This last optimization objective coincides with <cite>Xing et al. (2015)</cite> , but <cite>their work</cite> was motivated by an hypothetical inconsistency in Mikolov et al. (2013b) , where the optimization objective to learn word embeddings uses dot product, the objective to learn mappings uses Euclidean distance and the similarity computations use cosine. However, the fact is that, as long as W is orthogonal, optimizing the squared Euclidean distance of length-normalized embeddings is equivalent to optimizing the cosine, and therefore, the mapping objective proposed by <cite>Xing et al. (2015)</cite> is equivalent to that used by Mikolov et al. (2013b) with orthogonality constraint and unit vectors. In fact, our experiments show that orthogonality is more relevant than length normalization, in contrast to <cite>Xing et al. (2015)</cite> , <cite>who introduce</cite> orthogonality only to ensure that unit length is preserved after mapping.