Embedding entities and relations of a knowledge graph in a low-dimensional space has shown impressive performance in predicting missing links between entities. Although progresses have been achieved, existing methods are heuristically motivated and theoretical understanding of such embeddings is comparatively underdeveloped. This paper extends the random walk model of word embeddings to Knowledge Graph Embeddings (KGEs) to derive a scoring function that evaluates the strength of a relation R between two entities h (head) and t (tail). Moreover, we show that marginal loss minimisation, a popular objective used in much prior work in KGE, follows naturally from the log-likelihood ratio maximisation under the probabilities estimated from the KGEs according to our theoretical relationship. We propose a learning objective motivated by the theoretical analysis to learn KGEs from a given knowledge graph.Using the derived objective, accurate KGEs are learnt from FB15K237 and WN18RR benchmark datasets, providing empirical evidence in support of the theory.

Representing the semantic relations that exist between two given words (or entities) is an important first step in a wide-range of NLP applications such as analogical reasoning, knowledge base completion and relational information retrieval. A simple, yet surprisingly accurate method for representing a relation between two words is to compute the vector offset (PairDiff) between their corresponding word embeddings. Despite the empirical success, it remains unclear as to whether PairDiff is the best operator for obtaining a relational representation from word embeddings. We conduct a theoretical analysis of generalised bilinear operators that can be used to measure the l2 relational distance between two word-pairs. We show that, if the word embed- dings are standardised and uncorrelated, such an operator will be independent of bilinear terms, and can be simplified to a linear form, where PairDiff is a special case. For numerous word embedding types, we empirically verify the uncorrelation assumption, demonstrating the general applicability of our theoretical result. Moreover, we experimentally discover PairDiff from the bilinear relational compositional operator on several benchmark analogy datasets.