  The technique of Cross-Lingual Word Embedding (CLWE) plays a fundamental role
in tackling Natural Language Processing challenges for low-resource languages.
Its dominant approaches assumed that the relationship between embeddings could
be represented by a linear mapping, but there has been no exploration of the
conditions under which this assumption holds. Such a research gap becomes very
critical recently, as it has been evidenced that relaxing mappings to be
non-linear can lead to better performance in some cases. We, for the first
time, present a theoretical analysis that identifies the preservation of
analogies encoded in monolingual word embeddings as a necessary and sufficient
condition for the ground-truth CLWE mapping between those embeddings to be
linear. On a novel cross-lingual analogy dataset that covers five
representative analogy categories for twelve distinct languages, we carry out
experiments which provide direct empirical support for our theoretical claim.
These results offer additional insight into the observations of other
researchers and contribute inspiration for the development of more effective
cross-lingual representation learning strategies.
