Learning representations of entities and relations

Learning to represent factual knowledge about the world in a succinct and accessible manner is a fundamental machine learning problem. 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 (i.e. link prediction), 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. This thesis introduces three knowledge graph representation methods, each applied to the link prediction task. The first contribution 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. Drawing inspiration from the tensor factorisation view of HypER, the second contribution is TuckER, a relatively straightforward linear knowledge graph representation model, which, at the time of its introduction, obtained state-of-the-art link prediction performance across standard datasets. With a specific focus on representing hierarchical knowledge graph relations, 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. Since their publication, all above mentioned models have influenced a range of subsequent developments in the knowledge graph representation field. 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. This increased theoretical understanding can be used to aid future knowledge graph representation model design, as well as to improve models which incorporate logical rules between relations into their representations or those that jointly learn from multiple data sources (e.g. knowledge graphs and text)