The Framework

Geometric Deep Learning

The paradigm

This is the theoretical heart of the book: a single framework from which the dominant architectures fall out as special cases. Its organising principle is symmetry — the choice of a geometric domain, a symmetry group acting on it, and maps that commute with that action.

The five Gs, in depth

  • Geometry — the domains of data. Differentiable and Riemannian manifolds, tangent spaces and curvature give a precise language for the space a signal lives on.
  • Groups — the symmetries of data. Lie groups and their algebras describe symmetries; representation theory — the Peter–Weyl theorem and Schur’s lemma — decomposes them into irreducible pieces, the raw material for equivariant operators.
  • Graphs — the discretisation of geometries. The graph Laplacian and spectral analysis turn continuous domains into computable, neighbourhood-preserving discretisations.
  • Geodesics — information propagation. Metrics, the exponential map and parallel transport formalise how information travels — the geometric root of message passing.
  • Gauges — local invariances. Principal bundles and connections handle the freedom to choose local coordinate frames.

Equivariance and invariance

Equivariance and invariance are formalised as compatibility conditions between a function and a group action. Equivariance,

f(g⋅x)=ρ(g) f(x),f(g \cdot x) = \rho(g)\, f(x),

is the guiding design principle: it is what lets a network respect the structure of its data by construction rather than by training.

The unifying synthesis

The chapter’s synthesis is equivariant message passing as a common denominator:

  • plain neural networks are message passing over a complete graph;
  • CNNs are message passing over a regular grid, with weights tied by the translation group;
  • Transformers are adaptive message passing, where communication weights are computed from the data.

The Weisfeiler–Leman test pins down the expressive limit of this family: the graph-distinguishing power of message-passing networks coincides with the 1-WL algorithm — a precise ceiling, and a map of how the architectural families include one another.