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Title: Understanding Composition of Word Embeddings via Tensor Decomposition

Paper content:
Word embedding is a powerful tool in natural language processing.
In this paper we consider the problem of word embedding composition \--- given vector representations of two words, compute a vector for the entire phrase.
We give a generative model that can capture specific syntactic relations between words.
Under our model, we prove that the correlations between three words (measured by their PMI) form a tensor that has an approximate low rank Tucker decomposition.
The result of the Tucker decomposition gives the word embeddings as well as a core tensor, which can be used to produce better compositions of the word embeddings.
We also complement our theoretical results with experiments that verify our assumptions, and demonstrate the effectiveness of the new composition method.
Word embeddings have become one of the most popular techniques in natural language processing.
A word embedding maps each word in the vocabulary to a low dimensional vector.
Several algorithms (e.g., Mikolov et al. (2013) ; Pennington et al. (2014) ) can produce word embedding vectors whose distances or inner-products capture semantic relationships between words.
The vector representations are useful for solving many NLP tasks, such as analogy tasks (Mikolov et al., 2013) or serving as features for supervised learning problems (Maas et al., 2011) .While
word embeddings are good at capturing the semantic information of a single word, a key challenge is the problem of composition: how to combine the embeddings of two co-occurring, syntactically related words to an embedding of the entire phrase. In practice
composition is often done by simply adding the embeddings of the two words, but this may not be appropriate when the combined meaning of the two words differ significantly from the meaning of individual words (e.g., "complex number" should not just be "complex"+"number").In this paper
, we try to learn a model for word embeddings that incorporates syntactic information and naturally leads to better compositions for syntactically related word pairs. Our model is
motivated by the principled approach for understanding word embeddings initiated by Arora et al. (2015) , and models for composition similar to Coecke et al. (2010) . Arora et al.
(2015) gave a generative model (RAND-WALK) for word embeddings, and showed several previous algorithms can be interpreted as finding the hidden parameters of this model. However, the
RAND-WALK model does not treat syntactically related word-pairs differently from other word pairs. We give a generative
model called syntactic RAND-WALK (see Section 3) that is capable of capturing specific syntactic relations (e.g., adjective-noun or verb-object pairs). Taking adjective-noun
pairs as an example, previous works (Socher et al., 2012; Baroni & Zamparelli, 2010; Maillard & Clark, 2015) have tried to model the adjective as a linear operator (a matrix) that can act on the embedding of the noun. However, this would require
learning a d × d matrix for each adjective while the normal embedding only has dimension d. In our model, we use a core
tensor T ∈ R d×d×d to capture the relations between a pair of words and its context. In particular, using the tensor
T and the word embedding for the adjective, it is possible to define a matrix for the adjective that can be used as an operator on the embedding of the noun. Therefore our model allows the
same interpretations as many previous models while having much fewer parameters to train.One salient feature of our model is that it makes good use of high order statistics. Standard word embeddings are based
on the observation that the semantic information of a word can be captured by words that appear close to it. Hence most algorithms use pairwise
co-occurrence between words to learn the embeddings. However, for the composition problem
, the phrase of interest already has two words, so it would be natural to consider co-occurrences between at least three words (the two words in the phrase and their neighbors).Based on the model, we can prove an
elegant relationship between high order co-occurrences of words and the model parameters. In particular, we show that if we measure
the Pointwise Mutual Information (PMI) between three words, and form an n × n × n tensor that is indexed by three words a, b, w, then the tensor has a Tucker decomposition that exactly matches our core tensor T and the word embeddings (see Section 2, Theorem 1, and Corollary 1). This suggests a natural way of learning our
model using a tensor decomposition algorithm.Our model also allows us to approach the composition problem with more theoretical insights. Based on our model, if words a, b have the
particular syntactic relationships we are modeling, their composition will be a vector v a + v b + T (v a , v b , ·). Here v a , v b are the embeddings for word
a and b, and the tensor gives an additional correction term. By choosing different core tensors it is possible
to recover many previous composition methods. We discuss this further in Section 3.Finally, we
train our new model on a large corpus and give experimental evaluations. In the experiments, we show that the model learned
satisfies the new assumptions that we need. We also give both qualitative and quantitative results
for the new embeddings. Our embeddings and the novel composition method can capture
the specific meaning of adjective-noun phrases in a way that is impossible by simply "adding" the meaning of the individual words. Quantitative experiment also shows that our composition vector
are better correlated with humans on a phrase similarity task.

assistant: text: We present a generative model for compositional word embeddings that captures syntactic relations, and provide empirical verification and evaluation.