More formally, at each step of stochastic gradient descent, given a source space vector x i , its target gold label/translation y i in Y T r and the mapped vectorŷ i , we compute s j = cos(ŷ i , y j ) − cos(y i , y j ), for all vectors y j in Y T r s.t.
given the vector representation of a word in language A, we apply the mapping to obtain an estimate of the vector representation of its meaning in language B, returning the nearest neighbour of the mapped vector in the B space as candidate translation.