Why PairDiff works? -- A Mathematical Analysis of Bilinear Relational Compositional Operators for Analogy Detection

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 $\ell_{2}$ relational distance between two word-pairs. We show that, if the word embeddings 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 relation composition operator on several benchmark analogy datasets.