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Reasoning with knowledge graphs (KGs) has primarily focused on triple-shapedfacts. Recent advancements have been explored to enhance the semantics of thesefacts by incorporating more potent representations, such as hyper-relationalfacts. However, these approaches are limited to \emph{atomic facts}, whichdescribe a single piece of information. This paper extends beyond \emph{atomicfacts} and delves into \emph{nested facts}, represented by quoted triples wheresubjects 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 nestedfacts enable the expression of complex semantics like \emph{situations} overtime and \emph{logical patterns} over entities and relations. In response, weintroduce NestE, a novel KG embedding approach that captures the semantics ofboth atomic and nested factual knowledge. NestE represents each atomic fact asa $1\times3$ matrix, and each nested relation is modeled as a $3\times3$ matrixthat rotates the $1\times3$ atomic fact matrix through matrix multiplication.Each element of the matrix is represented as a complex number in thegeneralized 4D hypercomplex space, including (spherical) quaternions,hyperbolic quaternions, and split-quaternions. Through thorough analysis, wedemonstrate the embedding's efficacy in capturing diverse logical patterns overnested facts, surpassing the confines of first-order logic-like expressions.Our experimental results showcase NestE's significant performance gains overcurrent baselines in triple prediction and conditional link prediction. Thecode and pre-trained models are open available athttps://github.com/xiongbo010/NestE.

NestE: Modeling Nested Relational Structures for Knowledge Graph Reasoning