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Generalizing Knowledge Graph Embedding with Universal Orthogonal Parameterization

Rui Li, Chaozhuo Li, Yanming Shen, Zeyu Zhang, Xu Chen

TL;DR

GoldE introduces a universal orthogonal parameterization for knowledge graph embedding by leveraging generalized Householder reflections to realize orthogonal transformations across multiple geometries. By forming a product manifold that blends elliptic and hyperbolic spaces (and optionally Euclidean components), GoldE captures both cyclical and hierarchical graph structures while enabling dimensional extension. Empirically, it achieves state-of-the-art link prediction performance on WN18RR, FB15k-237, and YAGO3-10, with robust gains even under restricted embedding sizes. This framework opens avenues for more flexible, geometry-aware KG representations and can extend to hyper-relational or temporal knowledge graphs.

Abstract

Recent advances in knowledge graph embedding (KGE) rely on Euclidean/hyperbolic orthogonal relation transformations to model intrinsic logical patterns and topological structures. However, existing approaches are confined to rigid relational orthogonalization with restricted dimension and homogeneous geometry, leading to deficient modeling capability. In this work, we move beyond these approaches in terms of both dimension and geometry by introducing a powerful framework named GoldE, which features a universal orthogonal parameterization based on a generalized form of Householder reflection. Such parameterization can naturally achieve dimensional extension and geometric unification with theoretical guarantees, enabling our framework to simultaneously capture crucial logical patterns and inherent topological heterogeneity of knowledge graphs. Empirically, GoldE achieves state-of-the-art performance on three standard benchmarks. Codes are available at https://github.com/xxrep/GoldE.

Generalizing Knowledge Graph Embedding with Universal Orthogonal Parameterization

TL;DR

GoldE introduces a universal orthogonal parameterization for knowledge graph embedding by leveraging generalized Householder reflections to realize orthogonal transformations across multiple geometries. By forming a product manifold that blends elliptic and hyperbolic spaces (and optionally Euclidean components), GoldE captures both cyclical and hierarchical graph structures while enabling dimensional extension. Empirically, it achieves state-of-the-art link prediction performance on WN18RR, FB15k-237, and YAGO3-10, with robust gains even under restricted embedding sizes. This framework opens avenues for more flexible, geometry-aware KG representations and can extend to hyper-relational or temporal knowledge graphs.

Abstract

Recent advances in knowledge graph embedding (KGE) rely on Euclidean/hyperbolic orthogonal relation transformations to model intrinsic logical patterns and topological structures. However, existing approaches are confined to rigid relational orthogonalization with restricted dimension and homogeneous geometry, leading to deficient modeling capability. In this work, we move beyond these approaches in terms of both dimension and geometry by introducing a powerful framework named GoldE, which features a universal orthogonal parameterization based on a generalized form of Householder reflection. Such parameterization can naturally achieve dimensional extension and geometric unification with theoretical guarantees, enabling our framework to simultaneously capture crucial logical patterns and inherent topological heterogeneity of knowledge graphs. Empirically, GoldE achieves state-of-the-art performance on three standard benchmarks. Codes are available at https://github.com/xxrep/GoldE.
Paper Structure (32 sections, 4 theorems, 50 equations, 4 figures, 11 tables)

This paper contains 32 sections, 4 theorems, 50 equations, 4 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Setup
  4. Geometric Background
  5. Methodology
  6. Generalized Orthogonal Matrix and Mapping
  7. GoldE: Universal Orthogonal Parameterization
  8. Elliptic Orthogonal Parameterization
  9. Hyperbolic Orthogonal Parameterization
  10. Mixed Orthogonal Parameterization
  11. Optimization
  12. Experiments
  13. Experimental Setup
  14. Performance Comparison
  15. Effect of Universal Orthogonal Parameterization
  16. ...and 17 more sections

Key Result

Theorem 3.1

When $n=k$, the image of $\mathrm{Orth}$ is the set of all $k\times k$ generalized orthogonal matrices, i.e., ${\rm Image( \mathrm{Orth})}=\mathbf{O}_{\mathbf{w}}(k)$. (See proof in Appendix proof-thm-1)

Figures (4)

  • Figure 1: Illustrations of logical patterns (a-d) sun2019rotate and topological structures (e-f) RotH.
  • Figure 2: (a) Illustration of generalized Householder reflection; (b) Three types of reflections of $\mathbf{x}$ about the hyperplanes $\zeta$ orthogonal to $\mathbf{u}$ under different weighting vectors, i.e., Euclidean (green lines), elliptic (purple lines) and hyperbolic (blue lines). Note that all quantities are displayed in the uniformly weighted Euclidean space, thus the non-Euclidean hyperplanes $\zeta_{\mathbb{P}}$ and $\zeta_{\mathbb{Q}}$ do not appear perpendicular to $\mathbf{u}$; (c) Elliptic orthogonal parameterization in $2$-dimensional space; (d) Hyperbolic orthogonal parameterization in $2$-dimensional space.
  • Figure 3: MRR results of GoldE under the product of $\mathbb{X}$ with varying dimension $k_\mathbb{X}$$(\mathbb{X}\in\{\mathbb{E},\mathbb{P},\mathbb{Q}\})$ on WN18RR and FB15k-237.
  • Figure 4: MRR results of the models with embedding size $k\in\{10,16,20,32,50,200,500,800\}$ on WN18RR.

Theorems & Definitions (12)

  • Theorem 3.1
  • Claim 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Claim 3.5
  • Theorem 2.1
  • proof
  • proof
  • proof
  • proof
  • ...and 2 more