  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.
