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

Marc Finzi, Samuel Stanton, Pavel Izmailov, Andrew Gordon Wilson
Proceedings of the 37th International Conference on Machine Learning, PMLR 119:3165-3176, 2020.

Abstract

The translation equivariance of convolutional layers enables CNNs 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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v119-finzi20a, title = {Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data}, author = {Finzi, Marc and Stanton, Samuel and Izmailov, Pavel and Wilson, Andrew Gordon}, booktitle = {Proceedings of the 37th International Conference on Machine Learning}, pages = {3165--3176}, year = {2020}, editor = {III, Hal Daumé and Singh, Aarti}, volume = {119}, series = {Proceedings of Machine Learning Research}, month = {13--18 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v119/finzi20a/finzi20a.pdf}, url = {https://proceedings.mlr.press/v119/finzi20a.html}, abstract = {The translation equivariance of convolutional layers enables CNNs 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.} }
Endnote
%0 Conference Paper %T Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data %A Marc Finzi %A Samuel Stanton %A Pavel Izmailov %A Andrew Gordon Wilson %B Proceedings of the 37th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2020 %E Hal Daumé III %E Aarti Singh %F pmlr-v119-finzi20a %I PMLR %P 3165--3176 %U https://proceedings.mlr.press/v119/finzi20a.html %V 119 %X The translation equivariance of convolutional layers enables CNNs 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.
APA
Finzi, M., Stanton, S., Izmailov, P. & Wilson, A.G.. (2020). Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data. Proceedings of the 37th International Conference on Machine Learning, in Proceedings of Machine Learning Research 119:3165-3176 Available from https://proceedings.mlr.press/v119/finzi20a.html.

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