Adaptive Riemannian Graph Neural Networks

2025-08-04

AI Summary

Graph data often exhibit heterogeneous geometric structures—such as hierarchical tree-like patterns and dense community clusters—posing challenges for existing geometric graph neural networks (GNNs), which typically rely on single or piecewise fixed-curvature spaces and thus fail to capture local geometric diversity. To address this, we propose an adaptive Riemannian manifold learning framework that learns a continuous, anisotropic, differentiable metric tensor for each node, enabling autonomous adaptation to locally optimal curvature. We further introduce Ricci-flow-inspired curvature regularization to ensure geometric evolution convergence and unify theoretical foundations for both constant- and mixed-curvature GNNs. Our method employs covariant derivatives to achieve stable message passing and optimization on non-uniform manifolds. Extensive experiments demonstrate state-of-the-art performance on both homophilic and heterophilic graph benchmarks, while the learned geometric parameters remain interpretable—validating the theoretical soundness and practical efficacy of our approach.

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

Graph data often exhibits complex geometric heterogeneity, where structures with varying local curvature, such as tree-like hierarchies and dense communities, coexist within a single network. Existing geometric GNNs, which embed graphs into single fixed-curvature manifolds or discrete product spaces, struggle to capture this diversity. We introduce Adaptive Riemannian Graph Neural Networks (ARGNN), a novel framework that learns a continuous and anisotropic Riemannian metric tensor field over the graph. It allows each node to determine its optimal local geometry, enabling the model to fluidly adapt to the graph's structural landscape. Our core innovation is an efficient parameterization of the node-wise metric tensor, specializing to a learnable diagonal form that captures directional geometric information while maintaining computational tractability. To ensure geometric regularity and stable training, we integrate a Ricci flow-inspired regularization that smooths the learned manifold. Theoretically, we establish the rigorous geometric evolution convergence guarantee for ARGNN and provide a continuous generalization that unifies prior fixed or mixed-curvature GNNs. Empirically, our method demonstrates superior performance on both homophilic and heterophilic benchmark datasets with the ability to capture diverse structures adaptively. Moreover, the learned geometries both offer interpretable insights into the underlying graph structure and empirically corroborate our theoretical analysis.

Problem

• Capturing complex geometric heterogeneity in graph data
• Learning adaptive local geometry for each node
• Ensuring computational tractability and geometric regularity

Innovation

• Adaptive Riemannian metric tensor field learning
• Node-wise diagonal metric tensor parameterization
• Ricci flow-inspired geometric regularization