using a distance-dependent Chinese restaurant process

Kairit Sirts

Institute of Cybernetics at

Tallinn University of Technology

sirts@ioc.ee

&Jacob Eisenstein

School of Interactive Computing

Georgia Institute of Technology

jacobe@gatech.edu

Micha Elsner

Department of Linguistics

The Ohio State University

melsner0@gmail.com

&Sharon Goldwater

ILCC, School of Informatics

University of Edinburgh

sgwater@inf.ed.ac.uk

Institute of Cybernetics at

Tallinn University of Technology

sirts@ioc.ee

&Jacob Eisenstein

School of Interactive Computing

Georgia Institute of Technology

jacobe@gatech.edu

Micha Elsner

Department of Linguistics

The Ohio State University

melsner0@gmail.com

&Sharon Goldwater

ILCC, School of Informatics

University of Edinburgh

sgwater@inf.ed.ac.uk

We present a new approach to inducing the syntactic categories of words, combining their distributional and morphological properties in a joint nonparametric Bayesian model based on the distance-dependent Chinese Restaurant Process. The prior distribution over word clusterings uses a log-linear model of morphological similarity; the likelihood function is the probability of generating vector word embeddings. The weights of the morphology model are learned jointly while inducing part-of-speech clusters, encouraging them to cohere with the distributional features. The resulting algorithm outperforms competitive alternatives on English POS induction.

The morphosyntactic function of words is reflected in two ways: their distributional properties, and their morphological structure. Each information source has its own advantages and disadvantages. Distributional similarity varies smoothly with syntactic function, so that words with similar syntactic functions should have similar distributional properties. In contrast, there can be multiple paradigms for a single morphological inflection (such as past tense in English). But accurate computation of distributional similarity requires large amounts of data, which may not be available for rare words; morphological rules can be applied to any word regardless of how often it appears.

These observations suggest that a general approach to the induction of syntactic categories should leverage both distributional and morphological features [9, 7]. But these features are difficult to combine because of their disparate representations. Distributional information is typically represented in numerical vectors, and recent work has demonstrated the utility of continuous vector representations, or “embeddings” [16, 15, 13, 24]. In contrast, morphology is often represented in terms of sparse, discrete features (such as morphemes), or via pairwise measures such as string edit distance. Moreover, the mapping between a surface form and morphology is complex and nonlinear, so that simple metrics such as edit distance will only weakly approximate morphological similarity.

In this paper we present a new approach for inducing part-of-speech (POS) classes, combining morphological and distributional information in a non-parametric Bayesian generative model based on the *distance-dependent Chinese restaurant process* (ddCRP; Blei and Frazier, 2011).
In the ddCRP, each data point (word type) selects another point to “follow”; this chain of following links corresponds to a partition of the data points into clusters. The probability of word ${w}_{1}$ following ${w}_{2}$ depends on two factors:
1) the distributional similarity between all words in the proposed partition containing ${w}_{1}$ and ${w}_{2}$, which is encoded using a Gaussian likelihood function over the word embeddings; and
2) the morphological similarity between ${w}_{1}$ and ${w}_{2}$, which
acts as a prior distribution on the induced clustering.
We use a log-linear model to capture suffix similarities between words, and learn the feature weights by iterating between sampling and weight learning.

We apply our model to the English section of the the Multext-East corpus [11] in order to evaluate both against the coarse-grained and fine-grained tags, where the fine-grained tags encode detailed morphological classes. We find that our model effectively combines morphological features with distributional similarity, outperforming comparable alternative approaches.

Unsupervised POS tagging has a long history in NLP. This paper focuses on the POS induction problem (i.e., no tag dictionary is available), and here we limit our discussion to very recent systems. A review and comparison of older systems is provided by Christodoulopoulos et al. (2010), who found that imposing a one-tag-per-word-type constraint to reduce model flexibility tended to improve system performance; like other recent systems, we impose that constraint here.
Recent work also shows that the combination of morphological and distributional information yields the best results, especially cross-linguistically [9, 3]. Since then, most systems have incorporated morphology in some way, whether as an initial step to obtain prototypes for clusters [1], or as features in a generative model [14, 8, 21], or a representation-learning algorithm [27]. Several of these systems use a small fixed set of orthographic and/or suffix features, sometimes obtained from an unsupervised morphological segmentation system [1, 14, 8, 27]. Blunsom and Cohn’s [6] model learns an $n$-gram character model over the words in each cluster; we learn a log-linear model, which can incorporate arbitrary features. Berg-Kirkpatrick et al. (2010) also include a log-linear model of morphology in POS induction, but they use morphology in the likelihood term of a parametric sequence model, thereby encouraging all elements that share a tag to have the same morphological features. In contrast, we use *pairwise morphological similarity* as a prior in a non-parametric clustering model. This means that the membership of a word in a cluster requires only morphological similarity to some other element in the cluster, not to the cluster centroid; which may be more appropriate for languages with multiple morphological paradigms. Another difference is that our non-parametric formulation makes it unnecessary to know the number of tags in advance.

The ddCRP [5] is an extension of the CRP; like the CRP, it defines a distribution over partitions (“table assignments”) of data points (“customers”). Whereas in the regular CRP each customer chooses a table with probability proportional to the number of customers already sitting there, in the ddCRP each customer chooses another customer to follow, and sits at the same table with that customer. By identifying the connected components in this graph, the ddCRP equivalently defines a prior over clusterings.

If ${c}_{i}$ is the index of the customer followed by customer $i$, then the ddCRP prior can be written

$P({c}_{i}=j)\propto \{\begin{array}{cc}f\left({d}_{ij}\right)\hfill & \text{if}i\ne j\hfill \\ \alpha \hfill & \text{if}i=j,\hfill \end{array}$ | (1) |

where ${d}_{ij}$ is the distance between customers $i$ and $j$ and $f$ is a decay function. A ddCRP is sequential if customers can only follow previous customers, i.e., ${d}_{ij}=\mathrm{\infty}$ when $i>j$ and $f\left(\mathrm{\infty}\right)=0$. In this case, if ${d}_{ij}=1$ for all $$ then the ddCRP reduces to the CRP.

Separating the distance and decay function makes sense for “natural” distances (e.g., the number of words between word $i$ and $j$ in a document, or the time between two events), but they can also be collapsed into a single similarity function. We wish to assign higher similarities to pairs of words that share meaningful suffixes. Because we do not know which suffixes are meaningful a priori, we use a maximum entropy model whose features include all suffixes up to length three that are shared by at least one pair of words. Our prior is then:

$P({c}_{i}=j|\mathbf{w},\alpha )\propto \{\begin{array}{cc}{e}^{{\mathbf{w}}^{\text{\U0001d5b3}}\mathbf{g}\left(i,j\right)}\hfill & \text{if}i\ne j\hfill \\ \alpha \hfill & \text{if}i=j,\hfill \end{array}$ | (2) |

where ${g}_{s}\left(i,j\right)$ is 1 if suffix $s$ is shared by $i$th and $j$th words, and 0 otherwise.

We can create an infinite mixture model by combining the ddCRP prior with a likelihood function defining the probability of the data given the cluster assignments. Since we are using continuous-valued vectors (word embeddings) to represent the distributional characteristics of words, we use a multivariate Gaussian likelihood. We will marginalize over the mean $\mu $ and covariance $\mathrm{\Sigma}$ of each cluster, which in turn are drawn from Gaussian and inverse-Wishart (IW) priors respectively:

$$\mathrm{\Sigma}\sim IW({\nu}_{0},{\mathrm{\Lambda}}_{0})\mathit{\hspace{1em}\hspace{1em}}\mu \sim \mathcal{N}({\mu}_{0},{}^{\mathrm{\Sigma}}/_{{\kappa}_{0}})$$ | (3) |

The full model is then:

$P(\mathbf{X},$ | $\mathbf{c},\bm{\mu},\mathbf{\Sigma}|\mathrm{\Theta},\mathbf{w},\alpha )$ | (4) | ||

$=$ | $\prod _{k=1}^{K}}P\left({\mathrm{\Sigma}}_{k}\right|\mathrm{\Theta})p\left({\mu}_{k}\right|{\mathrm{\Sigma}}_{k},\mathrm{\Theta})$ | |||

$\times {\displaystyle \prod _{i=1}^{n}}\left(P\left({c}_{i}\right|\mathbf{w},\alpha )P\left({\mathbf{x}}_{i}\right|{\mu}_{{z}_{i}},{\mathrm{\Sigma}}_{{z}_{i}})\right),$ |

where $\mathrm{\Theta}$ are the hyperparameters for $\left(\bm{\mu},\mathbf{\Sigma}\right)$ and ${z}_{i}$ is the (implicit) cluster assignment of the $i$th word ${\mathbf{x}}_{i}$. With a CRP prior, this model would be an infinite Gaussian mixture model (IGMM; Rasmussen, 2000), and we will use the IGMM as a baseline.

The Gibbs sampler for the ddCRP integrates over
the Gaussian parameters,
sampling only follower variables. At each step, the follower link ${c}_{i}$ for a single customer $i$ is sampled, which can implicitly shift the entire block of $n$ customers $\text{fol}\left(i\right)$ who follow $i$ into a new cluster. Since we marginalize over the cluster parameters, computing $P({c}_{i}=j)$ requires computing the likelihood $P(\text{fol}\left(i\right),{\mathbf{X}}_{j}|\mathrm{\Theta})$, where ${\mathbf{X}}_{j}$ are the $k$ customers already clustered with $j$.
However, if we do *not* merge $\text{fol}\left(i\right)$ with ${\mathbf{X}}_{j}$, then we have $P\left({\mathbf{X}}_{j}\right|\mathrm{\Theta})$ in the overall joint probability. Therefore, we can decompose $P(\text{fol}\left(i\right),{\mathbf{X}}_{j}|\mathrm{\Theta})=P\left(\text{fol}\left(i\right)\right|{\mathbf{X}}_{j},\mathrm{\Theta})P\left({\mathbf{X}}_{j}\right|\mathrm{\Theta})$ and need only compute the change in likelihood due to merging in $\text{fol}\left(i\right)$:^{1}^{1}http://www.stats.ox.ac.uk/~teh/re- search/notes/GaussianInverseWishart.pdf:

$P\left(\text{fol}\left(i\right)\right|{\mathbf{X}}_{j},\mathrm{\Theta})$ | $={\pi}^{-nd/2}{\displaystyle \frac{{\kappa}_{k}^{d/2}{\left|{\mathrm{\Lambda}}_{k}\right|}^{{\nu}_{k}/2}}{{\kappa}_{n+k}^{d/2}{\left|{\mathrm{\Lambda}}_{n+k}\right|}^{{\nu}_{n+k}/2}}}$ | |||

$\times {\displaystyle \prod _{i=1}^{d}}{\displaystyle \frac{\mathrm{\Gamma}\left(\frac{{\nu}_{n+k}+1-i}{2}\right)}{\mathrm{\Gamma}\left(\frac{{\nu}_{k}+1-i}{2}\right)}},$ | (5) |

where the hyperparameters are updated as ${\kappa}_{n}={\kappa}_{0}+n$, ${\nu}_{n}={\nu}_{0}+n$, and

${\mu}_{n}$ | $={\displaystyle \frac{{\kappa}_{0}{\mu}_{0}+\overline{x}}{{\kappa}_{0}+n}}$ | (6) | ||

${\mathrm{\Lambda}}_{n}$ | $={\mathrm{\Lambda}}_{0}+Q+{\kappa}_{0}{\mu}_{\mathrm{\U0001d7ce}}\mu _{\mathrm{\U0001d7ce}}{}^{T}-{\kappa}_{n}{\mu}_{n}{\mu}_{n}^{T},$ | (7) |

where $Q={\sum}_{i=1}^{n}{\mathbf{x}}_{i}{\mathbf{x}}_{i}^{T}$.

Combining this likelihood term with the prior, the probability of customer $i$ following $j$ is

$$P({c}_{i}=j|{\mathbf{X}}_{,}\mathrm{\Theta},\mathbf{w},\alpha )\propto P\left(\text{fol}\left(i\right)\right|{\mathbf{X}}_{j},\mathrm{\Theta})P({c}_{i}=j|\mathbf{w},\alpha ).$$ | (8) |

Our non-sequential ddCRP introduces cycles into the follower structure, which are handled in the sampler as described by Socher et al. (2011). Also, the block of customers being moved around can potentially be very large, which makes it easy for the likelihood term to swamp the prior. In practice we found that introducing an additional parameter $a$ (used to exponentiate the prior) improved results—although we report results without this exponent as well. This technique was also used by Titov and Klementiev (2012) and Elsner et al. (2012).

Inference also includes optimizing the feature weights for the log-linear model in the ddCRP prior [23]. We interleave L-BFGS optimization within sampling, as in Monte Carlo Expectation-Maximization (Wei and Tanner, 1990). We do not apply the exponentiation parameter $a$ when training the weights because this procedure affects the follower structure only, and we do not have to worry about the magnitude of the likelihood. Before the first iteration we initialize the follower structure: for each word, we choose randomly a word to follow from amongst those with the longest shared suffix of up to 3 characters. The number of clusters starts around 750, but decreases substantially after the first sampling iteration.

For our experiments we used the English word embeddings from the Polyglot project [2]^{2}^{2}https://sites.google.com/site/rmyeid/projects/polyglot, which provides embeddings trained on Wikipedia texts for 100,000 of the most frequent words in many languages.

We evaluate on the English part of the Multext-East (MTE) corpus [11], which provides both coarse-grained and fine-grained POS labels for the text of Orwell’s “1984”. Coarse labels consist of 11 main word classes, while the fine-grained tags (104 for English) are sequences of detailed morphological attributes. Some of these attributes are not well-attested in English (e.g. gender) and some are mostly distinguishable via semantic analysis (e.g. 1st and 2nd person verbs). Many tags are assigned only to one or a few words. Scores for the fine-grained tags will be lower for these reasons, but we argue below that they are still informative.

Since Wikipedia and MTE are from different domains their lexicons do not fully overlap; we take the intersection of these two sets for training and evaluation. Table 1 shows corpus statistics.

Wikipedia tokens | 1843M |
---|---|

Multext-East tokens | 118K |

Multext-East types | 9193 |

Multext-East & Wiki types | 7540 |

With a few exceptions [4, 25], POS induction systems normally require the user to specify the number of desired clusters, and the systems are evaluated with that number set to the number of tags in the gold standard. For corpora such as MTE with both fine-grained and coarse-grained tages, previous evaluations have scored against the coarse-grained tags. Though coarse-grained tags have their place [17], in many cases the distributional and morphological distinctions between words are more closely aligned with the fine-grained tagsets, which typically distinguish between verb tenses, noun number and gender, and adjectival scale (comparative, superlative, etc.), so we feel that the evaluation against fine-grained tagset is more relevant here. For better comparison with previous work, we also evaluate against the coarse-grained tags; however, these numbers are not strictly comparable to other scores reported on MTE because we are only able to train and evaluate on the subset of words that also have Polyglot embeddings. To provide some measure of the difficulty of the task, we report baseline scores using K-means clustering, which is relatively strong baseline in this task [8].

There are several measures commonly used for unsupervised POS induction. We report greedy one-to-one mapping accuracy (1-1) [12] and the information-theoretic score V-measure (V-m), which also varies from 0 to 100% [20]. In previous work it has been common to also report many-to-one (m-1) mapping but this measure is particularly sensitive to the number of induced clusters (more clusters yield higher scores), which is variable for our models. V-m can be somewhat sensitive to the number of clusters [19] but much less so than m-1 [7]. With different number of induced and gold standard clusters the 1-1 measure suffers because some induced clusters cannot be mapped to gold clusters or vice versa. However, almost half the gold standard clusters in MTE contain just a few words and we do not expect our model to be able to learn them anyway, so the 1-1 measure is still useful for telling us how well the model learns the bigger and more distinguishable classes.

In unsupervised POS induction it is standard to report accuracy on tokens even when the model itself works on types. Here we report also type-based measures because these can reveal differences in model behavior even when token-based measures are similar.

For baselines we use K-means and the IGMM, which both only learn from the word embeddings. The CRP prior in the IGMM has one hyperparameter (the concentration parameter $\alpha $); we report results for $\alpha =5$ and $20$. Both the IGMM and ddCRP have four hyperparameters controlling the prior over the Gaussian cluster parameters: ${\mathrm{\Lambda}}_{0}$, ${\mu}_{0}$, ${\nu}_{0}$ and ${\kappa}_{0}$. We set the prior scale matrix ${\mathrm{\Lambda}}_{0}$ by using the average covariance from a K-means run with $K=200$. When setting the average covariance as the expected value of the IW distribution the suitable scale matrix can be computed as ${\mathrm{\Lambda}}_{0}=E\left[X\right]\left({\nu}_{0}-d-1\right)$, where ${\nu}_{0}$ is the prior degrees of freedom (which we set to $d$ + 10) and $d$ is the data dimensionality (64 for the Polyglot embeddings). We set the prior mean ${\mu}_{0}$ equal to the sample mean of the data and ${\kappa}_{0}$ to 0.01.

Fine types | Fine tokens | Coarse tokens | |||||
---|---|---|---|---|---|---|---|

i3-4 i5-6 i7-8 Model | K | Model | K-means | Model | K-means | Model | K-means |

K-means | 104 or 11 | 16.1 / 47.3 | - | 39.2 / 62.0 | - | 44.4 / 45.5 | - |

IGMM, $\alpha =5$ | 55.6 | 41.0 / 45.9 | 23.1 / 49.5 | 48.0 / 64.8 | 37.2 / 61.0 | 48.3 / 58.3 | 40.8 / 55.0 |

IGMM, $\alpha =20$ | 121.2 | 35.0 / 47.1 | 14.7 / 46.9 | 50.6 / 67.8 | 44.7 / 65.5 | 48.7 / 60.0 | 48.3 / 57.9 |

ddCRP uniform | 80.4 | 50.5 / 52.9 | 18.6 / 48.2 | 52.4 / 68.7 | 35.1 / 60.3 | 52.1 / 62.2 | 40.3 / 54.2 |

ddCRP learned | 89.6 | 50.1 / 55.1 | 17.6 / 48.0 | 51.1 / 69.7 | 39.0 / 63.2 | 48.9 / 62.0 | 41.1 / 55.1 |

ddCRP exp, $a=5$ | 47.2 | 64.0 / 60.3 | 25.0 / 50.3 | 55.1 / 66.4 | 33.0 / 59.1 | 47.8 / 55.1 | 36.9 / 53.1 |

We experiment with three different priors for the ddCRP model.
All our ddCRP models are non-sequential [22], allowing cycles to be formed.
The simplest model, *ddCRP uniform*, uses a uniform prior that sets the distance between any two words equal to one.^{3}^{3}In the sequential case this model would be equivalent to the IGMM [5]. Due to the nonsequentiality this equivalence does not hold, but we do expect to see similar results to the IGMM.
The second model, *ddCRP learned*, uses the log-linear prior with weights learned between each two Gibbs iterations as explained in section 4.
The final model, *ddCRP exp*, adds the prior exponentiation.
The $\alpha $
parameter for the ddCRP is set to 1 in all experiments.
For *ddCRP exp*, we report results with the exponent $a$ set to 5.

Table 2 presents all results. Each number is an average of 5 experiments with different random initializations. For each evaluation setting we provide two sets of scores—first are the 1-1 and V-m scores for the given model, second are the comparable scores for K-means run with the same number of clusters as induced by the non-parametric model.

These results show that all non-parametric models perform better than K-means, which is a strong baseline in this task [8]. The poor performace of K-means can be explained by the fact that it tends to find clusters of relatively equal size, although the POS clusters are rarely of similar size. The common noun singular class is by far the largest in English, containing roughly a quarter of the word types. Non-parametric models are able to produce cluster of different sizes when the evidence indicates so, and this is clearly the case here.

From the token-based evaluation it is hard to say which IGMM hyperparameter value is better even though the number of clusters induced differs by a factor of 2. The type-base evaluation, however, clearly prefers the smaller value with fewer clusters. Similar effects can be seen when comparing IGMM and ddCRP uniform. We expected these two models perform on the same level, and their token-based scores are similar, but on the type-based evaluation the ddCRP is clearly superior. The difference could be due to the non-sequentiality, or becuase the samplers are different—IGMM enabling resampling only one item at a time, ddCRP performing blocked sampling.

Further we can see that the ddCRP uniform and learned perform roughly the same. Although the prior in those models is different they work mainly using the the likelihood. The ddCRP with learned prior does produce nice follower structures within each cluster but the prior is in general too weak compared to the likelihood to influence the clustering decisions. Exponentiating the prior reduces the number of induced clusters and improves results, as it can change the cluster assignment for some words where the likelihood strongly prefers one cluster but the prior clearly indicates another.

The last column shows the token-based evaluation against the coarse-grained tagset. This is the most common evaluation framework used previously in the literature. Although our scores are not directly comparable with the previous results, our V-m scores are similar to the best published 60.5 [7] and 66.7 [21].

In preliminary experiments, we found that directly applying the best-performing English model to other languages is not effective. Different languages may require different parametrizations of the model. Further study is also needed to verify that word embeddings effectively capture syntax across languages, and to determine the amount of unlabeled text necessary to learn good embeddings.

This paper demonstrates that morphology and distributional features can be combined in a flexible, joint probabilistic model, using the distance-dependent Chinese Restaurant Process. A key advantage of this framework is the ability to include arbitrary features in the prior distribution. Future work may exploit this advantage more thoroughly: for example, by using features that incorporate prior knowledge of the language’s morphological structure. Another important goal is the evaluation of this method on languages beyond English.

Acknowledgments: KS was supported by the Tiger University program of the Estonian Information Technology Foundation for Education. JE was supported by a visiting fellowship from the Scottish Informatics & Computer Science Alliance. We thank the reviewers for their helpful feedback.

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