Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data

TL;DR

This paper introduces a versatile method to create convolutional layers equivariant to any Lie group, enabling improved modeling of diverse data types like images, molecular structures, and physical systems with conserved quantities.

Contribution

The authors present a general framework for constructing Lie group equivariant convolutional layers applicable across various data domains, simplifying the incorporation of symmetry priors.

Findings

Models achieve exact conservation of momentum in Hamiltonian systems.

Applicable to images, molecular data, and dynamical systems.

Facilitates rapid prototyping of equivariant neural networks.

Abstract

The translation equivariance of convolutional layers enables convolutional neural networks to generalize well on image problems. While translation equivariance provides a powerful inductive bias for images, we often additionally desire equivariance to other transformations, such as rotations, especially for non-image data. We propose a general method to construct a convolutional layer that is equivariant to transformations from any specified Lie group with a surjective exponential map. Incorporating equivariance to a new group requires implementing only the group exponential and logarithm maps, enabling rapid prototyping. Showcasing the simplicity and generality of our method, we apply the same model architecture to images, ball-and-stick molecular data, and Hamiltonian dynamical systems. For Hamiltonian systems, the equivariance of our models is especially impactful, leading to exact conservation of linear and angular momentum.