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The emergence of unsupervised word embeddings, pre-trained on very largemonolingual text corpora, is at the core of the ongoing neural revolution inNatural Language Processing (NLP). Initially introduced for English, suchpre-trained word embeddings quickly emerged for a number of other languages.Subsequently, there have been a number of attempts to align the embeddingspaces across languages, which could enable a number of cross-language NLPapplications. Performing the alignment using unsupervised cross-linguallearning (UCL) is especially attractive as it requires little data and oftenrivals supervised and semi-supervised approaches. Here, we analyze popularmethods for UCL and we find that often their objectives are, intrinsically,versions of the Wasserstein-Procrustes problem. Hence, we devise an approach tosolve Wasserstein-Procrustes in a direct way, which can be used to refine andto improve popular UCL methods such as iterative closest point (ICP),multilingual unsupervised and supervised embeddings (MUSE) and supervisedProcrustes methods. Our evaluation experiments on standard datasets showsizable improvements over these approaches. We believe that our rethinking ofthe Wasserstein-Procrustes problem could enable further research, thus helpingto develop better algorithms for aligning word embeddings across languages. Ourcode and instructions to reproduce the experiments are available athttps://github.com/guillemram97/wp-hungarian.

On a Novel Application of Wasserstein-Procrustes for Unsupervised Cross-Lingual Learning