  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.
