Geometric Deep Learning

Data is never shapeless. Images live on grids, molecules on graphs, signals on spheres, poses on groups — every dataset carries a geometry, and that geometry comes with properties: symmetries first of all, but also locality, curvature and scale. A convolutional network already honours one such symmetry, translation; a graph network, permutation; a spherical network, rotation.

Geometric Deep Learning takes that geometry as the starting point rather than an afterthought. Read the properties of a domain — beginning with its symmetries — and they tell you how a network on it must be built: which operations are allowed, which are ruled out, and why. From this small set of geometric principles the architectures that dominate practice today follow as special cases, and the ones still missing come into view.

The shape of the data decides the shape of the network — and that reasoning begins with symmetry.

1. 01 Introduction Why Euclidean deep learning hits a wall on graphs, manifolds and groups — and how geometric priors invert the data–architecture relationship. MotivationThe 5 GsEquivariance →
2. 02 Artificial Neural Networks The perceptron, layers as measurable maps, and a full proof of Cybenko's universal approximation theorem — with the parametric limits of Euclidean architectures. PerceptronUniversal approximationDeep learning →
3. 03 Geometric Deep Learning The unifying framework — the 5 Gs, equivariance and invariance, and equivariant message passing that recovers CNNs, GNNs and Transformers as special cases. EquivarianceMessage passingWeisfeiler–Leman →
4. 04 Convolutional Models From classical CNNs to group, steerable, spherical, E(3)/SE(3), gauge and Clifford convolutions — culminating in an equivariant universality theorem. Group convolutionSteerable CNNsGauge equivariance →
5. 05 Attention Models Transformers as adaptive message passing, then equivariant, hyperbolic and topological attention — GATr, Hodge Laplacians and persistent homology. TransformersEquivariant attentionHyperbolic geometry →
6. 06 Conclusions Geometry as an organizing principle — how domain, symmetry group and connectivity separate ANNs, CNNs, GNNs and Transformers, and where the open problems lie. SynthesisOpen problemsFuture directions →

Six chapters — from neural-network foundations through the geometric framework of the Five Gs to its convolutional and attention models. Read the chapters online →