Reasoning with knowledge graphs (KGs) has primarily focused on triple-shaped facts. Recent advancements have been explored to enhance the semantics of these facts by incorporating more potent representations, such as hyper-relational facts. However, these approaches are limited to \emph{atomic facts}, which describe a single piece of information. This paper extends beyond \emph{atomic facts} and delves into \emph{nested facts}, represented by quoted triples where subjects and objects are triples themselves (e.g., ((\emph{BarackObama}, \emph{holds\_position}, \emph{President}), \emph{succeed\_by}, (\emph{DonaldTrump}, \emph{holds\_position}, \emph{President}))). These nested facts enable the expression of complex semantics like \emph{situations} over time and \emph{logical patterns} over entities and relations. In response, we introduce NestE, a novel KG embedding approach that captures the semantics of both atomic and nested factual knowledge. NestE represents each atomic fact as a $1\times3$ matrix, and each nested relation is modeled as a $3\times3$ matrix that rotates the $1\times3$ atomic fact matrix through matrix multiplication. Each element of the matrix is represented as a complex number in the generalized 4D hypercomplex space, including (spherical) quaternions, hyperbolic quaternions, and split-quaternions. Through thorough analysis, we demonstrate the embedding's efficacy in capturing diverse logical patterns over nested facts, surpassing the confines of first-order logic-like expressions. Our experimental results showcase NestE's significant performance gains over current baselines in triple prediction and conditional link prediction. The code and pre-trained models are open available at https://github.com/xiongbo010/NestE.

翻译：知识图谱推理主要聚焦于三元组形式的事实。近期的研究通过引入更强大的表示方式（如超关系事实）来增强这些事实的语义表达能力。然而，这些方法仅限于描述单一信息的**原子事实**。本文突破原子事实的局限，深入研究了**嵌套事实**，即通过引号标记的三元组来表示，其中主客体本身也是三元组（例如（（奥巴马，担任职务，总统），继任者，（特朗普，担任职务，总统）））。嵌套事实能够表达随时间变化的**情境**以及实体和关系间的**逻辑模式**等复杂语义。为此，我们提出NestE——一种新颖的知识图谱嵌入方法，能够同时捕获原子事实和嵌套事实的语义。NestE将每个原子事实表示为$1\times3$矩阵，每个嵌套关系建模为$3\times3$矩阵，通过矩阵乘法旋转$1\times3$原子事实矩阵。矩阵中的每个元素采用广义四维超复数空间（包括球面四元数、双曲四元数和分裂四元数）中的复数表示。通过深入分析，我们证明了该嵌入方法在捕获嵌套事实的多样化逻辑模式方面的有效性，超越了传统一阶逻辑表达式的局限。实验结果表明，NestE在三元组预测和条件链接预测任务上均显著优于现有基线方法。代码和预训练模型已开源：https://github.com/xiongbo010/NestE。