In (<cite>Guo and Diab, 2012b</cite>) <cite>we</cite> use Alternating Least Square [ALS] for inference, which is to set the derivative of equation 1 for P/Q to 0 and iteratively compute P/Q by fixing the other matrix (Srebro and Jaakkola, 2003) . However, it is no longer applicable with the new term (equation 2) involving the length of word vectors |P ·,i |. Therefore we approximate the objective function by treating the vector length |P ·,i | as fixed values during the ALS iterations: where P ·,s(i) are the latent vectors of similar words of word i; the length of these vectors in the current iteration are stored in L s(i) (similarly L i is the current length of P ·,i ) (cf. (Steck, 2010; <cite>Guo and Diab, 2012b</cite>) for optimization details).