# A Latent Variable Model Approach to PMI-based Word Embeddings (Arora 2016)

## Background

There are two popular classes of word embedding techniques:

1. Compute a low-rank approximation of a re-weighted co-occurrence matrix (i.e., PCA via SVD).
2. Use a neural network language model’s representation of a word (e.g., word2vec, GloVe).

A re-weighting scheme used in (1) is to replace the co-occurrence statistics with the pointwise mutual information between two words.

The pointwise mutual information (PMI) of a pair of outcomes $x \in X$, $y \in Y$, $X$, $Y$ discrete random variables measures the extent to which their joint distribution differs from the product of the marginal distributions:

Note that $\pmi(x;y)$ attains its maximum when $p(x \mid y) = 1$ or $p(y \mid x) = 1$.

It is observed empirically that

## Key Contribution

This paper proposes a generative model for word embeddings that provides a theoretical justification of $(\ref{low-rank-pmi})$ and word2vec and GloVe. The key assumption it makes is that word vectors, the latent variables of the model, are spatially isotropic (intuition: “no preferred direction in space”). Isotropy of low-dimensional vectors helps explain the linear structure of word vectors as well.

## The Generative Model

A time-step model: at time $t$, word $t$ is produced by a random walk of a discourse vector $c_t \in \mathbb{R}^d$ that represents the topic of conversation. Each generated word has a latent vector $v_w \in \mathbb{R}^d$ that measures the correlation with the discourse vector. In particular:

$\Pb(w$ is the $t$-th word $\mid c_t) \propto \exp \inner{c_t}{v_w}$.

$c_{t+1} := c_t +$ a small random displacement. Under this model, the authors prove that the co-occurrence probabilities and marginal probabilities are functions of the word vectors; this is useful when optimizing the likelihood function $\ell = \log \prod_{w, w'} p(w, w')^{X_{w,w'}}$.

## Commentary

• This paper answers an interesting question: Why is it that a nonlinear model like word2vec produces outputs that have linear structures (e.g., king - man = woman)?
• It’s really cool that a relatively simple generative model grounded in a solid theoretical foundation produces results that are competitive with neural network models.