Models

Convolutional Models

Convolution as the equivariant map

The first family of geometric architectures generalises the convolution. On a regular grid, convolution is precisely the linear map that is equivariant to translations — the reason CNNs are so effective on images. The chapter builds a ladder of generalisations that replace translation with richer symmetry groups.

Harmonic analysis and spectral convolutions

Harmonic analysis is the unifying tool: the Fourier transform, spherical harmonics and Chebyshev decompositions connect the spatial and spectral views of convolution, and make group convolution computable.

A ladder of generalisations

• Group convolutions (G-CNNs) for finite groups — equivariance to rotations and reflections.
• Steerable CNNs for continuous $\mathrm{SO}(2)$, built from irreducible representations.
• Spherical CNNs for $\mathrm{SO}(3)$, for signals on the sphere.
• $\mathrm{E}(3)$/$\mathrm{SE}(3)$-equivariant networks for 3D point clouds and molecules.
• Gauge-equivariant convolutions on manifolds, using local frames and parallel transport.
• Clifford / geometric-algebra convolutions acting directly on multivectors.

Equivariant universality

The ladder culminates in an equivariant universality theorem: equivariant CNNs can approximate any continuous equivariant function — the geometric counterpart of the classical universal approximation result. Topological extensions via Hodge Laplacians push convolution onto simplicial and cellular complexes, capturing higher-order relations beyond pairwise edges.