Group Convolutions (G-Convolutions) are the mathematical generalization of standard convolution from the translation group to arbitrary symmetry groups — including rotation, reflection, scaling, and permutation — enabling neural networks to achieve equivariance with respect to any specified transformation group — the foundational theoretical framework that unifies standard CNNs, steerable CNNs, spherical CNNs, and graph neural networks as special cases of convolution over different symmetry groups.

What Are Group Convolutions?

• Definition: Standard convolution is defined on the translation group $mathbb{Z}^2$ — the filter slides (translates) across the 2D grid and computes a correlation at each position. Group convolution generalizes this to an arbitrary group $G$ — the filter slides and simultaneously applies all group transformations (rotations, reflections, etc.) at each position, producing a function on $G$ rather than just on the spatial grid.
• Standard CNN as Group Convolution: A standard 2D CNN performs convolution over the translation group $G = mathbb{Z}^2$. The output $(f * g)(t) = sum_x f(x) g(t^{-1}x)$ where $t$ is a translation. This is automatically equivariant to translations — shifting the input shifts the output by the same amount. Group convolution extends this to $G = mathbb{Z}^2 times H$ where $H$ is an additional symmetry group (rotations, reflections).
• Lifting Layer: The first layer of a group CNN "lifts" the input from the spatial domain to the group domain. For a rotation group CNN ($p4$ with 4 rotations), the lifting layer applies the filter at each spatial position and each of the 4 orientations, producing a feature map indexed by both position and rotation — $f(x, r)$ rather than just $f(x)$.

Why Group Convolutions Matter

• Theoretical Foundation: Group convolution provides the rigorous mathematical answer to "how do you build equivariant neural networks?" — the convolution theorem for groups guarantees that group convolution is equivariant by construction. Every equivariant linear map between feature spaces can be expressed as a group convolution, making it the universal building block for equivariant architectures.
• Weight Sharing: Standard convolution shares weights across spatial positions (translation weight sharing). Group convolution additionally shares weights across group transformations — a single filter handles all rotations simultaneously, rather than learning separate copies for each orientation. This dramatically reduces parameter count while guaranteeing equivariance across the entire transformation group.
• Systematic Construction: Given any symmetry group $G$, group convolution theory provides a systematic recipe for constructing an equivariant architecture: (1) identify the group, (2) define feature types by irreducible representations, (3) construct equivariant kernel spaces, (4) implement group convolution layers. This recipe eliminates ad-hoc architectural decisions and ensures mathematical correctness.
• Hierarchy of Groups: Group convolution naturally supports hierarchies — starting with a large group (many symmetries) and progressively relaxing to smaller groups as the network deepens. Early layers can be fully rotation-equivariant (capturing low-level features at all orientations), while deeper layers relax to translation-only equivariance (capturing high-level semantics that may have preferred orientations).

Group Convolution Spectrum

Group Convolutions are scanning all the symmetry possibilities — sliding and transforming filters through every element of the symmetry group to ensure that no orientation, reflection, or permutation is missed, providing the mathematical bedrock on which all equivariant neural network architectures are built.