DELE: Deductive $\mathcal{EL}^{++}$ Embeddings for Knowledge Base Completion

TL;DR

This work tackles knowledge base completion for normalized $\mathcal{EL}^{++}$ ontologies by integrating deductive closure into geometry-based embeddings. It introduces negative losses across all normal forms and a fast approximate closure algorithm to guide negative sampling and evaluation, applied to ELEmbeddings, ELBE, and Box$^2$EL on GO/STRING, Food, and GALEN datasets. The results show improved performance in many KB completion tasks and underscore the importance of accounting for entailed axioms during evaluation and training. The approach clarifies how deductive closure shapes model behavior and provides practical guidance for robust ontology embeddings with publicly released code and data.

Abstract

Ontology embeddings map classes, roles, and individuals in ontologies into $\mathbb{R}^n$, and within $\mathbb{R}^n$ similarity between entities can be computed or new axioms inferred. For ontologies in the Description Logic $\mathcal{EL}^{++}$, several optimization-based embedding methods have been developed that explicitly generate models of an ontology. However, these methods suffer from some limitations; they do not distinguish between statements that are unprovable and provably false, and therefore they may use entailed statements as negatives. Furthermore, they do not utilize the deductive closure of an ontology to identify statements that are inferred but not asserted. We evaluated a set of embedding methods for $\mathcal{EL}^{++}$ ontologies, incorporating several modifications that aim to make use of the ontology deductive closure. In particular, we designed novel negative losses that account both for the deductive closure and different types of negatives and formulated evaluation methods for knowledge base completion. We demonstrate that our embedding methods improve over the baseline ontology embedding in the task of knowledge base or ontology completion.

Figures (3)

• Figure 1: ELEmbeddings example. Dashed circles represent translated classes by role vector corresponding to $has\_function$ role. The normalized theory $\{\{GO_1\} \sqcap \{GO_2\} \sqsubseteq \bot, A \sqcap B \sqsubseteq \bot, \exists has\_function.\{GO_1\} \sqsubseteq B, \exists has\_function.\{GO_2\} \sqsubseteq A\}$ is better preserved when negative losses are incorporated to all normal forms (Figure b) rather than only to GCI2 normal form (Figure a).
• Figure 2: ELEmbeddings example. Dashed circles represent translated classes by role vector corresponding to $has\_function$ role. 'Red' classes represent proteins $\{Q_1\}, \dots, \{Q_5\}$, 'green' classes represent proteins $\{P_1\}, \dots, \{P_5\}$. Axioms $\{Q_i\} \sqsubseteq \exists has\_function.\{GO_2\}, i = 1, \dots, 5$ are better preserved when negatives are filtered based on precomputed deductive closure (Figure b) rather than when random negatives are sampled (Figure a). The same applies for the axiom $\exists has\_function.\{GO_2\} \sqsubseteq A$.
• Figure 3: Metrics reported for biased fraction of random negatives combined with entailed axioms from the precomputed deductive closure.