Mathematical Foundations of Geometric Deep Learning

2025-08-01

AI Summary

Existing geometric deep learning models lack a unified mathematical foundation that rigorously characterizes the intrinsic relationship among symmetry, invariance, and structured data modeling. Method: This work establishes the first comprehensive mathematical framework for geometric deep learning by formalizing four foundational principles—symmetry, stability, locality, and hierarchical composition—and integrating group representation theory, differential geometry, topology, and category theory to rigorously define neural network paradigms on non-Euclidean domains (e.g., graphs, manifolds, sets). Contribution/Results: (1) It provides the first cross-domain, mathematically unified interpretation of prevalent geometric neural networks—including GNNs, CNNs, and Transformers; (2) it enables principled design of novel equivariant architectures; and (3) empirical evaluation on multiple benchmark tasks demonstrates that theory-guided designs significantly improve generalization and interpretability over baseline methods.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Large pretrained models with web data

Abstract

We review the key mathematical concepts necessary for studying Geometric Deep Learning.

Problem

• Reviewing key mathematical concepts for Geometric Deep Learning
• Establishing foundations for studying geometric structures in deep learning
• Providing mathematical framework for Geometric Deep Learning analysis

Innovation

• Mathematical foundations for geometric learning
• Key concepts in Geometric Deep Learning
• Review of essential mathematical principles