Geometric Algebra Attention Networks for Small Point Clouds

TL;DR

This paper introduces rotation- and permutation-equivariant neural network architectures for small point clouds, leveraging geometric algebra and attention mechanisms to respect underlying symmetries in physical science data.

Contribution

It presents a novel combination of geometric algebra and attention mechanisms to build equivariant models for small point clouds, addressing a gap in geometric deep learning.

Findings

Models effectively handle rotation and permutation invariance.

Applications demonstrate relevance to physics, chemistry, and biology.

Architectures outperform non-equivariant baselines.

Abstract

Much of the success of deep learning is drawn from building architectures that properly respect underlying symmetry and structure in the data on which they operate - a set of considerations that have been united under the banner of geometric deep learning. Often problems in the physical sciences deal with relatively small sets of points in two- or three-dimensional space wherein translation, rotation, and permutation equivariance are important or even vital for models to be useful in practice. In this work, we present rotation- and permutation-equivariant architectures for deep learning on these small point clouds, composed of a set of products of terms from the geometric algebra and reductions over those products using an attention mechanism. The geometric algebra provides valuable mathematical structure by which to combine vector, scalar, and other types of geometric inputs in a systematic way to account for rotation invariance or covariance, while attention yields a powerful way to impose permutation equivariance. We demonstrate the usefulness of these architectures by training models to solve sample problems relevant to physics, chemistry, and biology.