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Capturing Knowledge Graphs and Rules with Octagon Embeddings

Victor Charpenay, Steven Schockaert

TL;DR

This work addresses the need for region-based knowledge graph embeddings that support relational composition and explicit rule capture. It introduces axis-aligned octagons as the building blocks in a coordinate-wise region model, proving that regions are closed under intersection and composition and that octagon parameters can be normalised for stability. The authors prove that octagon embeddings can represent arbitrary knowledge graphs and can capture a wide class of rules, while identifying theoretical limits (e.g., hexagons lack expressive power for composition). Empirically, octagon embeddings achieve competitive link prediction performance on standard benchmarks, with ablations showing the importance of the $v$-constraint and attention; the work also discusses practical learning with soft boundaries and potential extensions including cross-coordinate comparisons to broaden rule coverage.

Abstract

Region based knowledge graph embeddings represent relations as geometric regions. This has the advantage that the rules which are captured by the model are made explicit, making it straightforward to incorporate prior knowledge and to inspect learned models. Unfortunately, existing approaches are severely restricted in their ability to model relational composition, and hence also their ability to model rules, thus failing to deliver on the main promise of region based models. With the aim of addressing these limitations, we investigate regions which are composed of axis-aligned octagons. Such octagons are particularly easy to work with, as intersections and compositions can be straightforwardly computed, while they are still sufficiently expressive to model arbitrary knowledge graphs. Among others, we also show that our octagon embeddings can properly capture a non-trivial class of rule bases. Finally, we show that our model achieves competitive experimental results.

Capturing Knowledge Graphs and Rules with Octagon Embeddings

TL;DR

This work addresses the need for region-based knowledge graph embeddings that support relational composition and explicit rule capture. It introduces axis-aligned octagons as the building blocks in a coordinate-wise region model, proving that regions are closed under intersection and composition and that octagon parameters can be normalised for stability. The authors prove that octagon embeddings can represent arbitrary knowledge graphs and can capture a wide class of rules, while identifying theoretical limits (e.g., hexagons lack expressive power for composition). Empirically, octagon embeddings achieve competitive link prediction performance on standard benchmarks, with ablations showing the importance of the -constraint and attention; the work also discusses practical learning with soft boundaries and potential extensions including cross-coordinate comparisons to broaden rule coverage.

Abstract

Region based knowledge graph embeddings represent relations as geometric regions. This has the advantage that the rules which are captured by the model are made explicit, making it straightforward to incorporate prior knowledge and to inspect learned models. Unfortunately, existing approaches are severely restricted in their ability to model relational composition, and hence also their ability to model rules, thus failing to deliver on the main promise of region based models. With the aim of addressing these limitations, we investigate regions which are composed of axis-aligned octagons. Such octagons are particularly easy to work with, as intersections and compositions can be straightforwardly computed, while they are still sufficiently expressive to model arbitrary knowledge graphs. Among others, we also show that our octagon embeddings can properly capture a non-trivial class of rule bases. Finally, we show that our model achieves competitive experimental results.
Paper Structure (27 sections, 43 theorems, 94 equations, 5 figures, 4 tables)

This paper contains 27 sections, 43 theorems, 94 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Coordinate-wise region embeddings.
  4. Cross-coordinate region embeddings.
  5. Capturing rules.
  6. Modelling Relations with Octagons
  7. Parameterisation.
  8. Capturing relational properties.
  9. Expressivity of Octagon Embeddings
  10. Capturing Knowledge Graphs
  11. Capturing Rules
  12. Learning Octagon Embeddings
  13. Experimental Results
  14. Related Work
  15. Conclusions
  16. ...and 12 more sections

Key Result

Proposition 1

Consider the following set of parameters: Then $\mathsf{Octa}(x^-,x^+,y^-,y^+,u^-,u^+,v^-,v^+)=\mathsf{Octa}(x_1^-, x_1^+, y_1^-, y_1^+, u_1^-, u_1^+, v_1^-, v_1^+)$. Furthermore, if we have $x_1^->x_1^+$, $y_1^->y_1^+$, $u_1^- > u_1^+$ or $v_1^- > v_1^+$ then $\mathsf{Octa}(x^-, x^+, y^-, y^+, u^-, u^+, v^-, v^+)=\emptyset$. Otherwise, the

Figures (5)

  • Figure 1: Basic region based embeddings: TransE bands (left) and hexagons (right).
  • Figure 2: Region-based embeddings with octagons.
  • Figure 3: Example of an octagon encoding a transitive relation.
  • Figure 4: Distribution of scale factors in 40-dimensional ExpressivE embeddings for FB15k-237 (left) and WN18RR (right).
  • Figure 5: Illustration of the octagons constructed in the proof of Proposition \ref{['propFullyExpressive']}.

Theorems & Definitions (71)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • Proposition 8
  • Proposition 9
  • Definition 1
  • ...and 61 more