Question: What are the tasks that this method has shown improvements?
Evidence: Our experimental results show that the proposed additional transformation does not only benefit cross-lingual evaluation tasks, but, perhaps surprisingly, also monolingual ones. In particular, we perform an extensive set of experiments on standard benchmarks for bilingual dictionary induction and monolingual and cross-lingual word similarity, as well as on an extrinsic task: cross-lingual hypernym discovery.
Tables 2 and 3 show the monolingual and cross-lingual word similarity results, respectively. For both the monolingual and cross-lingual settings, we can notice that our models generally outperform the corresponding baselines. Moreover, in cases where no improvement is obtained, the differences tend to be minimal, with the exception of RG-65, but this is a very small test set for which larger variations can thus be expected. In contrast, there are a few cases where substantial gains were obtained by using our model. This is most notable for English WordSim and SimLex in the monolingual setting.
As can be seen in Table 1 , our refinement method consistently improves over the baselines (i.e., VecMap and MUSE) on all language pairs and metrics. The higher scores indicate that the two monolingual embedding spaces become more tightly integrated because of our additional transformation. It is worth highlighting here the case of English-Finnish, where the gains obtained in $P@5$ and $P@10$ are considerable. This might indicate that our approach is especially useful for morphologically richer languages such as Finnish, where the limitations of the previous bilingual mappings are most apparent.
The results listed in Table 4 indicate several trends. First and foremost, in terms of model-wise comparisons, we observe that our proposed alterations of both VecMap and MUSE improve their quality in a consistent manner, across most metrics and data configurations. In Italian our proposed model shows an improvement across all configurations. However, in Spanish VecMap emerges as a highly competitive baseline, with our model only showing an improved performance when training data in this language abounds (in this specific case there is an increase from 17.2 to 19.5 points in the MRR metric). This suggests that the fact that the monolingual spaces are closer in our model is clearly beneficial when hybrid training data is given as input, opening up avenues for future work on weakly-supervised learning. Concerning the other baseline, MUSE, the contribution of our proposed model is consistent for both languages, again becoming more apparent in the Italian split and in a fully cross-lingual setting, where the improvement in MRR is almost 3 points (from 10.6 to 13.3). Finally, it is noteworthy that even in the setting where no training data from the target language is leveraged, all the systems based on cross-lingual embeddings outperform the best unsupervised baseline, which is a very encouraging result with regards to solving tasks for languages on which training data is not easily accessible or not directly available.
In particular, we perform an extensive set of experiments on standard benchmarks for bilingual dictionary induction and monolingual and cross-lingual word similarity, as well as on an extrinsic task: cross-lingual hypernym discovery.
For both the monolingual and cross-lingual settings, we can notice that our models generally outperform the corresponding baselines. 
As can be seen in Table 1 , our refinement method consistently improves over the baselines (i.e., VecMap and MUSE) on all language pairs and metrics.
First and foremost, in terms of model-wise comparisons, we observe that our proposed alterations of both VecMap and MUSE improve their quality in a consistent manner, across most metrics and data configurations.
Answer: bilingual dictionary induction, monolingual and cross-lingual word similarity, and cross-lingual hypernym discovery