processing complex mesh data with thousands of nodes efficiently.
• Robotic motion planning: Within a robotic planning context, GATr served as the backbone of an $E$-invariant diffusion model. Its performance was superior to other baseline approaches, underscoring its practical utility in model-based reinforcement learning and planning tasks.

Computational Efficiency

GATr maintains favorable scaling characteristics similar to traditional Transformers, owing to the efficient computation of pairwise interactions via dot-product attention. This allows GATr to be applied to large systems with numerous interacting components, an essential capability for real-world applications involving complex geometric data.

Implications and Future Work

The integration of geometric algebra with transformer architectures represents a significant step forward in processing geometric data, offering a potent combination of structure and scalability. The implications are substantial for fields that require sophisticated geometric computations, like robotics and molecular modeling.

Future work could explore further refinement of the architecture to improve efficiency, investigate its universal approximation capabilities, and extend its application to other types of symmetry groups or higher-dimensional geometric spaces. Additionally, enhancing the understanding and accessibility of geometric algebra within the machine learning community could broaden the adoption of such innovative approaches.