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Publication# Further results on latent discourse models and word embeddings

Abstract

We discuss some properties of generative models for word embeddings. Namely, (Arora et al., 2016) proposed a latent discourse model implying the concentration of the partition function of the word vectors. This concentration phenomenon led to an asymptotic linear relation between the pointwise mutual information (PMI) of pairs of words and the scalar product of their vectors. Here, we first revisit this concentration phenomenon and prove it under slightly weaker assumptions, for a set of random vectors symmetrically distributed around the origin. Second, we empirically evaluate the relation between PMI and scalar products of word vectors satisfying the concentration property. Our empirical results indicate that, in practice, this relation does not hold with arbitrarily small error. This observation is further supported by two theoretical results: (i) the error cannot be exactly zero because the corresponding shifted PMI matrix cannot be positive semidefinite; (ii) under mild assumptions, there exist pairs of words for which the error cannot be close to zero. We deduce that either natural language does not follow the assumptions of the considered generative model, or the current word vector generation methods do not allow the construction of the hypothesized word embeddings.

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Related concepts (16)

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Concentration

In chemistry, concentration is the abundance of a constituent divided by the total volume of a mixture. Several types of mathematical description can be distinguished: mass concentration, molar concentration, number concentration, and volume concentration. The concentration can refer to any kind of chemical mixture, but most frequently refers to solutes and solvents in solutions. The molar (amount) concentration has variants, such as normal concentration and osmotic concentration. Dilution is reduction of concentration, e.

Dot product

In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used. It is often called the inner product (or rarely projection product) of Euclidean space, even though it is not the only inner product that can be defined on Euclidean space (see Inner product space for more).

Word embedding

In natural language processing (NLP), a word embedding is a representation of a word. The embedding is used in text analysis. Typically, the representation is a real-valued vector that encodes the meaning of the word in such a way that words that are closer in the vector space are expected to be similar in meaning. Word embeddings can be obtained using language modeling and feature learning techniques, where words or phrases from the vocabulary are mapped to vectors of real numbers.

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