  Encoding facts as representations of entities and binary relationships
between them, as learned by knowledge graph representation models, is useful
for various tasks, including predicting new facts, question answering, fact
checking and information retrieval. The focus of this thesis is on (i)
improving knowledge graph representation with the aim of tackling the link
prediction task; and (ii) devising a theory on how semantics can be captured in
the geometry of relation representations. Most knowledge graphs are very
incomplete and manually adding new information is costly, which drives the
development of methods which can automatically infer missing facts. The first
contribution of this thesis is HypER, a convolutional model which simplifies
and improves upon the link prediction performance of the existing convolutional
state-of-the-art model ConvE and can be mathematically explained in terms of
constrained tensor factorisation. The second contribution is TuckER, a
relatively straightforward linear model, which, at the time of its
introduction, obtained state-of-the-art link prediction performance across
standard datasets. The third contribution is MuRP, first multi-relational graph
representation model embedded in hyperbolic space. MuRP outperforms all
existing models and its Euclidean counterpart MuRE in link prediction on
hierarchical knowledge graph relations whilst requiring far fewer dimensions.
Despite the development of a large number of knowledge graph representation
models with gradually increasing predictive performance, relatively little is
known of the latent structure they learn. We generalise recent theoretical
understanding of how semantic relations of similarity, paraphrase and analogy
are encoded in the geometric interactions of word embeddings to how more
general relations, as found in knowledge graphs, can be encoded in their
representations.
