The first term in (2) corresponds to the pairwise interactions between h and t. P, Q ∈ R d×d are the nonsingular 2 projection matrices involving first-order contributions respectively of h and t towards r. Let us consider the problem of learning the simplest bilinear functional form according to (2) from a given dataset of analogous word-pairs D + = {((h, t), (h , t ))}.
we would like to learn the parameters A, P and Q such that some distance (loss) between analogous word-pairs is minimised.