Gauge Equivariant CNNs

We conclude Module 3 by extending our reach beyond Euclidean spaces to general manifolds (like curved surfaces) using Gauge Equivariant CNNs.

1. The Problem of Coordinates

On a curved manifold (like a sphere), there is no global grid or canonical coordinate system. To define a convolution kernel (which has valid directions like "up" and "right"), we must choose a Local Reference Frame (Gauge) at each point.

However, this choice is arbitrary. If we rotate our local frame, the numerical values of our feature vectors change.

2. Gauge Equivariance

A network is Gauge Equivariant if a change in the local frame results in a predictable linear transformation of the feature vectors. This is the manifold generalization of rotation equivariance.

• Feature Field: A collection of feature vectors, each defined relative to its local frame.
• Gauge Transformation: Rotating the local frame by $g \in SO(d)$ causes the feature vector scalar components to transform via representation $\rho(g)$.

3. Parallel Transport

To aggregate information from a neighbor $j$ to a central node $i$, we cannot simply sum their vectors because they live in different coordinate frames. We must first transport the vector from frame $j$ to frame $i$ along the edge connecting them. This operation is determined by the Connection (or Parallel Transport) on the manifold.