Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data
This work provides a general framework for incorporating equivariance to various transformations, which is incremental but useful for researchers in fields like physics and chemistry dealing with continuous data.
The authors tackled the problem of extending convolutional neural networks to be equivariant to any Lie group with a surjective exponential map, enabling applications beyond images to data like molecules and Hamiltonian systems, resulting in exact conservation of linear and angular momentum for Hamiltonian systems.
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.