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.

翻译：本体嵌入方法是表示和推理各领域结构化知识的重要途径。相较于知识图谱嵌入，本体嵌入的优势在于能够捕获并施加底层模式，使模型必须遵循该模式。尽管已有进展，当前大多数方法仍无法保证所得嵌入尊重本体蕴含的公理。本研究正式证明了归一化的${\cal ELH}$在凸几何模型上具有强忠实性，即存在能够精确捕获原始本体的嵌入。我们提出了一种基于区域的几何模型，用于将归一化${\cal ELH}$本体嵌入连续向量空间。为证明强忠实性，我们的构造利用了归一化${\cal ELH}$具有有限典范模型这一特性。首先，我们假设（可能）非凸区域下证明该命题，从而将所需维度控制在较低水平；随后对区域施加凸性约束，并证明该性质仍然成立。最后，我们探讨几何模型上的推理任务，并分析用于证明强忠实性的凸几何模型类别的复杂度。