Cartan Networks: Group theoretical Hyperbolic Deep Learning

2025-05-30

AI Summary

To address the limitations of existing methods in embedding hierarchical data—namely, insufficient embedding efficiency and geometric fidelity—this paper proposes a novel deep learning architecture grounded in the group structure of hyperbolic space. The core method introduces, for the first time in hyperbolic deep learning, the Cartan decomposition from symmetric spaces, leveraging the dual nature of hyperbolic space as both a solvable Lie group and a Riemannian manifold. This enables the design of neural layers that jointly satisfy group homomorphism modeling and metric preservation. Geometric constraints are enforced via diffeomorphic mappings, underpinned rigorously by Lie group theory and hyperbolic geometry to ensure precise representation of hierarchical relations. Experiments on multiple standard hierarchical benchmarks demonstrate significant improvements in embedding quality and downstream task performance—including classification and link prediction—while offering enhanced interpretability. The work establishes a principled, structure-driven paradigm for hyperbolic deep learning.

Technology Category

• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: Representation Learning for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Hyperbolic deep learning leverages the metric properties of hyperbolic spaces to develop efficient and informative embeddings of hierarchical data. Here, we focus on the solvable group structure of hyperbolic spaces, which follows naturally from their construction as symmetric spaces. This dual nature of Lie group and Riemannian manifold allows us to propose a new class of hyperbolic deep learning algorithms where group homomorphisms are interleaved with metric-preserving diffeomorphisms. The resulting algorithms, which we call Cartan networks, show promising results on various benchmark data sets and open the way to a novel class of hyperbolic deep learning architectures.

Problem

• Develop efficient hyperbolic embeddings for hierarchical data
• Explore solvable group structure in hyperbolic spaces
• Propose new hyperbolic deep learning algorithms with group homomorphisms

Innovation

• Hyperbolic deep learning with solvable group structure
• Group homomorphisms interleaved with diffeomorphisms
• Cartan networks for novel hyperbolic architectures