Strong Faithfulness for ELH Ontology Embeddings

TL;DR

This work formally proves that normalized ELH has the strong faithfulness property on convex geometric models, which means that there is an embedding that precisely captures the original ontology.

Abstract

Ontology embedding methods are powerful approaches to represent and reason over structured knowledge in various domains. One advantage of ontology embeddings over knowledge graph embeddings is their ability to capture and impose an underlying schema to which the model must conform. Despite advances, most current approaches do not guarantee that the resulting embedding respects the axioms the ontology entails. In this work, we formally prove that normalized ${\cal ELH}$ has the strong faithfulness property on convex geometric models, which means that there is an embedding that precisely captures the original ontology. We present a region-based geometric model for embedding normalized ${\cal ELH}$ ontologies into a continuous vector space. To prove strong faithfulness, our construction takes advantage of the fact that normalized ${\cal ELH}$ has a finite canonical model. We first prove the statement assuming (possibly) non-convex regions, allowing us to keep the required dimensions low. Then, we impose convexity on the regions and show the property still holds. Finally, we consider reasoning tasks on geometric models and analyze the complexity in the class of convex geometric models used for proving strong faithfulness.

Figures (5)

• Figure 1: A partial visualization (showing only the positive section of the real line) of a geometric interpretation $\bar{\eta}_{\mathcal{I}}$ where elements $d_0 \ldots d_3$ are mapped to their respective intervals, and where $\bar{\mu}(d_0), \bar{\mu}(d_2), \bar{\mu}(d_3) \in \bar{\eta}_{\mathcal{I}}(A)$ and $\bar{\mu}(d_2) \in \bar{\eta}_{\mathcal{I}}(B)$.
• Figure 2: An illustration of the region $\eta_\mathcal{I}(A) \cap \eta_\mathcal{I}(B)$.
• Figure 3: The three possible cases when there is an element in the intersection of $A,B,C$.
• Figure 4: A mapping to the binary vector $\mu(d)$ when $d\in\Delta^\mathcal{I}\xspace$, where $d \in a_0^{\mathcal{I}}$, $d \in A_0^{\mathcal{I}}$ and $(d, d_0)\in r^\mathcal{I}\xspace_0$.
• Figure 5: A mapping of $\mu(d)$ and $\mu(e)$ according to interpretation $\mathcal{I}\xspace$. The axes colored in red, blue, and green correspond to the dimensions associated with $a$, $A$, and $B$, respectively.

Theorems & Definitions (74)

• Example 1
• Definition 2: Geometric Interpretation
• Definition 3
• Definition 4
• Proposition 4
• Theorem 5
• Corollary 5
• Definition 6: Strong Faithfulness
• Example 7
• Definition 8