Gauge Equivariant Convolutional Networks and the Icosahedral CNN

TL;DR

This work develops a general theory of gauge-equivariant convolution on manifolds, enabling neural networks to respect local geometric frames rather than rely on global symmetries. It introduces a concrete instantiation, the Icosahedral CNN, which leverages a regular hexagonal grid and an atlas of charts to realize gauge-equivariant convolutions efficiently via a single conv2d operation. The method demonstrates strong performance on omnidirectional image segmentation and climate pattern tasks, while maintaining scalability superior to prior spherical CNN approaches. By unifying frame-bundle concepts with practical neural architectures and showing effective weight sharing through kernel constraints, the paper provides a versatile, scalable framework for learning on curved and irregular geometries with intrinsic geometric awareness.

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.

Figures (8)

• Figure 1: A gauge is a smoothly varying choice of tangent frame on a subset $U$ of a manifold $M$. A gauge is needed to represent geometrical quantities such as convolutional filters and feature maps (i.e. fields), but the choice of gauge is ultimately arbitrary. Hence, the network should be equivariant to gauge transformations, such as the change between red and blue gauge pictured here.
• Figure 2: On curved spaces, parallel transport is path dependent. The black vector is transported to the same point via two different curves, yielding different results. The same phenomenon occurs for other geometric objects, including filters.
• Figure 3: The exponential map and the gauge $w_p$.
• Figure 4: The Icosahedron with grid $\mathcal{H}_r$ for $r=2$ (left). We define $5$ overlapping charts that cover the grid (center). Chart $V_5$ is highlighted in gray (left). Colored edges that appear in multiple charts are to be identified. In each chart, we define the gauge by the standard axis aligned basis vectors $e_1, e_2 \in V_i$. For points $p \in U_i \cap U_j$, the transition between charts involves a change of gauge, shown as $+1 \cdot 2\pi/6$ and $-1 \cdot 2\pi /6$ (elements of $G=C_6$). On the right we show how the signal is represented in a padded array of shape $5 \cdot (2^r + 2) \times (2^{r+1}+2)$.
• Figure 5: G-Padding (scalar signal)