Isotropy in the Contextual Embedding Space: Clusters and Manifolds

1 INTRODUCTION . The polysemous English word “ bank ” has two common senses : 1. the money sense , a place that people save or borrow money ; 2. the river sense , a slope of earth that prevents the flooding . In modern usage , the two senses are very different from one another , though interestingly , both senses share similar etymologies ( and both can be traced back to the same word in Proto-Germanic ) . In the static embedding , multiple instances of the same word ( e.g . “ bank ” ) will be represented using the same vector . On the contrary , the contextual embedding assigns different vectors to different instances of the same word , depending on the context . Historically , static embedding models like Word2vec ( Mikolov et al. , 2013b ) and GloVe ( Pennington et al. , 2014 ) , predated contextual embedding models such as ELMo ( Peters et al. , 2018 ) , GPT ( Radford et al. , 2018 ) , BERT ( Devlin et al. , 2018 ) and ERNIE ( Sun et al. , 2019 ) . Much of the literature on language modeling has moved to contextual embeddings recently , largely because of their superior performance on the downstreaming tasks . 1.1 RELATED WORK . The static embeddings are often found to be easier to interpret . For example , the Word2Vec and GloVe papers discuss adding and subtracting vectors , such as : vec ( king ) - vec ( man ) + vec ( women ) = vec ( queen ) . Inspired by this relationship , researchers started to explore geometric properties of static embedding spaces . For example , Mu & Viswanath ( 2018 ) proposed a very counter-intuitive method that removes the top principle components ( the dominating directions in the transformed embedding space ) , which surprisingly improved the word representations . Rather than completely discarding the principle components , Liu et al . ( 2019 ) proposed to use a technique called Conceptor Negation , to softly suppress transformed dimensions with larger variances . Both approaches , simply removing certain principle components as well as Conceptor Negation , produce significant improvements over vanilla embeddings obtained by static language models . In Huang et al . ( 2020 ) , the authors studied how to effectively transform static word embeddings from one language to another . Unfortunately , the strong illustrative representation like the king-queen example above , is no longer obvious in a general contextual embedding space . Arguing that syntax structure indeed exists in the contextual embeddings , Hewitt & Manning ( 2019 ) proposed a structural probe to identify the syntax trees buried in the space , and found the evidence of implicit syntax tree in BERT and ELMo . The advantage of contextual embedding over the static counterpart , mainly come from its capability to assign different vectors to the same word , depending on the word sense in the context . Researchers in ( Reif et al. , 2019 ) found such a geometric representation of word senses in the BERT model . These papers reveal the existence of linguistic features embedded implicitly in the contextual vector spaces . The geometric properties of contextual embedding space are also investigated and compared with the static embedding space . Mimno & Thompson ( 2017 ) found anisotropy when negative sampling is used . In ( Ethayarajh , 2019 ) , the authors characterize how vectors are distributed in the contextual space . They found that most vectors occupy in a relatively narrow cone in the space . Pairs of vectors within this cone have large cosines . This phenomenon can be found in most state-of-the-art contextual embedding models . In ( Gao et al. , 2019 ) , the authors named this phenomenon ” representation degeneration ” , and attempted to mitigate the problem by introducing a regularization term that minimizes cosine similarities between vectors . In a very recent work , Demeter et al . ( 2020 ) suggest there is a structure weakness in the space that leads to bias when using soft-max , as is common with deep language models . 1.2 MOTIVATION AND CONTRIBUTIONS . Isotropy often makes the space more effectively utilized and more robust to perturbations ( no extreme directions that lead to high condition number ) . It is counter-intuitive and not clear why those contextual embedding models perform remarkably well on many tasks given their anisotropic embeddings bring all the vectors close together , hard to distinguish one from another . On one hand , it is widely believed that contextual embeddings encode the relevant linguistic information ( e.g . ( Reif et al. , 2019 ) ) , but on the other hand , it is also widely believed that the contextual space is anisotropic that representations become degenerated ( e.g . ( Mimno & Thompson , 2017 ) , ( Gao et al. , 2019 ) , ( Ethayarajh , 2019 ) ) . These motivate us to find a reasonable understanding that bridges this gap . This paper is similar in spirit to ( Mu & Viswanath , 2018 ) , but different in three aspects . First , we generalize their work on traditional static embeddings to more modern contextual embeddings . Second , we introduce clustering methods to isolate the space , whereas they used PCA to remove dominant dimensions ( that tend to dominate the variance ) . Finally , we identify low dimensional manifolds in the space , and introduce an alternative approach ( LID ) to characterize local subspaces . Key Contributions : This paper takes a deeper look into the contextual embedding spaces of popular pre-trained models . It identifies the following facts that were misunderstood or not known before : 1 ) We find isotropy within clusters in the contextual embedding space , in contrast to previous reports of anisotropy ( caused by misleading isolated clusters ) . We introduce clustering and center shifting to reveal the isotropy , and show more consistent layer-wise behavior across models . 2 ) We find a Swiss-Roll manifold in GPT/GPT2 embeddings , but not in BERT/DistilBERT embeddings . The manifold is related to word frequency , suggesting a difference in how models evolve as they see more data . We use approximate Local Intrinsic Dimension ( LID ) to characterize the manifold , and find contextual embedding models , including all BERT , GPT families and ELMo , often have small LIDs . The small LIDs can be viewed as the local anisotropy of the space . The code for this paper could be found at https : //github.com/TideDancer/IsotropyContxt . 2 ANALYSIS SETTINGS . 2.1 MODELS AND DATASETS . In this paper , we consider popular pre-trained contextual embedding models , including BERT , DistilBERT ( Sanh et al. , 2019 ) ( or denoted as D-BERT in the rest of the paper ) , GPT , GPT2 ( Radford et al. , 2019 ) and ELMo . For the BERT and GPT families , we perform our evaluations on the pretrained uncased base models from Huggingface ( https : //huggingface.co/transformers/index.html # ) . The pre-trained ELMo model is from AllenNLP ( https : //docs.allennlp.org/v1.0.0/ ) . BERT and DBERT are non-causal models because of their attention mechanism , where tokens can attend to any token in the input , regardless of their relative positions . In contrast , GPT and GPT2 are causal models because attention is limited to the tokens previously seen in the input . Different models achieve contextual embedding in different ways . For instance , BERT adds positional embeddings to the token embeddings , while ELMo performs vector concatenation . Most models start with an initial layer that maps token ids to vectors . This paper is not concerned with that lookup table layer , and only focuses on the layers after that . The base BERT , GPT and GPT2 models have 12 layers of interest , indexed from 0 to 11 , while D-BERT has 6 layers and ELMo has two . We use Penn Tree Bank ( PTB ) ( Marcus et al. , 1993 ) and WikiText-2 ( Merity et al. , 2016 ) datasets . The PTB has 0.88 million words and WikiText-2 has 2 million . Both of them are the standard datasets for language models . In the rest of the paper , we report on PTB since we see similar results with both datasets . Details on WikiText-2 analysis could be found in Appendix . 2.2 NOTATION . For each position in a corpus , we have a word . Words are converted into tokens , using the appropriate tokenizer for the model . Tokenizers could split some words into subwords , therefore , the number of obtained tokens ( denoted as n ) could be more than number of words in the corpus . PTB , for example , contains 0.88 million words , but has n = 1.2 million tokens , when processed by BERT ’ s tokenizer . Let V be the vocabulary , a set of distinct tokens . For any element in the vocabulary V , we call it a type . For example , BERT has a vocabulary of roughly 30 , 000 types . We may mix using “ word ” and “ type ” for ease of reading . We denote the i-th type in V as ti . Let Φ ( ti ) = { φ1 ( ti ) , φ2 ( ti ) , . . . } be the set of all embedding instances of ti ( note that different contexts in the corpus yield different embeddings of ti ) . By construction , ∑ t |Φ ( t ) | = n. We define the inter-type cosine similarity as Sinter , Ei 6=j [ cos ( φ ( ti ) , φ ( tj ) ) ] ( 1 ) where φ ( ti ) is one random sample from Φ ( ti ) , and the same for φ ( tj ) ∈ Φ ( tj ) . The expectation is taken over all pairs of different types . Similarly , we define the intra-type cosine similarity as Sintra , Ei [ Ek 6=l [ cos ( φk ( ti ) , φl ( ti ) ) ] ] ( 2 ) where the inner expectation is over different embeddings φ ( ti ) for the same type ti , and the outer expectation is over all types . Both Sinter and Sintra take values between −1 and 1 . Note that for i.i.d . Gaussian random samples x , y , the expected cosine similarity E [ cos ( x , y ) ] = 0 . A cosine value closer to 0 often indicates strong isotropy . Clearly , the inter-type metric describes the similarity between different types , where the intra-type one measures similarity between same type ’ s embedding instances . Our definitions of Sinter and Sintra are similar to the measures used in Ethayarajh ( 2019 ) , but at the corpus level . Note that some types are more frequent than others , especially under a Zipfian distribution ( Piantadosi , 2014 ) , and therefore , the size of Φ ( t ) varies dramatically with the frequency of type t . 2.3 AN INITIAL LOOK AT ANISOTROPY . Inspired by Ethayarajh ( 2019 ) , we follow their procedure and take a first look at the anisotropy identified by Mimno & Thompson ( 2017 ) and Ethayarajh ( 2019 ) , in the contextual embedding space . Figure 1 shows strong anisotropy effects in a number of models . These findings are consistent with Ethayarajh ( 2019 ) , though we use slightly different metrics . The plots show expected cosine ( Sinter and Sintra ) as a function of layer . For efficiency , we approximate Sintra by imposing a limit of 1,000 samples for frequent types , t , if |Φ ( t ) | > 1000 . From the figure we can see the following : • Both Sinter and Sintra are high ( 0 ) across almost all the layers and all the models . In particular , the same as reported in Ethayarajh ( 2019 ) , GPT2 is relatively more anisotropic . • Sinter tends to increase with layer , in contrast with Sintra which in general decreases but with fluctuations . This means that embeddings for different types are moving closer to one another at deeper layers , while embeddings for the same type ’ s instances are spreading away . • The last layer is often special . Note that the last layer has smaller cosines than the second last in most cases , with the notable exception of GPT2 . In summary , we observe large cosines ( across layers/models ) , especially for the GPT2 model . When cosines are close to 1 , embeddings lie in a subspace defined by a very narrow cone ( Ethayarajh , 2019 ) . One might expect embeddings to be more effective if they took advantage of a larger subspace . Are these models missing an opportunity to have the benefits from isotropy ( Mu & Viswanath , 2018 ) ? We answer this question in the following sections .

Summary: The authors investigate the token embedding space of a variety of contextual embedding models for natural language. Using techniques based on nearest neighbors, clustering, and PCA, they report a variety of results on local dimensionality / anisotropy / clustering / manifold structure in these embedding models which are of general interest to scientists and practitioners hoping to understand these models. These include findings of (local) isotropy in the embeddings when appropriately clustered and shifted, and an apparent manifold structure in the GPT models.