Gauge Equivariant Convolutional Networks and the Icosahedral CNN
This work addresses the challenge of developing scalable and practical equivariant networks for manifold-based data, such as spherical signals, which is incremental by building on existing equivariant and geometric deep learning methods.
The authors tackled the problem of extending equivariant neural networks to local gauge transformations, enabling general convolutional networks on manifolds that depend only on intrinsic geometry, and demonstrated substantial improvements in segmenting omnidirectional images and global climate patterns.
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.