Gauge Equivariant Convolutional Networks and the Icosahedral CNN

TL;DR

This paper introduces gauge equivariant CNNs that extend symmetry principles to local gauge transformations on manifolds, enabling scalable, geometry-dependent neural networks with improved performance on spherical and climate data.

Contribution

It develops a general framework for gauge equivariant CNNs on manifolds, including a practical implementation on the icosahedral surface, enhancing scalability and performance.

Findings

Outperforms previous methods on omnidirectional image segmentation

Achieves better results on global climate pattern analysis

Uses a single conv2d call for gauge equivariant convolution

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.