Gauge Equivariant Convolutional Networks and the Icosahedral CNN

Taco Cohen, Maurice Weiler, Berkay Kicanaoglu, Max Welling
Proceedings of the 36th International Conference on Machine Learning, PMLR 97:1321-1330, 2019.

Abstract

The principle of equivariance to symmetry transformations enables a theoretically grounded approach to neural network architecture design. Equivariant networks have shown excellent performance and data efficiency on vision and medical imaging problems that exhibit symmetries. Here we show how this principle can be extended beyond global symmetries to local gauge transformations. This enables the development of a very general class of convolutional neural networks on manifolds that depend only on the intrinsic geometry, and which includes many popular methods from equivariant and geometric deep learning. We implement gauge equivariant CNNs for signals defined on the surface of the icosahedron, which provides a reasonable approximation of the sphere. By choosing to work with this very regular manifold, we are able to implement the gauge equivariant convolution using a single conv2d call, making it a highly scalable and practical alternative to Spherical CNNs. Using this method, we demonstrate substantial improvements over previous methods on the task of segmenting omnidirectional images and global climate patterns.

Cite this Paper

BibTeX
@InProceedings{pmlr-v97-cohen19d, title = {Gauge Equivariant Convolutional Networks and the Icosahedral {CNN}}, author = {Cohen, Taco and Weiler, Maurice and Kicanaoglu, Berkay and Welling, Max}, booktitle = {Proceedings of the 36th International Conference on Machine Learning}, pages = {1321--1330}, year = {2019}, editor = {Chaudhuri, Kamalika and Salakhutdinov, Ruslan}, volume = {97}, series = {Proceedings of Machine Learning Research}, month = {09--15 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v97/cohen19d/cohen19d.pdf}, url = {https://proceedings.mlr.press/v97/cohen19d.html}, abstract = {The principle of equivariance to symmetry transformations enables a theoretically grounded approach to neural network architecture design. Equivariant networks have shown excellent performance and data efficiency on vision and medical imaging problems that exhibit symmetries. Here we show how this principle can be extended beyond global symmetries to local gauge transformations. This enables the development of a very general class of convolutional neural networks on manifolds that depend only on the intrinsic geometry, and which includes many popular methods from equivariant and geometric deep learning. We implement gauge equivariant CNNs for signals defined on the surface of the icosahedron, which provides a reasonable approximation of the sphere. By choosing to work with this very regular manifold, we are able to implement the gauge equivariant convolution using a single conv2d call, making it a highly scalable and practical alternative to Spherical CNNs. Using this method, we demonstrate substantial improvements over previous methods on the task of segmenting omnidirectional images and global climate patterns.} }
Endnote
%0 Conference Paper %T Gauge Equivariant Convolutional Networks and the Icosahedral CNN %A Taco Cohen %A Maurice Weiler %A Berkay Kicanaoglu %A Max Welling %B Proceedings of the 36th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2019 %E Kamalika Chaudhuri %E Ruslan Salakhutdinov %F pmlr-v97-cohen19d %I PMLR %P 1321--1330 %U https://proceedings.mlr.press/v97/cohen19d.html %V 97 %X The principle of equivariance to symmetry transformations enables a theoretically grounded approach to neural network architecture design. Equivariant networks have shown excellent performance and data efficiency on vision and medical imaging problems that exhibit symmetries. Here we show how this principle can be extended beyond global symmetries to local gauge transformations. This enables the development of a very general class of convolutional neural networks on manifolds that depend only on the intrinsic geometry, and which includes many popular methods from equivariant and geometric deep learning. We implement gauge equivariant CNNs for signals defined on the surface of the icosahedron, which provides a reasonable approximation of the sphere. By choosing to work with this very regular manifold, we are able to implement the gauge equivariant convolution using a single conv2d call, making it a highly scalable and practical alternative to Spherical CNNs. Using this method, we demonstrate substantial improvements over previous methods on the task of segmenting omnidirectional images and global climate patterns.
APA
Cohen, T., Weiler, M., Kicanaoglu, B. & Welling, M.. (2019). Gauge Equivariant Convolutional Networks and the Icosahedral CNN. Proceedings of the 36th International Conference on Machine Learning, in Proceedings of Machine Learning Research 97:1321-1330 Available from https://proceedings.mlr.press/v97/cohen19d.html.

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