  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.
