  Recent research in cross-lingual word embeddings has almost exclusively
focused on offline methods, which independently train word embeddings in
different languages and map them to a shared space through linear
transformations. While several authors have questioned the underlying
isomorphism assumption, which states that word embeddings in different
languages have approximately the same structure, it is not clear whether this
is an inherent limitation of mapping approaches or a more general issue when
learning cross-lingual embeddings. So as to answer this question, we experiment
with parallel corpora, which allows us to compare offline mapping to an
extension of skip-gram that jointly learns both embedding spaces. We observe
that, under these ideal conditions, joint learning yields to more isomorphic
embeddings, is less sensitive to hubness, and obtains stronger results in
bilingual lexicon induction. We thus conclude that current mapping methods do
have strong limitations, calling for further research to jointly learn
cross-lingual embeddings with a weaker cross-lingual signal.
