Principles of GDL
Geometric Deep Learning (GDL), popularized by Michael Bronstein and others, is a framework that unifies diverse neural network architectures (such as CNNs, GNNs, and Transformers) via the concepts of symmetry groups and invariance. It posits that network architecture should be determined by the geometric structure of the data domain.
Theoretical Concepts
- Invariance: Properties where model output remains constant under specific transformations of the input (e.g., translation invariance in CNNs).
- Equivariance: Transformations of the input result in predictable transformations of the output representation.
- Graph Neural Networks (GNNs): Architectures designed for non-Euclidean data structures like social graphs or molecular manifolds.
Algebraic & Geometric Extensions
- Clifford Algebras (Geometric Algebra): A mathematical framework for generalizing rotations and reflections to higher dimensions, increasingly used in 3D-aware neural networks.
- Harmonic Analysis on Graphs: Utilizing spectral graph theory and graph Laplacians to perform signal processing on complex relational data.
- Gauge Equivariant CNNs: Generalizing equivariance to arbitrary manifolds where there is no global coordinate system.
- Topological Data Analysis (TDA): Utilizing persistent homology to extract shape-based features from high-dimensional datasets.
Operational Utility
GDL enables the construction of Knowledge Graph Embeddings that adhere to the topological structure of facts. This facilitates relational reasoning that is robust to structural permutations and captures high-level relational invariants.