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Lattice-preserving $\mathcal{ALC}$ ontology embeddings with saturation

Fernando Zhapa-Camacho, Robert Hoehndorf

TL;DR

This work proposes an ontology embedding method for the A L C DL language that considers the lattice structure of concept descriptions and outperforms state-of-the-art methods in several knowledge base completion tasks.

Abstract

Generating vector representations (embeddings) of OWL ontologies is a growing task due to its applications in predicting missing facts and knowledge-enhanced learning in fields such as bioinformatics. The underlying semantics of OWL ontologies are expressed using Description Logics (DLs). Initial approaches to generate embeddings relied on constructing a graph out of ontologies, neglecting the semantics of the logic therein. Recent semantic-preserving embedding methods often target lightweight DL languages like $\mathcal{EL}^{++}$, ignoring more expressive information in ontologies. Although some approaches aim to embed more descriptive DLs like $\mathcal{ALC}$, those methods require the existence of individuals, while many real-world ontologies are devoid of them. We propose an ontology embedding method for the $\mathcal{ALC}$ DL language that considers the lattice structure of concept descriptions. We use connections between DL and Category Theory to materialize the lattice structure and embed it using an order-preserving embedding method. We show that our method outperforms state-of-the-art methods in several knowledge base completion tasks. Furthermore, we incoporate saturation procedures that increase the information within the constructed lattices. We make our code and data available at \url{https://github.com/bio-ontology-research-group/catE}.

Lattice-preserving $\mathcal{ALC}$ ontology embeddings with saturation

TL;DR

This work proposes an ontology embedding method for the A L C DL language that considers the lattice structure of concept descriptions and outperforms state-of-the-art methods in several knowledge base completion tasks.

Abstract

Generating vector representations (embeddings) of OWL ontologies is a growing task due to its applications in predicting missing facts and knowledge-enhanced learning in fields such as bioinformatics. The underlying semantics of OWL ontologies are expressed using Description Logics (DLs). Initial approaches to generate embeddings relied on constructing a graph out of ontologies, neglecting the semantics of the logic therein. Recent semantic-preserving embedding methods often target lightweight DL languages like , ignoring more expressive information in ontologies. Although some approaches aim to embed more descriptive DLs like , those methods require the existence of individuals, while many real-world ontologies are devoid of them. We propose an ontology embedding method for the DL language that considers the lattice structure of concept descriptions. We use connections between DL and Category Theory to materialize the lattice structure and embed it using an order-preserving embedding method. We show that our method outperforms state-of-the-art methods in several knowledge base completion tasks. Furthermore, we incoporate saturation procedures that increase the information within the constructed lattices. We make our code and data available at \url{https://github.com/bio-ontology-research-group/catE}.
Paper Structure (26 sections, 2 theorems, 4 equations, 3 figures, 6 tables)

This paper contains 26 sections, 2 theorems, 4 equations, 3 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Description Logics
  4. Construction of the Lattice Structure
  5. Intersection of concepts:
  6. Union of concepts:
  7. Negation of concepts:
  8. Existential restriction of concepts:
  9. Universal restriction of concepts:
  10. Subsumption axioms:
  11. Class assertion axioms:
  12. Role assertion axioms:
  13. Saturation procedures
  14. Embedding into an ordered-vector space
  15. Evaluation
  16. ...and 11 more sections

Key Result

Theorem 1

The pair $(X, \preceq)$ over $\mathbb{R}^{n}$, where for elements $a, b \in X$ with $a=(a_1,..., a_n)$ and $b=(b_1,..., b_n)$, $a\preceq b$ if and only if $a_1\leq b_1, ..., a_n\leq b_n$, is a partial order.

Figures (3)

  • Figure 1: Lattice representation. $\bot$ is the bottom element and $\top$ is to top element. Arrows represent the $\sqsubseteq$ operator.
  • Figure 2: Lattice representations of complex concept descriptions.
  • Figure 3: Impact of embedding size and number of negatives on the Hits@100 and ROC AUC over different datasets.

Theorems & Definitions (4)

  • Theorem 1: $(X, \preceq)$ is a partial order
  • proof
  • Theorem 2: Lattice-preserving embeddings
  • proof