Let’s Play Mono-Poly: BERT Can Reveal Words’ Polysemy Level and Partitionability into Senses

Pre-trained language models (LMs) encode rich information about linguistic structure but their knowledge about lexical polysemy remains unclear. We propose a novel experimental setup for analyzing this knowledge in LMs specifically trained for different languages (English, French, Spanish, and Greek) and in multilingual BERT. We perform our analysis on datasets carefully designed to reflect different sense distributions, and control for parameters that are highly correlated with polysemy such as frequency and grammatical category. We demonstrate that BERT-derived representations reflect words’ polysemy level and their partitionability into senses. Polysemy-related information is more clearly present in English BERT embeddings, but models in other languages also manage to establish relevant distinctions between words at different polysemy levels. Our results contribute to a better understanding of the knowledge encoded in contextualized representations and open up new avenues for multilingual lexical semantics research.


Introduction
Pre-trained contextual language models have advanced the state of the art in numerous natural language understanding tasks (Devlin et al., 2019;Peters et al., 2018). Their success has motivated a large number of studies exploring what these models actually learn about language (Voita et al., 2019a;Clark et al., 2019;Voita et al., 2019b;Tenney et al., 2019). The bulk of this interpretation work relies on probing tasks which serve to predict linguistic properties from the representations generated by the models (Linzen, 2018;Rogers et al., 2020). The focus was initially put on linguistic aspects pertaining to grammar and syntax (Linzen et al., 2016;Hewitt and Manning, 2019;Hewitt Figure 1: BERT distinguishes monosemous (mono) from polysemous (poly) words in all layers. Representations for a poly word are obtained from sentences reflecting up to ten different senses (poly-bal), the same sense (poly-same), or natural occurrence in a corpus (poly-rand). and Liang, 2019). The first probing tasks addressing semantic knowledge explored phenomena in the syntax-semantics interface, such as semantic role labelling and coreference (Tenney et al., 2019;Kovaleva et al., 2019), and the symbolic reasoning potential of LM representations (Talmor et al., 2020).
Lexical meaning was largely overlooked in early interpretation work, but is now attracting increasing attention. Pre-trained LMs have been shown to successfully leverage sense annotated data for disambiguation (Wiedemann et al., 2019;Reif et al., 2019). The interplay between word type and token-level information in the hidden representations of LSTM LMs has also been explored , as well as the similarity estimates that can be drawn from contextualised representations without directly addressing word meaning (Ethayarajh, 2019). In recent work, Vulić et al. (2020) probe BERT representations for lexical semantics, addressing out-of-context word similarity. Whether these models encode knowledge about lexical polysemy and sense distinc-tions is, however, still an open question. Our work aims to fill this gap.
We propose methodology for exploring the knowledge about word senses in contextualised representations. Our approach follows a rigorous experimental protocol proper to lexical semantic analysis, which involves the use of datasets carefully designed to reflect different sense distributions. This allows us to investigate the knowledge models acquire during training, and the influence of context variation on token representations. We account for the strong correlation between word frequency and number of senses (Zipf, 1945), and for the relationship between grammatical category and polysemy, by balancing the frequency and part of speech (PoS) distributions in our datasets and applying a frequency-based model to polysemy prediction.
Importantly, our investigation encompasses monolingual models in different languages (English, French, Spanish and Greek) and multilingual BERT (mBERT). We demonstrate that BERT contextualised representations encode an impressive amount of knowledge about polysemy, and are able to distinguish monosemous (mono) from polysemous (poly) words in a variety of settings and configurations (cf. Figure 1). Importantly, we demonstrate that representations derived from contextual LMs encode knowledge about words' polysemy acquired through pretraining which is combined with information from new contexts of use (Sections 3-6). Additionally, we show that BERT representations can serve to determine how easy it is to partition a word's semantic space into senses (Section 7).
Our methodology can serve for the analysis of words and datasets from different topics, domains and languages. Knowledge about words' polysemy and sense partitionability has numerous practical implications: it can guide decisions towards a sense clustering or a perinstance approach in applications (Reisinger and Mooney, 2010;Neelakantan et al., 2014;Camacho-Collados and Pilehvar, 2018); point to words with stable semantics which can be safe cues for disambiguation in running text (Leacock et al., 1998;Agirre and Martinez, 2004;Loureiro and Camacho-Collados, 2020); determine the needs in terms of context size for disambiguation (e.g. in queries, chatbots); help lexicographers define the number of entries for a word to be present in a resource, and plan the time and effort needed in semantic annotation tasks (Mc-Carthy et al., 2016). It could also guide crosslingual transfer, serving to identify polysemous words for which transfer might be harder. Finally, analysing words' semantic space can be highly useful for the study of lexical semantic change (Rosenfeld and Erk, 2018;Dubossarsky et al., 2019;Giulianelli et al., 2020;Schlechtweg et al., 2020). We make our code and datasets available to enable comparison across studies and to promote further research in these directions. 1

Related Work
The knowledge pre-trained contextual LMs encode about lexical semantics has only recently started being explored. Works by Reif et al. (2019) and Wiedemann et al. (2019) propose experiments using representations built from Wikipedia and the SemCor corpus (Miller et al., 1993), and show that BERT can organise word usages in the semantic space in a way that reflects the meaning distinctions present in the data. It is also shown that BERT can perform well in the word sense disambiguation (WSD) task by leveraging the senserelated information available in these resources. These works address the disambiguation capabilities of the model but do not show what BERT actually knows about words' polysemy, which is the main axis of our work. In our experiments, sense annotations are not used to guide the models into establishing sense distinctions, but rather for creating controlled conditions that allow us to analyse BERT's inherent knowledge of lexical polysemy.
Probing has also been proposed for lexical semantics analysis, but addressing different questions than the ones posed in our work. Aina et al. (2019) probe the hidden representations of a bidirectional (bi-LSTM) LM for lexical (type-level) and contextual (token-level) information. They specifically train diagnostic classifiers on the tasks of retrieving the input embedding of a word and a representation of its contextual meaning (as reflected in its lexical substitutes). The results show that the information about the input word that is present in LSTM representations is not lost after contextualisation; however, the quality of the information available for a word is assessed through the model's ability to identify the corresponding embedding, as in Adi et al. (2017) and Conneau et al. (2018). Also, lexical ambiguity is only viewed through the lens of contextualisation. In our work, on the contrary, it is given a central role: we explicitly address the knowledge BERT encodes about words' degree of polysemy and partitionability into senses. Vulić et al. (2020) also propose to probe contextualised models for lexical semantics, but they do so using "static" word embeddings obtained through pooling over several contexts, or extracting representations for words in isolation and from BERT's embedding layer, before contextualisation. These representations are evaluated on tasks traditionally used for assessing the quality of static embeddings, such as out-ofcontext similarity and word analogy, which are not tailored for addressing lexical polysemy. Other contemporaneous work explores lexical polysemy in static embeddings (Jakubowski et al., 2020), and the relation of ambiguity and context uncertainty as approximated in the space constructed by mBERT using information-theoretic measures (Pimentel et al., 2020). Finally, work by Ethayarajh (2019) provides useful observations regarding the impact of context on the representations, without explicitly addressing the semantic knowledge encoded by the models. Through an exploration of BERT, ELMo and GPT-2 (Radford et al., 2019), the author highlights the highly distorted similarity of the obtained contextualised representations which is due to the anisotropy of the vector space built by each model. 2 The question of meaning is not addressed in this work, making it hard to draw any conclusions about lexical polysemy.
Our proposed experimental setup is aimed at investigating the polysemy information encoded in the representations built at different layers of deep pre-trained LMs. Our approach basically relies on the similarity of contextualised represen-tations, which amounts to word usage similarity (Usim) estimation, a classical task in lexical semantics (Erk et al., 2009;Huang et al., 2012;Erk et al., 2013). The Usim task precisely involves predicting the similarity of word instances in context without use of sense annotations. BERT has been shown to be particularly good at this task (Garí Soler et al., 2019;Pilehvar and Camacho-Collados, 2019). Our experiments allow to explore and understand what this ability is due to.

Dataset Creation
We build our English dataset using SemCor 3.0 (Miller et al., 1993), a corpus manually annotated with WordNet senses (Fellbaum, 1998). It is important to note that we do not use the annotations for training or evaluating any of the models. These only serve to control the composition of the sentence pools that are used for generating contextualised representations, and to analyse the results. We form sentence pools for monosemous (mono) and polysemous (poly) words that occur at least ten times in SemCor. 3 For each mono word, we randomly sample ten of its instances in the corpus. For each poly word, we form three sentence pools of size ten reflecting different sense distributions: • Balanced (poly-bal). We sample a sentence for each sense of the word in SemCor until a pool of ten sentences is formed.
• Random (poly-rand). We randomly sample ten poly word instances from SemCor. We expect this pool to be highly biased towards a specific sense due to the skewed frequency distribution of word senses (Kilgarriff, 2004;McCarthy et al., 2004). This con-figuration is closer to the expected natural occurrence of senses in a corpus, it thus serves to estimate the behaviour of the models in a real-world setting.
• Same sense (poly-same). We sample ten sentences illustrating only one sense of the poly word. Although the composition of this pool is similar to that of the mono pool (i.e. all instances describe the same sense) we call it poly-same because it describes one sense of a polysemous word. 4 Specifically, we want to explore whether BERT representations derived from these instances can serve to distinguish mono from poly words.
The controlled composition of the poly sentence pools allows us to investigate the behaviour of the models when they are exposed to instances of polysemous words describing the same or different senses. There are 1,765 poly words in Sem-Cor with at least 10 sentences available. 5 We randomly subsample 418 from these in order to balance the mono and poly classes. Our English dataset is composed of 836 mono and poly words, and their instances in 8,195 unique sentences.  (Navigli and Ponzetto, 2012). 6 We extract sentences from the high precision version 7 of Eurosense, and create sentence pools in the same way as in English, balancing the number of monosemous and polysemous words (418). We determine the number of senses for a word as the number of its Babelnet senses that are mapped to a WordNet sense. 8

Contextualised Word Representations
We experiment with representations generated by three English models: BERT (Devlin et al.,4 The polysemous words are the same as in poly-bal and poly-rand. 5 We use sentences of up to 100 words. 6 BabelNet is a multilingual semantic network built from multiple lexicographic and encyclopedic resources, such as WordNet and Wikipedia. 7 The high coverage version of Eurosense is larger than the high precision one, but disambiguation is less accurate. 8 This filtering serves to exclude BabelNet senses that correspond to named entities and are not useful for our purposes (such as movie or album titles), and to run these experiments under similar conditions across languages. 2019) 9 , ELMo (Peters et al., 2018) and con-text2vec (Melamud et al., 2016). BERT is a Transformer architecture (Vaswani et al., 2017) that is jointly trained for a masked LM and a next sentence prediction task. Masked LM inolves a Cloze-style task, where the model needs to guess randomly masked words by jointly conditioning on their left and right context. We use the bert-base-uncased and bert-base-cased models, pre-trained on the BooksCorpus (Zhu et al., 2015) and English Wikipedia. ELMo is a bi-LSTM LM trained on Wikipedia and news crawl data from WMT 2008-2012. We use 1024-d representations from the 5.5B model. 10 Context2vec is a neural model based on word2vec's CBOW architecture (Mikolov et al., 2013) which learns embeddings of wide sentential contexts using a bi-LSTM. The model produces representations for words and their context. We use the context representations from a 600-d context2vec model pre-trained on the ukWaC corpus (Baroni et al., 2009). 11 For French, Spanish and Greek, we use BERT models specifically trained for each language: Flaubert (flaubert_base_uncased) , BETO (bert-base-spanishwwm-uncased) (Cañete et al., 2020), and Greek BERT (bert-base-greek-uncased-v1) (Koutsikakis et al., 2020). We also use the bert-base-multilingual-cased model for each of the four languages. mBERT was trained on Wikipedia data of 104 languages. 12 All BERT models generate 768-d representations.

The Self-Similarity Metric
All models produce representations that describe word meaning in specific contexts of use. For each instance i of a target word w in a sentence, we extract its representation from: (i) each of the 12 layers of a BERT model; 13 (ii) each of the three context2vec 12 The mBERT model developers recommend using the cased version of the model rather than the uncased one, especially for languages with non-Latin alphabets, because it fixes normalisation issues. More details about this model can be found here: https://github.com/google-research/bert/ blob/master/multilingual.md. 13 We also tried different combinations of the last four layers, but this did not improve the results. When a word is split ELMo layers; (iii) context2vec. We calculate selfsimilarity (Self Sim) (Ethayarajh, 2019) for w in a sentence pool p and a layer l, by taking the average of the pairwise cosine similarities of the representations of its instances in l: (1) In formula 1, |I| is the number of instances for w (ten in our experiments); x wli and x wlj are the representations for instances i and j of w in layer l. The Self Sim score is in the range [-1, 1]. We report the average Self Sim for all w's in a pool p. We expect it to be higher for monosemous words and words with low polysemy than for highly polysemous words. We also expect the poly-same pool to have a higher average Self Sim score than the other poly pools which contain instances of different senses.
Contextualisation has a strong impact on Self Sim since it introduces variation in the token-level representations, making them more dissimilar. The Self Sim value for a word would be 1 with non-contextualised (or static) embeddings, as all its instances would be assigned the same vector. In contextual models, Self Sim is lower in layers where the impact of the context is stronger (Ethayarajh, 2019). It is, however, important to note that contextualisation in BERT models is not monotonic, as shown by previous studies of the models' internal workings (Voita et al., 2019a;Ethayarajh, 2019). Our experiments presented in the next section provide additional evidence in this respect. Figure 2 shows the average Self Sim value obtained for each sentence pool with representations produced by BERT models. The thin lines in the first plot illustrate the average Self Sim score calculated for mono and poly words using representations from each layer of the uncased English BERT model. We observe a clear distinction of words according to their polysemy: Self Sim is higher for mono than for poly words across all layers and sentence pools. BERT establishes a clear distinction even between the mono and into multiple wordpieces (WPs), we obtain its representation by averaging the WPs.

Mono-Poly in English
poly-same pools, which contain instances of only one sense. This distinction is important; it suggests that BERT encodes information about a word's monosemous or polysemous nature regardless of the sentences that are used to derive the contextualised representations. Specifically, BERT produces less similar representations for word instances in the poly-same pool compared to mono, reflecting that poly words can have different meanings.
We also observe a clear ordering of the three poly sentence pools: average Self Sim is higher in poly-same, which only contains instances of one sense, followed by mid-range values in poly-rand, and gets its lowest values in the balanced setting (poly-bal). This is noteworthy given that poly-rand contains a mix of senses but with a stronger representation of w's most frequent sense than poly-bal (71% vs. 47%). 14 Our results demonstrate that BERT representations encode two types of lexical semantic knowledge: information about the polysemous nature of words acquired through pre-training (as reflected in the distinction between mono and poly-same), and information from the particular instances of a word used to create the contextualised representations (as shown by the finergrained distinctions between different poly settings). BERT's knowledge about polysemy can be due to differences in the types of context where words of different polysemy levels occur. We expect poly words to be seen in more varied contexts than mono words, reflecting their different senses. BERT encodes this variation with the LM objective through exposure to large amounts of data, and this is reflected in the representations. The same ordering pattern is observed with mBERT (lower part of Figure 2) and with ELMo (Figure 3 (a)). With context2vec, average Self Sim in mono is 0.40, 0.38 in poly-same, 0.37 in poly-rand, and 0.35 in poly-bal. This suggests that these models also have some inherent knowledge about lexical polysemy, but differences are less clearly marked than in BERT.
Using the cased model leads to an overall increase in Self Sim and to smaller differences between bands, as shown by the thick lines in the first plot of Figure 2. Our explanation for the lower distinction ability of the bert-base-cased model is that it encodes sparser information about  words than the uncased model. It was trained on a more diverse set of strings, so many WPs are present in both their capitalised and noncapitalised form in the vocabulary. In spite of that, it has a smaller vocabulary size (29K WPs) than the uncased model (30.5K). Also, a higher number of WPs correspond to word parts than in the uncased model (6,478 vs 5,829).
We test the statistical significance of the mono/poly-rand distinction using unpaired two-samples t-tests when the normality assumption is met (as determined with Shapiro Wilk's tests). Otherwise, we run a Mann Whitney U test, the non-parametrical alternative of this t-test. In order to lower the probability of type I errors (false positives) that increases when performing multiple tests, we correct p-values using the Benjamini-Hochberg False Discovery Rate (FDR) adjustment (Benjamini and Hochberg, 1995). Our results show that differences are significant across all embedding types and layers (α = 0.01).
The decreasing trend in Self Sim observed for BERT in Figure 2, and the peak in layer 11, confirm the phases of context encoding and token reconstruction observed by Voita et al. (2019a). 15 In earlier layers, context variation makes representations more dissimilar and Self Sim decreases. In the last layers, information about the input token is recovered for LM prediction and similarity scores are boosted. Our results show clear distinctions across all BERT and ELMo layers. This suggests that lexical information is spread throughout the layers of the models, and contributes new evidence to the discussion on the localisation of semantic information inside the models (Rogers et al., 2020;Vulić et al., 2020).

Mono-Poly in Other Languages
The top row of Figure 2 shows the average Self Sim obtained for French, Spanish and Greek words using monolingual models.
Flaubert, BETO and Greek BERT representations clearly distinguish mono and poly words, but average Self Sim values for different poly pools are much closer than in English. BETO seems to capture these fine-grained distinctions slightly better than the French and Greek models. The second row of the Figure shows results obtained with mBERT representations. We observe the highly similar average Self Sim values assigned to different poly pools, which show that distinction is harder than in monolingual models.  Statistical tests show that the difference between Self Sim values in mono and poly-rand is significant in all layers of BETO, Flaubert, Greek BERT, and mBERT for Spanish and French. 16 The magnitude of the difference in Greek BERT is, however, smaller compared to the other models (0.03 vs. 0.09 in BETO at the layers with the biggest difference in average Self Sim).

SelfSim-based Ranking
In this set of experiments, we explore the impact of words' degree of polysemy on the representations. We control for this factor by grouping words into three polysemy bands as in McCarthy et al. (2016), which correspond to a specific number of senses (k): low: 2 ≤ k ≤ 3, mid: 4 ≤ k ≤ 16 In mBERT for Greek, the difference is significant in ten layers. 6, high: k > 6. For English, the three bands are populated with a different number of words: low: 551, mid: 663, high: 551. In the other languages, we form bands containing 300 words each. 17 In Figure 4, we compare mono words with words in each polysemy band in terms of their average Self Sim. Values for mono words are taken from Section 3. For poly words, we use representations from the poly-rand sentence pool which better approximates natural occurrence in a corpus. For comparison, we report in Figure 5 results obtained in English using sentences from the poly-same and poly-bal pools. 18 In English, the pattern is clear in all plots: Self Sim is higher for mono than for poly words in any band, confirming that BERT is able to distinguish mono from poly words at different polysemy levels. The range of Self Sim values for a band is inversely proportional to its k: words in low get higher values than words in high. The results denote that the meaning of highly polysemous words is more variable (lower Self Sim) than the meaning of words with fewer senses. As expected, scores are higher and inter-band similarities are closer in poly-same (cf. Figure 5 (b)) compared to poly-rand and poly-bal, where distinctions are clearer. The observed differences confirm that BERT can predict the pol- Figure 6: The left plots show the similarity between random words in models for each language. Plots on the right side show the difference between the similarity of random words and Self Sim in poly-rand. ysemy level of words, even from instances describing the same sense.
We observe similar patterns with ELMo (cf. Figure 3 (b)) and context2vec representations in poly-rand, 19 but smaller absolute inter-band differences. In poly-same, both models fail to correctly order the bands. Overall, our results highlight that BERT encodes higher quality knowledge about polysemy. We test the significance of the inter-band differences detected in poly-rand using the same approach as in Section 3.4.1. These are significant in all but a few 20 layers of the models.
The bands are also correctly ranked in the other three languages but with smaller inter-band differences than in English, especially in Greek where clear distinctions are only made in a few middle layers. This variation across languages can 19 Average Self Sim values for context2vec in the poly-rand setting: low: 0.37, mid: 0.36, high: 0.36. 20 low→mid in ELMo's third layer, and mid→high in context2vec and in BERT's first layer. be explained to some extent by the quality of the automatic EuroSense annotations, which has a direct impact on the quality of the sentence pools. Results of a manual evaluation conducted by Delli Bovi et al. (2017) showed that WSD precision is ten points higher in English (81.5) and Spanish (82.5) than in French (71.8). The Greek portion, however, has not been evaluated.
Plots in the second row of Figure 4 show results obtained using mBERT. Similarly to the previous experiment (Section 3.4), mBERT overall makes less clear distinctions than the monolingual models. The low and mid bands often get similar Self Sim values, which are close to mono in French and Greek. Still, inter-band differences are significant in most layers of mBERT and the monolingual French, Spanish and Greek models. 21

Anisotropy Analysis
In order to better understand the reasons behind the smaller inter-band differences observed with mBERT, we conduct an additional analysis of the models' anisotropy. We create 2,183 random word pairs from the English mono, low, mid and high bands, and 1,318 pairs in each of the other languages. 22 We calculate the cosine similarity between two random instances of the words in each pair and take the average over all pairs (RandSim). The plots in the left column of Figure 6 show the results. We observe a clear difference in the scores obtained by monolingual models (solid lines) and mBERT (dashed lines). Clearly, mBERT assigns higher similarities to random words, an indication that its semantic space is more anisotropic than the one built by monolingual models. High anisotropy means that representations occupy a narrow cone in the vector space, which results in lower quality similarity estimates and in the model's limited potential to establish clear semantic distinctions.
We also compare RandSim to the average Self Sim obtained for poly words in the poly-rand sentence pool (cf. Section 3.1). In a quality semantic space, we would expect Self Sim (between same word instances) to be much higher than RandSim. The right column of Figure 6 shows the difference between these two scores. diff in a layer l is calculated as in Equation 2: diff l = Avg Self Sim l (poly−rand) − RandSim l (2) We observe that the difference is smaller in the space built by mBERT, which is more anisotropic than the space built by monolingual models. This is particularly obvious in the upper layers of the model. This result confirms the lower quality of mBERT's semantic space compared to monolingual models.
Finally, we believe that another factor behind the worse mBERT performance is that the multilingual WP vocabulary is mostly English-driven, resulting in arbitrary partitionings of words in the other languages. This word splitting procedure must have an impact on the quality of the lexical information in mBERT representations.

Analysis by Frequency and PoS
Given the strong correlation between word frequency and number of senses (Zipf, 1945), we explore the impact of frequency on BERT representations. Our goal is to determine the extent to which it influences the good mono/poly detection results obtained in Sections 3.4 and 4.1.

Dataset Composition
We perform this analysis in English using frequency information from Google Ngrams (Brants and Franz, 2006). For French, Spanish and Greek, we use frequency counts gathered from the OS-CAR corpus (Suárez et al., 2019). We split the words into four ranges (F ) corresponding to the quartiles of frequencies in each dataset. Each range f in F contains the same number of words. We provide detailed information about the composition of the English dataset in Figure 7. 23 Figure 7(a) shows that mono words are much less frequent than poly words. Figure 7(b) shows the distribution of different PoS categories in each band. Nouns are the prevalent category in all bands and verbs are less present among mono words (10.8%), as expected. Finally, adverbs are hardly represented in the high polysemy band (1.2% of all words).

Self-Sim by Frequency Range and PoS
We examine the average BERT Self Sim per frequency range in poly-rand. Due to space constraints, we only report detailed results for the English BERT model in Figure 8 (plot (a)). The clear ordering by range suggests that BERT can successfully distinguish words by their frequency, especially in the last layers. Plot (b) in Figure 8 shows the average Self Sim for words of each PoS category. Verbs have the lowest Self Sim which is not surprising given that they are highly polysemous (as shown in Figure 7(b)). We observe the same trend for monolingual models in the other three languages.

Controlling for Frequency and PoS
We conduct an additional experiment where we control for the composition of the poly bands Figure 9: Average Self Sim inside the poly bands balanced for frequency (FREQ-bal) and part of speech (POSbal). Self Sim is calculated using representations generated by monolingual BERT models from sentences in each language-specific pool. We do not balance the Greek dataset for PoS because it only contains nouns.
in terms of grammatical category and word frequency. We call these two settings POS-bal and FREQ-bal. We define n pos , the smallest number of words of a specific PoS that can be found in a band. We form the POS-bal bands by subsampling from each band the same number of words (n pos ) of that PoS. For example, all POS-bal bands have n n nouns and n v verbs. We follow a similar procedure to balance the bands by frequency in the FREQ-bal setting. In this case, n f is the minimum number of words of a specific frequency range f that can be found in a band. We form the FREQ-bal dataset by subsampling from each band the same number of words (n f ) of a given range f in F . Table 2 shows the distribution of words per PoS and frequency range in the POS-bal and FREQbal settings for each language. The table reads as follows: the English POS-bal bands contain 198 nouns, 45 verbs, 64 adjectives and 7 adverbs; similarly for the other two languages. Greek is not included in this POS-based analysis because all sense-annotated Greek words in EuroSense are nouns. In FREQ-bal, each English band contains 40 words that occur less than 7.1M times in Google Ngrams, and so on and so forth.
We examine the average Self Sim values obtained for words in each band in poly-rand. Figure 9 shows the results for monolingual models. We observe that the mono and poly words in the POS-bal and FREQ-bal bands are ranked similarly to Figure 4. This shows that BERT's polysemy predictions do not rely on frequency  or part of speech. The only exception is Greek BERT which cannot establish correct inter-band distinctions when the influence of frequency is neutralised in the FREQ-bal setting. A general observation that applies to all models is that although inter-band distinctions become less clear, the ordering of the bands is preserved. We observe the same trend with ELMo and context2vec. Statistical tests show that all inter-band distinctions established by English BERT are still signif-icant in most layers of the model. 24 This is not the case for ELMo and context2vec, which can distinguish between mono and poly words but fail to establish significant distinctions between polysemy bands in the balanced settings. For French and Spanish, the statistical analysis shows that all distinctions in POS-bal are significant in at least one layer of the models. The same applies to the mono→poly distinction in FREQ-bal but finergrained distinctions disappear. 25

Classification by Polysemy Level
Our finding that word instance similarity differs across polysemy bands suggests that this feature can be useful for classification. In this Section, we probe the representations for polysemy using a classification experiment where we test their ability to guess whether a word is polysemous, and which poly band it falls in. We use the poly-rand sentence pools and a standard train/dev/test split (70/15/15%) of the data. For the mono/poly distinction (i.e. the data used in Section 3), this results in 584/126/126 words per set in each language. To guarantee a fair evaluation, we make sure there is no overlap between the lemmas in the three sets. We use two types of features: (i) the average Self Sim for a word; (ii) all pairwise cosine similarities collected for its instances, which results in 45 features per word (pairCos). We train a binary logistic regression classifier for each type of representation and feature.
As explained in Section 4, the three poly bands (low, mid and high) and mono contain a different number of words. For classification into polysemy bands, we balance each class by randomly subsampling words from each band. In total, we use 1,168 words for training, 252 for development and 252 for testing (70/15/15%) in English. In the other languages, we use a split of 840/180/180 words. We train multi-class logistic regression classifiers with the two types of features, Self Sim and pairCos. We compare the results of the classifiers to a baseline that predicts always the same class, and to a frequencybased classifier which only uses the words' log frequency in Google Ngrams, or in the OSCAR corpus, as a feature. 24 Note that the sample size in this analysis is smaller compared to that used in Sections 3.4 and 4.1.
25 With a few exceptions: mono→low and mid→high are significant in all BETO layers.  Table 3: Accuracy of binary (mono/poly) and multiclass (poly bands) classifiers using Self Sim and pairCos features on the test sets. Comparison to a baseline that predicts always the same class and a classifier that only uses log frequency as feature. Subscripts denote the layers used. Table 3 presents classification accuracy on the test set. We report results obtained with the best layer for each representation type and feature as determined on the development sets. In English, best accuracy is obtained by BERT in both the binary (0.79) and multiclass settings (0.49), followed by mBERT (0.77 and 0.46). Despite its simplicity, the frequency-based classifier obtains better results than context2vec and ELMo, and performs on par with mBERT in the binary setting. This shows that frequency information is highly relevant for the mono-poly distinction. All classifiers outperform the same class baseline. These results are very encouraging, showing that BERT embeddings can be used to determine whether a word has multiple meanings, and provide a rough indication of its polysemy level. Results in the other three languages are not as high as those obtained in English, but most models give higher results than the frequency-based classifier. 26

Word Sense Clusterability
We have shown that representations from pretrained LMs encode rich information about words' degree of polysemy. They can successfully distinguish mono from poly lemmas, and predict the polysemy level of words. Our previous experi-ments involved a set of controlled settings representing different sense distributions and polysemy levels. In this Section, we explore whether these representations can also point to the clusterability of poly words in an uncontrolled setting.

Task Definition
Instances of some poly words are easier to group into interpretable clusters than others. This is, for example, a simple task for the ambiguous noun rock which can express two clearly separate senses (STONE and MUSIC), but harder for book which might refer to the CONTENT or OBJECT senses of the word (e.g. I read a book vs. I bought a book).
In what follows, we test the ability of contextualised representations to estimate how easy this task is for a specific word, i.e. its partitionability into senses.
Following McCarthy et al. (2016), we use the clusterability metrics proposed by Ackerman and Ben-David (2009)  That's not a very high bar, present no overlap in their in-context substitutes: {snack, biscuit, block, slab} vs. {pole, marker, hurdle, barrier, level, ob-struction}. Semantically related instances, on the contrary, share a different number of substitutes depending on their proximity. The need for manual annotations, however, constrains the method's applicability to specific datasets.
We propose to extend and scale up the Mc-Carthy et al. (2016) clusterability approach using contextualised representations, in order to make it applicable to a larger vocabulary. These experiments are carried out in English due to the lack of evaluation data in other languages.

Data
We run our experiments on the usage similarity (Usim) dataset (Erk et al., 2013) for comparison with previous work. Usim contains ten instances for 56 target words of different PoS from the Se-mEval Lexical Substitution dataset (McCarthy and Navigli, 2007). Word instances are manually annotated with pairwise similarity scores on a scale from 1 (completely different) to 5 (same meaning).
We represent target word instances in Usim in two ways: using contextualised represen-tations generated by BERT, context2vec and ELMo (BERT-REP, c2v-REP, ELMo-REP); 27 using substitute-based representations with automatically generated substitutes. The substitutebased approach allows for a direct comparison with the method of McCarthy et al. (2016). They represent each instance i of a word w in Usim as a vector i, where each substitute s assigned to w over all its instances (i ∈ I) becomes a dimension (d s ). For a given i, the value for each d s is the number of annotators who proposed substitute s. d s contains a zero entry if s was not proposed for i. We refer to this type of representation as Gold-SUB. We generate our substitutebased representations with BERT using the simple "word similarity" approach in Zhou et al. (2019). For an instance i of word w in context C, we rank a set of candidate substitutes S = {s 1 , s 2 , ..., s n } based on the cosine similarity of the BERT representations for i and for each substitute s j ∈ S in the same context C. We use representations from the last layer of the model. As candidate substitutes, we use the unigram paraphrases of w in the Paraphrase Database (PPDB) XXL package (Ganitkevitch et al., 2013;Pavlick et al., 2015). 28 For each instance i of w, we obtain a ranking R of all substitutes in S. We remove low-quality substitutes (i.e. noisy paraphrases or substitutes referring to a different sense of w) using the filtering approach proposed by Garí Soler et al. (2019). We check each pair of substitutes in subsequent positions in R, starting from the top; if a pair is unrelated in PPDB, all substitutes from that position onwards are discarded. The idea is that good quality substitutes should be both high-ranked and semantically related. We build vectors as in Mc-Carthy et al. (2016), using the cosine similarity assigned by BERT to each substitute as a value. We call this representation BERT-SUB.

Sense Clustering
The clusterability metrics that we use are metrics initially proposed for estimating the quality of the optimal clustering that can be obtained from a dataset; the better the quality of this clustering, the higher the clusterability of the dataset it is derived Gold Metric BERT-REP c2v-REP ELMo-REP BERT-SUB Gold-SUB BERT-AGG  Table 4: Spearman's ρ correlation between automatic metrics and gold standard clusterability estimates. Significant correlations (where the null hypothesis ρ = 0 is rejected with α < 0.05) are marked with *. The arrows indicate the expected direction of correlation for each metric. Subscripts indicate the layer that achieved best performance. The two strongest correlations obtained with each gold standard measure are in boldface.
from (Ackerman and Ben-David, 2009). In order to estimate the clusterability of a word w, we thus need to first cluster its instances in the data. We use the k-means algorithm which requires the number of senses for a lemma. This is, of course, different for every lemma in our dataset. We define the optimal number of clusters k for a lemma in a data-driven manner using the Silhouette coefficient (SIL) (Rousseeuw, 1987), without recourse to external resources. 29 For a data point i, SIL compares the intra-cluster distance (i.e. the average distance from i to every other data point in the same cluster) with the average distance of i to all points in its nearest cluster. The SIL value for a clustering is obtained by averaging SIL for all data points, and it ranges from -1 to 1. We cluster each type of representation for w using k-means with a range of k values (2 ≤ k ≤ 10), and retain the k of the clustering with the highest mean SIL. Additionally, since BERT representations' cosine similarity correlates well with usage similarity (Garí Soler et al., 2019), we experiment with Agglomerative Clustering with average linkage directly on the cosine distance matrix obtained with BERT representations (BERT-AGG). For comparison, we also use Agglomerative Clustering on the gold usage similarity scores from Usim, transformed into distances (Gold-AGG).

Clusterability Metrics
We use in our experiments the two best performing metrics from McCarthy et al. (2016): Variance Ratio (VR) (Zhang, 2001) and Separability (SEP) (Ostrovsky et al., 2012). VR calculates the ratio of the within-and between-cluster variance for a given clustering solution. SEP measures the difference in loss between two clusterings with k−1 and 29 We do not use McCarthy et al.'s graph-based approach because it is not compatible with all our representation types. k clusters and its range is [0,1). We use k-means' sum of squared distances of data points to their closest cluster center as the loss. Details about these two metrics are given in Appendix A. 30 We also experiment with SIL as a clusterability metric, as it can assess cluster validity. For VR and SIL, a higher value indicates higher clusterability. The inverse applies to SEP. We calculate Spearman's ρ correlation between the results of each clusterability metric and two gold standard measures derived from Usim: Uiaa and Umid. Uiaa is the inter-annotator agreement for a lemma in terms of average pairwise Spearman's correlation between annotators' judgments. Higher Uiaa values indicate higher clusterability, meaning that sense partitions are clearer and easier to agree upon. Umid is the proportion of midrange judgments (between 2 and 4) assigned by annotators to all instances of a target word. It indicates how often usages do not have identical (5) or completely different (1) meaning. Therefore, higher Umid values indicate lower clusterability.

Results and Discussion
The clusterability results are given in Table 4. Agglomerative Clustering on the gold Usim similarity scores (Gold-AGG) gives best results on the Uiaa evaluation in combination with the SIL clusterability metric (ρ = 0.80). This is unsurprising, since Uiaa and Umid are derived from the same Usim scores. From our automatically generated representations, the strongest correlation with Uiaa (0.69) is obtained with BERT-AGG and the SIL clusterability metric. The SIL metric also works well with BERT-REP achieving the strongest correlation with Umid (-0.46). It consti- Figure 10: Spearman's ρ correlations between the gold standard Uiaa and Umid scores, and clusterability estimates obtained using Agglomerative Clustering on a cosine distance matrix of BERT representations. tutes, thus, a good alternative to the SEP and VR metrics used in previous studies when combined with BERT representations.
Interestingly, the correlations obtained using raw BERT contextualised representations are much higher than the ones observed with Mc-Carthy et al. (2016)'s representations which rely on manual substitutes (Gold-SUB). These were in the range of 0.20-0.34 for Uiaa and 0.16-0.38 for Umid (in absolute value). The results demonstrate that BERT representations offer good estimates of the partitionability of words into senses, improving over substitute annotations. As expected, the substitution-based approach performs better with clean manual substitutes (Gold-SUB) than with automatically generated ones (BERT-SUB).
We present a per layer analysis of the correlations obtained with the best performing BERT representations (BERT-AGG) and the SIL metric in Figure 10. We report the absolute values of the correlation coefficient for a more straightforward comparison. For Uiaa, the higher layers of the model make the best predictions: correlations increase monotonically up to layer 10, and then they show a slight decrease. Umid prediction shows a more irregular pattern: it peaks at layers 3 and 8, and decreases again in the last layers.

Conclusion
We have shown that contextualised BERT representations encode rich information about lexical polysemy. Our experimental results suggest that this high quality knowledge about words, which allows BERT to detect polysemy in different configurations and across all layers, is acquired during pre-training. Our findings hold for the English BERT as well as for BERT models in other lan-guages, as shown by our experiments on French, Spanish and Greek, and to a lesser extent for multilingual BERT. Moreover, English BERT representations can be used to obtain a good estimation of a word's partitionability into senses. These results open up new avenues for research in multilingual semantic analysis, and we can consider various theoretical and application-related extensions for this work.
The polysemy and sense-related knowledge revealed by the models can serve to develop novel methodologies for improved cross-lingual alignment of embedding spaces and cross-lingual transfer, pointing to more polysemous (or less clusterable) words for which transfer might be harder. Predicting the polysemy level of words can also be useful for determining the context needed for acquiring representations that properly reflect the meaning of word instances in running text. From a more theoretical standpoint, we expect this work to be useful for studies on the organisation of the semantic space in different languages and on lexical semantic change.