If we rewrite the bias terms as a i = a i − || x i || 2 2 and b j = b j − || c j || 2 2 , we obtain the equivalent representation: Together with Lemma 1, we recognize this as a weighted multidimensional scaling (MDS) objective with weights f (C ij ).
splitting the word vector x i and context vector c i is helpful in practice but not necessary under the assumptions of Lemma 1 since the true embedding x i = c i = x i /σ and a i , b i = 0 is a global minimum whenever dim( x) = d. GloVe can recover the true metric provided that we set d correctly.