Foundations

Introduction

From Euclidean success to its limits

Deep learning transformed artificial intelligence, but the architectures behind that success — convolutional, recurrent and attention networks — were conceived for data on regular Euclidean domains: pixel grids, temporal sequences, feature tables. As the field turned to molecules, social networks and 3D surfaces, the assumption of Euclidean regularity became a conceptual limitation, not merely a technical one. Forcing a graph into an adjacency matrix or a surface into voxels discards structure and wastes parameters.

Geometric Deep Learning

Geometric Deep Learning (GDL) inverts the traditional relationship between data and architecture: instead of forcing the data to fit a preexisting model, the geometric structure of the data dictates the structure of the model. By building symmetries directly into the network, GDL earns equivariance, robust generalisation and parameter efficiency.

The five Gs

The book is organised around five interlocking concepts — the 5 Gs:

  • Geometry — the intrinsic structure of the data domain (Riemannian manifolds, tangent spaces, curvature).
  • Groups — the symmetries that preserve the problem (Lie groups, representation theory, the equivariance principle).
  • Graphs — discretisations of geometric domains that capture neighbourhood relations (the graph Laplacian, spectral analysis).
  • Geodesics — optimal paths that drive information propagation (metrics, the exponential map, parallel transport — message passing in GNNs).
  • Gauges — local invariances in the choice of coordinate frames (principal bundles and connections).

A function ff is equivariant to a group G\mathfrak{G} acting on its input and output spaces when

f(g⋅x)=g⋅f(x)∀ g∈G.f(g \cdot x) = g \cdot f(x) \qquad \forall\, g \in \mathfrak{G}.

What this book sets out to do

  • Develop a rigorous mathematical framework for the 5 Gs, grounded in group theory, differential geometry and harmonic analysis.
  • Unify architectures: show formally how CNNs, GNNs and Transformers emerge as special cases of equivariant message passing.
  • Extend convolution and attention to non-Euclidean domains.
  • Connect theory to practice through representative modern architectures.

Roadmap

The remaining chapters build from foundations to advanced models: artificial neural networks and universal approximation, the unifying GDL framework, then its two great families — convolutional and attention models — before a closing synthesis of results, limitations and open problems.