Geometric Relational Embeddings
Bo Xiong
TL;DR
This work introduces Geometric Relational Embeddings, a framework that maps relational data to geometric objects (points, balls, boxes, cones) and non-Euclidean manifolds to faithfully capture discrete, symbolic structures such as hierarchies, cycles, and logical constraints. It develops several coordinated models, including pseudo-Riemannian Graph Convolutional Networks (Q-GCN), UltraE (ultrahyperbolic KG embeddings), BoxEL for EL++ ontologies, and Hyperbolic-structured approaches for structured ML constraints (HMI), along with high-order extensions ShrinkE and FactE for hyper-relational and nested facts. The methods provide soundness guarantees (BoxEL) and scalable inference (Manhattan-like distances, diffeomorphic mappings) while achieving strong empirical performance on graph reconstruction, link prediction, ontology reasoning, and structured multi-label classification tasks. Collectively, the contributions offer a unified geometric toolkit to model complex relational phenomena—hierarchies, cycles, logical rules, and high-order relations—across knowledge graphs, ontologies, and structured ML settings, with demonstrated advantages in low-dimensional representations. The work lays groundwork for integrating symbolic knowledge with geometric embeddings, enabling more reliable reasoning and interpretability in KG-based AI systems and downstream applications in biology and knowledge management.
Abstract
Relational representation learning transforms relational data into continuous and low-dimensional vector representations. However, vector-based representations fall short in capturing crucial properties of relational data that are complex and symbolic. We propose geometric relational embeddings, a paradigm of relational embeddings that respect the underlying symbolic structures. Specifically, this dissertation introduces various geometric relational embedding models capable of capturing: 1) complex structured patterns like hierarchies and cycles in networks and knowledge graphs; 2) logical structures in ontologies and logical constraints applicable for constraining machine learning model outputs; and 3) high-order structures between entities and relations. Our results obtained from benchmark and real-world datasets demonstrate the efficacy of geometric relational embeddings in adeptly capturing these discrete, symbolic, and structured properties inherent in relational data.