Knowledge base (KB) embeddings aim at combining the capability of classical knowledge graph embeddings to generalize the information present in facts, the ABox, with conceptual knowledge represented in an ontology language, the TBox. Several authors have recently explored the idea of mapping concepts to convex regions in a vector space. This is useful to represent hierarchies, typically present in TBoxes, since more general concepts can be mapped to larger regions, containing those regions associated with more specific concepts. However, the power of convexity is rarely leveraged during the actual learning tasks. Here, we introduce BoxLitE, a KB embedding model for DL-Lite$^{\mathcal{H}}$ that allows for convex optimization. We show that for any satisfiable DL-Lite$^{\mathcal{H}}$ KB, there is a BoxLitE embedding that is a weakly faithful model. As a proof of concept, we show how to formulate the KB embedding task as a convex optimization problem and how to obtain embeddings with such desirable faithfulness properties.

翻译：知识库嵌入旨在结合经典知识图谱嵌入概括事实信息（ABox）的能力与本体语言（TBox）中概念知识的表征。近年来，多位学者探索了将概念映射为向量空间中凸区域的方法。该策略对典型存在于TBox中的层级关系具有天然优势——更泛化的概念可映射至更大区域，并包含与更具体概念关联的子区域。然而，凸性的潜力在实际学习任务中鲜少被充分利用。本文提出BoxLitE模型，一种面向DL-Lite$^{\mathcal{H}}$的知识库嵌入方法，首次实现凸优化框架下的嵌入学习。我们证明：对于任意可满足的DL-Lite$^{\mathcal{H}}$知识库，存在具有弱忠实性质的BoxLitE嵌入。作为概念验证，我们展示了如何将知识库嵌入任务转化为凸优化问题，并获取具有理想忠实性特征的嵌入。