  Cross-Lingual Word Embeddings (CLWEs) encode words from two or more languages
in a shared high-dimensional space in which vectors representing words with
similar meaning (regardless of language) are closely located. Existing methods
for building high-quality CLWEs learn mappings that minimise the $\ell_{2}$
norm loss function. However, this optimisation objective has been demonstrated
to be sensitive to outliers. Based on the more robust Manhattan norm (aka.
$\ell_{1}$ norm) goodness-of-fit criterion, this paper proposes a simple
post-processing step to improve CLWEs. An advantage of this approach is that it
is fully agnostic to the training process of the original CLWEs and can
therefore be applied widely. Extensive experiments are performed involving ten
diverse languages and embeddings trained on different corpora. Evaluation
results based on bilingual lexicon induction and cross-lingual transfer for
natural language inference tasks show that the $\ell_{1}$ refinement
substantially outperforms four state-of-the-art baselines in both supervised
and unsupervised settings. It is therefore recommended that this strategy be
adopted as a standard for CLWE methods.
