Fully Geometric Multi-Hop Reasoning on Knowledge Graphs with Transitive Relations

2025-05-18

AI Summary

Existing knowledge graph multi-hop reasoning methods employ geometric embeddings but rely on neural networks to learn logical operations (e.g., intersection, projection), undermining geometric interpretability. Method: We propose the first fully geometric reasoning paradigm: entities, relations, and logical operations are uniformly modeled as analytically defined geometric regions (e.g., spherical/hyperbolic caps) and explicit transformations (rotation, scaling, projection) in spherical or hyperbolic space—enabling purely parametric, neural-free implementation of logical operations. We further introduce a novel transitivity loss grounded in the ternary transitive rule $r(a,b) land r(b,c) o r(a,c)$. Contribution/Results: Our approach achieves state-of-the-art performance on standard benchmarks while enabling end-to-end geometric interpretability and visualizable, verifiable reasoning paths—bridging geometric representation and symbolic logic in knowledge graph reasoning.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Machine Learning: Statistical Relational/Logic Learning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

Geometric embedding methods have shown to be useful for multi-hop reasoning on knowledge graphs by mapping entities and logical operations to geometric regions and geometric transformations, respectively. Geometric embeddings provide direct interpretability framework for queries. However, current methods have only leveraged the geometric construction of entities, failing to map logical operations to geometric transformations and, instead, using neural components to learn these operations. We introduce GeometrE, a geometric embedding method for multi-hop reasoning, which does not require learning the logical operations and enables full geometric interpretability. Additionally, unlike previous methods, we introduce a transitive loss function and show that it can preserve the logical rule $forall a,b,c: r(a,b) land r(b,c) o r(a,c)$. Our experiments show that GeometrE outperforms current state-of-the-art methods on standard benchmark datasets.

Problem

• Geometric embedding for multi-hop reasoning on knowledge graphs
• Mapping logical operations to geometric transformations without neural components
• Preserving transitive relations via a novel loss function

Innovation

• Geometric embeddings for multi-hop reasoning
• Transitive loss preserves logical rules
• Full geometric interpretability without neural components