This paper proposes a unified methodology to systematize the field of geometric deep learning, drawing inspiration from the symmetry and invariance principles of Felix Klein's Erlangen Program. Its central goal is to derive inductive biases and neural network architectures from geometric foundations, offering a coherent theoretical framework for models designed in complex domains such as unstructured sets, graphs, manifolds, and grids.

Who is it for?

Beginners: An accessible introduction to key concepts of geometric deep learning.

Experts: Innovative connections between well-known architectures (CNNs, GNNs, Transformers) and underlying geometric principles.

Practitioners: Practical perspectives for solving problems in real applications using symmetries and structural regularities.

Key topics covered:

Geometric principles:

Exploiting symmetries, invariance, and representations in data.

Stability to deformations, scale separation, and group actions.

Mathematical foundations:

Metric spaces, Riemannian manifolds, fiber bundles, and automorphisms.

Convolutions adapted to non-Euclidean domains.

Challenges and solutions:

The curse of dimensionality in generic function learning.

Designing equivariant models that preserve structure under perturbations.

Core contribution:

The paper transcends specific implementations to highlight how the low-dimensional geometry of the physical world can guide the design of efficient machine learning systems. By linking abstract concepts (such as group actions) to practical architectures, the authors demonstrate that the stability and generalizability of models such as CNNs or GNNs emerge naturally from universal geometric principles.

Relevance:

Essential reading for those seeking to understand why modern neural networks work, beyond how, opening doors to innovations in areas such as computer vision, graph processing, and learning on manifolds.