  There has recently been much interest in extending vector-based word
representations to multiple languages, such that words can be compared across
languages. In this paper, we shift the focus from words to documents and
introduce a method for embedding documents written in any language into a
single, language-independent vector space. For training, our approach leverages
a multilingual corpus where the same concept is covered in multiple languages
(but not necessarily via exact translations), such as Wikipedia. Our method,
Cr5 (Crosslingual reduced-rank ridge regression), starts by training a
ridge-regression-based classifier that uses language-specific bag-of-word
features in order to predict the concept that a given document is about. We
show that, when constraining the learned weight matrix to be of low rank, it
can be factored to obtain the desired mappings from language-specific
bags-of-words to language-independent embeddings. As opposed to most prior
methods, which use pretrained monolingual word vectors, postprocess them to
make them crosslingual, and finally average word vectors to obtain document
vectors, Cr5 is trained end-to-end and is thus natively crosslingual as well as
document-level. Moreover, since our algorithm uses the singular value
decomposition as its core operation, it is highly scalable. Experiments show
that our method achieves state-of-the-art performance on a crosslingual
document retrieval task. Finally, although not trained for embedding sentences
and words, it also achieves competitive performance on crosslingual sentence
and word retrieval tasks.
