RiemannGL: Riemannian Geometry Changes Graph Deep Learning

2026-02-11

AI Summary

Existing graph deep learning approaches are largely confined to Euclidean spaces or specific non-Euclidean manifolds—such as hyperbolic space—and thus struggle to systematically capture the intrinsic geometric properties of graph structures. This work proposes a unified framework grounded in Riemannian geometry, enhancing graph neural networks through intrinsic manifold modeling. It establishes Riemannian graph learning as a comprehensive research paradigm that encompasses manifold types, network architectures, and learning methodologies. By doing so, the study not only offers a novel theoretical perspective and methodological foundation but also delineates a critical pathway toward transcending current limitations and fully unlocking the potential of Riemannian geometry in graph representation learning.

Technology Category

• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches 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

Graphs are ubiquitous, and learning on graphs has become a cornerstone in artificial intelligence and data mining communities. Unlike pixel grids in images or sequential structures in language, graphs exhibit a typical non-Euclidean structure with complex interactions among the objects. This paper argues that Riemannian geometry provides a principled and necessary foundation for graph representation learning, and that Riemannian graph learning should be viewed as a unifying paradigm rather than a collection of isolated techniques. While recent studies have explored the integration of graph learning and Riemannian geometry, most existing approaches are limited to a narrow class of manifolds, particularly hyperbolic spaces, and often adopt extrinsic manifold formulations. We contend that the central mission of Riemannian graph learning is to endow graph neural networks with intrinsic manifold structures, which remains underexplored. To advance this perspective, we identify key conceptual and methodological gaps in existing approaches and outline a structured research agenda along three dimensions: manifold type, neural architecture, and learning paradigm. We further discuss open challenges, theoretical foundations, and promising directions that are critical for unlocking the full potential of Riemannian graph learning. This paper aims to provide a coherent viewpoint and to stimulate broader exploration of Riemannian geometry as a foundational framework for future graph learning research.

Problem

• Riemannian geometry
• graph representation learning
• manifold structures
• graph neural networks
• non-Euclidean data

Innovation

• Riemannian geometry
• graph representation learning
• intrinsic manifold
• graph neural networks
• non-Euclidean structure