Gauge Equivariant Convolutional Networks and the Icosahedral CNN

The idea of equivariance to symmetry transformations provides one of the first theoretically grounded principles for neural network architecture design. Equivariant networks have shown excellent performance and data efficiency on vision and medical imaging problems that exhibit symmetries. In this paper we show how the theory can be extended from global symmetries to local gauge transformations, which makes it possible in principle to develop equivariant networks on general manifolds.

We implement gauge equivariant CNNs for signals defined on the icosahedron, which provides a reasonable approximation of spherical signals. 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.

We evaluate the effectiveness of Icosahedral CNNs on a number of different problems, and show that they yield excellent accuracy and computational efficiency.