SlowGuess/NAACL_2021_OCR huggingface dataset:
To reduce randomness, we executed each model in each setup three times and the average accuracy (ACC, aka. precision at rank 1) is reported. Following Glavaš et al. (2019), by comparing scores achieved before and after  $\ell_1$  refinement, statistical significance is indicated via the  $p$ -value of two-tailed t-tests with Bonferroni correction (Dror et al., 2018) (note that  $p$ -values are not recorded for Tab. 2b given the small number of runs).
5.1 Bilingual Lexicon Induction
Refining unsupervised baselines. Tab. 1a follows the main setup of Lample et al. (2018), who tested six language pairs using Wiki-Embs<sup>4</sup>. After  $\ell_1$  refinement, MUSE- $\ell_1$ , JA-MUSE- $\ell_1$ , and VECMAP- $\ell_1$  all significantly ( $p < 0.01$ ) outperform their corresponding base algorithms, with an average  $1.1\%$  performance gain over MUSE,
<sup>3</sup>http://www.netlib.org/ode/vode.f
4Note that we are unable to report the result of English to Esperanto as the corresponding dictionary is missing, see https://git.io/en-eo-dict-issue.
EN-DE	EN-ES	EN-FR	EN-RU	EN-ZH
MUSE*	74.0	81.7	82.3	44.0	32.5
MUSE-ℓ2	74.0	82.1	82.6	43.8*	31.9*
MUSE-ℓ1	75.2	82.6	82.9	45.6*	33.8*
JA-MUSE*	74.2	81.4	82.8	45.0	36.1
JA-MUSE-ℓ2	74.1	81.6	82.7	45.1	36.2
JA-MUSE-ℓ1	75.4	82.0	83.1	46.3	38.1
VECMAP*	75.1	82.3	80.0	49.2	00.0
VECMAP-ℓ2	74.8	82.3	79.4	48.9	00.0
VECMAP-ℓ1	75.4	82.9	80.2	49.9	00.0
(a) Wiki-Embs (setup of Lample et al. (2018)).
EN-DE	EN-ES	EN-FI	EN-IT
MUSE	00.0	07.1	00.0	09.1
MUSE-ℓ2	00.0	00.0	00.0	00.0
MUSE-ℓ1	00.0	00.0	00.0	00.0
JA-MUSE	47.9	48.4	33.0	37.2
JA-MUSE-ℓ2	47.9	48.6	32.9	37.3
JA-MUSE-ℓ1	48.8	49.7	35.2	37.7
VECMAP	48.2	48.1	32.6	37.3
VECMAP-ℓ2	48.1	47.9	32.9	37.1
VECMAP-ℓ1	49.0	48.9	34.4	37.8
(b) News-Embs (setup of Artetxe et al. (2018a)).
Table 1: ACC (%) of unsupervised BLI. (a) Rows marked with  $\checkmark$ ,  $\clubsuit$  and  $\clubsuit$  are respectively from Lample et al. (2018), Wang et al. (2020) and Zhou et al. (2019). NB: for EN- $\left\{\mathrm{RU}, \mathrm{ZH}\right\}$  we observed one failed run (ACC < 10.0%), where we only record the average of successful scores with *. (b) Rows marked with  $\checkmark$  are from Artetxe et al. (2018a).
EN-DE	EN-FI	EN-FR	EN-HR	EN-IT	EN-RU	EN-TR
JA-RCSLS	50.9	33.9	63.0	29.1	58.3	41.3	29.4
JA-RCSLS-ℓ2	50.7	33.8	63.0	29.1	58.2	41.3	29.5
JA-RCSLS-ℓ1	51.6	34.5	63.4	30.4	59.0	41.9	30.2
PROC-B	52.1	36.0	63.3	29.6	60.5	41.9	30.1
PROC-B-ℓ2	51.8	34.4	63.1	28.2	60.5	39.8	28.0
PROC-B-ℓ1	52.6	36.3	63.7	30.5	60.5	42.3	30.9
(a) Wiki-Embs (setup of Glavaš et al. (2019)).
EN-DE	EN-FI	EN-IT
JA-RCSLS	46.8	42.0	37.4
JA-RCSLS-ℓ2	46.9	42.2	37.5
JA-RCSLS-ℓ1	48.3	44.6	39.0
PROC-B	47.5	41.4	37.3
PROC-B-ℓ2	47.1	41.7	37.4
PROC-B-ℓ1	52.6	43.3	41.1
(b) News-Embs.
Table 2: MRR (%) of supervised BLI. Rows marked with  $\clubsuit$  are from the supplementary of Glavaš et al. (2019).
1.1% over JA-MUSE, and 0.5% over VECMAP. To put these improvements in context, Heyman et al. (2019) reported an improvement of 0.4% for VECMAP on same dataset and language pairs.
Our method tends to work better on the more distant language pairs. For instance, for the distant pairs EN-{RU,ZH}, the increments achieved by MUSE-  $\ell_1$  are  $1.6\%$  and  $1.3\%$ , respectively; whereas for the close pairs EN-{DE, ES, FR} the average gain is a maximum of  $0.9\%$ . A similar trend can be observed for JA-MUSE-  $\ell_1$  and VECMAP-  $\ell_1$ . (As the VECMAP algorithm always collapses for EN-ZH, no result is reported for this language pair).
Another set of experiments were conducted to evaluate the robustness of our algorithm following the main setup of Artetxe et al. (2018a), who tested four language pairs based on the more homogeneous News-Embs. Tab. 1b shows that JA-MUSE- $\ell_{1}$  and VECMAP- $\ell_{1}$  consistently improves the original VECMAP with an average gain of  $1.2\%$  and  $1.0\%$  ( $p < 0.01$ ). Obtaining such substantial improvements over the state-of-the-art is nontrivial. For example, even a very recent weakly supervised method by Wang et al. (2019) is inferior to VECMAP by  $1.0\%$  average ACC. On the other hand, MUSE fails to produce any analysable result as it always collapses on the more challenging News-Embs. Improvement with  $\ell_{1}$  refinement is
also larger when language pairs are more distant, e.g., for VECMAP-  $\ell_{1}$  the ACC gain on EN-FI is  $1.8\%$  , more than double of the gain  $(0.7\%)$  on the close pairs EN-  $\{\mathrm{DE},\mathrm{IT}\}$  (cf. Tab. 1a and above).
We also conduct an ablation study by reporting the performance of  $\ell_2$  refinement scheme ( $\{\mathsf{MUSE},\mathsf{JAMUSE},\mathsf{VECMAP}\} -\ell_2$ ). This observation is in accordance with that of Lample et al. (2018), who reported that after performing  $\ell_2$  refinement in the first loop, applying further iterations only produces marginal precision gain, if any.
Overall, the  $\ell_1$  refinement consistently and significantly improve the CLWEs produced by base algorithms, regardless of the embeddings and setups used, thereby demonstrating the effectiveness and robustness of the proposed algorithm.
Refining supervised baselines. To test the generalisability of our method, we also applied it on state-of-the-art supervised CLWE models: PROC-B (Glavaš et al., 2019) and JA-RCSLS (Wang et al., 2020). Following the setup of Glavaš et al. (2019), we learn mappings using Wiki-Embs and 1K training splits of their dataset.
Their evaluation code retrieves bilingual word pairs using the classic nearest-neighbour algorithm and outputs the Mean Reciprocal Rank (MRR). As shown in Tab. 2a, both JA-RCSLS-  $\ell_1$  and PROCB-  $\ell_{1}$  outperform the baseline algorithms for all
Unsupervised	DE-IT	DE-TR	FI-HR	FI-IT	HR-RU	IT-FR	TR-IT
ICP	44.7	21.5	20.8	26.3	30.9	62.9	24.3
GWA	44.0	10.1	00.9	17.3	00.1	65.5	14.2
MUSE	49.6	23.7	22.8	32.7	00.0	66.2	30.6
MUSE-ℓ2	50.3	23.9	23.1	32.7	34.9	67.1	30.5*
MUSE-ℓ1	50.7	26.5	25.4	35.0	37.9	67.6	33.3*
JA-MUSE	50.9	25.6	23.4	34.9	36.9	68.3	34.7
JA-MUSE-ℓ2	50.9	25.5	23.4	34.7	36.9	68.4	34.7
JA-MUSE-ℓ1	51.5	28.4	26.1	36.0	37.6	68.7	36.1
VECMAP	49.3	25.3	28.0	35.5	37.6	66.7	33.2
VECMAP-ℓ2	48.8	25.7	28.5	35.8	38.4	67.0	33.5
VECMAP-ℓ1	50.1	28.2	30.3	37.1	40.1	67.6	35.9
Supervised
DLV	42.0	16.7	18.4	24.4	26.4	58.5	20.9
RCCLS	45.3	20.1	21.4	27.2	29.1	63.7	24.6
JA-RSCLS	46.6	20.9	22.1	29.0	29.9	65.2	25.3
JA-RSCLS-ℓ2	46.4	20.8	22.3	29.0	29.8	65.2	25.3
JA-RSCLS-ℓ1	47.3	22.2	23.8	30.1	31.2	65.9	26.6
PROC-B	50.7	25.0	26.3	32.8	34.8	66.5	29.8
PROC-B-ℓ2	50.0	24.1	25.6	31.8	34.3	66.4	29.6
PROC-B-ℓ1	51.1	25.6	26.9	33.6	35.0	67.4	30.5
Table 3: MRR (%) of BLI for non-EN language pairs. Rows marked with  $\triangledown$  are from the supplementary of Glavaš et al. (2019). MUSE yielded one unsuccessful run for TR-IT, and we only record the average of the two successful scores with *.
(c) MUSE on IT-FR Wiki-Embs (cf. Tab. 3).
Figure 1: Changes to  $||\mathbf{AM} - \mathbf{B}||_2$  after applying  $\ell_{1}$  (upper) and  $\ell_{2}$  (lower) refinement. Each word pairs is represented by a bar ordered on the x-axis by the distance between them. See Fig. A.1 for the alternative version.
language pairs (with the exception of EN-IT where the score of PROC-B is unchanged) with an average improvement of  $0.9\%$  and  $0.5\%$ , respectively  $(p < 0.01)$ .
JA-RCSLS- $\ell_1$  and PROC-B- $\ell_1$  were also tested using News-Embs with results shown in Tab. 2b<sup>5</sup>.  $\ell_1$  refinement achieves an impressive improvement for both close (EN-DE, IT) and distant (EN-FI) language pairs: average gain of  $1.9\%$  and  $3.9\%$  respectively and over  $5\%$  for EN-DE (PROC-B- $\ell_1$ ) in particular. The  $\ell_2$  refinement does not benefit the supervised baseline, similar to the lack of improvement observed in the unsupervised setups.
Comparison of unsupervised and supervised settings. This part provides a comparison of the effectiveness of  $\ell_1$  refinement in unsupervised and supervised scenarios. Unlike previous experiments where only alignments involving English were investigated, these tests focus on non-EN setups. Glavaš et al. (2019)'s dataset is used to construct seven representative pairs which cover every category of etymological combination, i.e., intralanguage-branch {HR-RU, IT-FR}, inter-language-branch {DE-IT}, and inter-language-family {DE-TR, FI-HR, FI-IT, TR-IT}. The 1K training splits are used as seed lexicons in supervised runs. Apart
from our main baselines, we further report the results of several other competitive CLWE models: Iterative Closest Point Model (ICP, Hoshen and Wolf, 2018), Gromov-Wasserstein Alignment Model (GWA, Alvarez-Melis and Jaakkola, 2018), Discriminative Latent-Variable Model (DLV, Ruder et al., 2018) and Relaxed CSLS Model (RCSLS, Joulin et al., 2018).
Results shown in Tab. 3 demonstrate that the main baselines (MUSE, JA-MUSE, VECMAP, JA-RCSLS, and PROC-B) outperform these other models by a large margin. For all these main baselines, post applying  $\ell_1$  refinement improves the mapping quality for all language pairs  $(p < 0.01)$ , with average improvements of  $1.7\%$ ,  $1.4\%$ ,  $1.8\%$ ,  $1.1\%$ , and  $0.8\%$ , respectively. Consistent with findings in the previous experiments,  $\ell_2$  refinement does not enhance performance. Improvement with  $\ell_1$  refinement is higher when language pairs are more distant, e.g., for all inter-language-family pairs such as FI-HR and TR-IT, even the minimum improvement of MUSE- $\ell_1$  over MUSE is  $2.3\%$ .
Comparing unsupervised and supervised approaches, it can be observed that MUSE, JA-MUSE and VECMAP achieve higher overall gain with  $\ell_1$  refinement than JA-RCSLS and PROC-B, where JA-MUSE- $\ell_1$  and VECMAP- $\ell_1$  give the best overall performance. One possible explanation to this
Note that results for EN-ES is not included, as no EN-ES dictionary is provided in Glavaš et al. (2019)'s dataset.
phenomenon is that there is only a single source of possible noise in unsupervised models (i.e. the embedding topology) but for supervised methods noise can also be introduced via the seed lexicons. Consequently unsupervised approaches drive more benefit from  $\ell_1$  refinement, which reduces the influence of topological outliers in CLWEs.
Topological behaviours of  $\ell_1$  and  $\ell_2$  refinements.
To validate our assumption that  $\ell_2$  refinement is more sensitive to outliers while its  $\ell_1$  counterpart is more robust, we analyse how each refinement strategy changes the distance between bilingual word vector pairs in the synthetic dictionary  $D$  (cf. Algorithm 1) constructed from trained CLWE models. Specifically, for each word vector pair we subtract its post-refinement distance from the original distance (i.e., without applying additional  $\ell_1$  or  $\ell_2$  refinement step). Fig. 1 shows visualisation examples for three algorithms and language pairs, where each bar represents one word pair. It can be observed that  $\ell_1$  refinement effectively reduces the distance for most word pairs, regardless of their original distance (i.e., indicated by bars with negative values in the figures). The conventional  $\ell_2$  refinement strategy, in contrast, exhibits very different behaviour and tends to be overly influenced by word pairs with large distance (i.e. by outliers). The reason for this is that the  $\ell_2$ -norm penalty increases quadratically, causing the solution to put much more weight on optimising distant word pairs (i.e., word pairs on the right end of the X-axis show sharp distance decrements). This observation is in line with Rousseeuw and Leroy (1987) and explains why  $\ell_1$  loss performs substantially stronger than  $\ell_2$  loss in the refinement.
Case study. After aligning EN-RU embeddings with unsupervised MUSE, we measured the distance between vectors corresponding to the groundtruth dictionary of Lample et al. (2018) (cf. Fig. 1a). We then detected large outliers by finding vector pairs whose distance falls above  $\mathrm{Q3} + 1.5 \cdot (\mathrm{Q3} - \mathrm{Q1})$ , where Q1 and Q3 respectively denote the lower and upper quartile based on the popular InterQuartile Range (Hoaglin et al., 1986). We found that many of the outliers correspond to polysemous entries, such as {state (2× noun meanings and 1× verb meaning), coctoHne (only means status)}, {type (2× nominal meanings and 1× verb meaning), tHII (only means kind)}, and {film (5× noun meanings), ΦHJBM (only means movie)}. We then
Unsupervised	EN-DE	EN-FR	EN-RU	EN-TR
ICP*	58.0	51.0	57.2	40.0
GWA*	42.7	38.3	37.6	35.9
MUSE*	61.1	53.6	36.3	35.9
MUSE-ℓ2	61.1	53.0	57.3*	48.9*
MUSE-ℓ1	63.5	55.3	58.9*	52.3*
JA-MUSE	61.3	55.2	58.1	55.0
JA-MUSE-ℓ2	61.2	55.2	57.6	55.1
JA-MUSE-ℓ1	62.9	57.9	59.4	57.5
VECMAP*	60.4	61.3	58.1	53.4
VECMAP-ℓ2	60.3	60.6	57.7	53.5
VECMAP-ℓ1	61.5	63.7	60.1	56.4
Supervised
RCSLS*	37.6	35.7	37.8	38.7
JA-RSCLS	50.2	48.9	51.0	51.7
JA-RSCLS-ℓ2	50.4	48.6	50.9	51.5
JA-RSCLS-ℓ1	51.3	50.1	53.2	52.6
PROC-B*	61.3	54.3	59.3	56.8
PROC-B-ℓ2	61.0	54.8	58.9	55.1
PROC-B-ℓ1	62.1	54.8	60.7	58.2
Table 4: ACC  $(\%)$  of NLI. Rows marked with  $\checkmark$  are from Glavaš et al. (2019). MUSE yielded one unsuccessful run for EN-RU and EN-TR respectively, which we exclude when calculating the average (with  $*$ ).
re-perform  $\ell_{2}$ -based mapping after removing these vector pairs, observing that the accuracy jumps to  $45.9\%$  (cf. the original  $\ell_{2}$ -norm alignment it is  $43.8\%$  and after  $\ell_{1}$  refinement it is  $45.6\%$ , cf. Tab. 1). This indicates that although all baselines already make use of preprocessing steps including vector normalization, outlier issues still exist and harms the  $\ell_{2}$  norm CLWEs. However, they can be alleviated by the proposed  $\ell_{1}$  refinement technique.
5.2 Natural Language Inference
Finally, we experimented with a downstream NLI task in which the aim is to determine whether a "hypothesis" is true (entailment), false (contradiction) or undetermined (neutral), given a "premise". Higher ACC indicates better encoding of semantics in the tested embeddings. The CLWEs used are those trained with Wiki-Embs for BLI. For MUSE, JA-MUSE and VECMAP, we also obtain CLWEs for EN-TR pair with the same configuration.
Following Glavaš et al. (2019), we first train the Enhanced Sequential Inference Model (Chen et al., 2017) based on the large-scale English MultiNLI corpus (Williams et al., 2018) using vectors of language  $\mathsf{L}_{\mathsf{A}}$  (EN) from an aligned bilingual embedding space (e.g., EN-DE). Next, we replace the  $\mathsf{L}_{\mathsf{A}}$  vectors with the vectors of language  $\mathsf{L}_{\mathsf{B}}$  (e.g., DE), and directly test the trained model on the language  $\mathsf{L}_{\mathsf{B}}$  portion of the XNLI corpus (Conneau et al., 2018).
Results in Tab. 4 show that the CLWEs refined
by our algorithm yield the highest ACC for all language pairs in both supervised and unsupervised settings. The  $\ell_2$  refinement, on the contrary, is not beneficial overall. Improvements in cross-lingual transfer for NLI exhibit similar trends to those in the BLI experiments, i.e. greater performance gain for unsupervised methods and more distant language pairs, consistent with previous observations (Glavaš et al., 2019). For instance, MUSE- $\ell_1$  JAMUSE- $\ell_1$  and VECMAP- $\ell_1$  outperform their baselines by at least  $2\%$  in ACC on average ( $p < 0.01$ ), whereas the improvements of JA-RSCLS- $\ell_1$  and PROC-B- $\ell_1$  over their corresponding base methods are  $2\%$  and  $2.1\%$  respectively ( $p < 0.01$ ). For both unsupervised and supervised methods,  $\ell_1$  refinement demonstrates stronger effect for more distant language pairs, e.g., MUSE- $\ell_1$  surpasses MUSE by  $1.2\%$  for EN-FR, whereas a more impressive  $2.7\%$  gain is achieved for EN-TR.
In summary, in addition to improving BLI performance, our  $\ell_{1}$  refinement method also produces a significant improvement for a downstream task (NLI), demonstrating its effectiveness in improving the CLWE quality.
This paper proposes a generic post-processing technique to enhance CLWE performance based on optimising  $\ell_1$  loss. This algorithm is motivated by successful applications in other research fields (e.g. computer vision and data mining) which exploit the  $\ell_1$  norm cost function since it has been shown to be more robust to noisy data than the commonly-adopted  $\ell_2$  loss. The approach was evaluated using ten diverse languages and word embeddings from different domains on the popular BLI benchmark, as well as a downstream task of cross-lingual transfer for NLI. Results demonstrated that our algorithm can significantly improve the quality of CLWEs in both supervised and unsupervised setups. It is therefore recommended that this straightforward technique be applied to improve performance of CLWEs.
The convergence speed of the optimiser prevented us from performing  $\ell_1$  loss optimisation over multiple iterations. Future work will focus on improving the efficiency of our  $\ell_1$  OPA solver, as well as exploring the application of other robust loss functions within CLWE training strategies.
This work provides an effective post-hoc method to improve CLWEs, advancing the state-of-the-art in both supervised and unsupervised settings. Our comprehensive empirical studies demonstrate that the proposed algorithm can facilitate researches in machine translation, cross-lingual transfer learning, etc, which have deep societal impact of bridging cultural gaps across the world.
Besides, this paper introduces and solves an optimisation problem based on an under-explored robust cost function, namely  $\ell_1$  loss. We believe it could be of interest for the wider community as outlier is a long-standing issue in many artificial intelligence applications.
One caveat with our method, as is the case for all word-embedding-based systems, is that various biases may exist in vector spaces. We suggest this problem should always be looked at critically. In addition, our implemented solver can be computationally expensive, leading to increased electricity consumption and the associated negative environmental repercussions.
This work is supported by the award made by the UK Engineering and Physical Sciences Research Council (Grant number: EP/P011829/1) and Baidu, Inc. We would also like to express our sincerest gratitude to Guanyi Chen, Ruizhe Li, Xiao Li, Shun Wang, and the anonymous reviewers for their insightful and helpful comments.
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