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Representing the semantic relations that exist between two given words (orentities) is an important first step in a wide-range of NLP applications suchas analogical reasoning, knowledge base completion and relational informationretrieval. A simple, yet surprisingly accurate method for representing arelation between two words is to compute the vector offset (\PairDiff) betweentheir corresponding word embeddings. Despite the empirical success, it remainsunclear as to whether \PairDiff is the best operator for obtaining a relationalrepresentation from word embeddings. We conduct a theoretical analysis ofgeneralised bilinear operators that can be used to measure the $\ell_{2}$relational distance between two word-pairs. We show that, if the wordembeddings are standardised and uncorrelated, such an operator will beindependent of bilinear terms, and can be simplified to a linear form, where\PairDiff is a special case. For numerous word embedding types, we empiricallyverify the uncorrelation assumption, demonstrating the general applicability ofour theoretical result. Moreover, we experimentally discover \PairDiff from thebilinear relation composition operator on several benchmark analogy datasets.

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