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The solution is obvious. Sexism can be thought of as a kind of warping of this vector space. Indeed, the gender bias itself is a property that the team can search for in the vector space. So fixing it is just a question of applying the opposite warp in a way that preserves the overall structure of the space.

Finally, they use the transformed vector space to produce a new list of gender analogies and then ask turkers to rate them again. This produces pairings such as: she:he::hen:cock; maid:housekeeper; gals:dudes; daughter:son, and so on.

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Clearly, language is filled with many examples of gender bias that are hard to justify. An interesting question is the extent to which this kind of vector space mathematics should be used to correct it.

This number-crunching shows exactly how often a word appears close to other words, an important factor in how they are used. So the word Olympics might appear close to words like running, jumping, and throwing but less often next to words like electron or stegosaurus. This set of relationships can be thought of as a multidimensional vector that describes how the word Olympics is used within a language, which itself can be thought of as a vector space.

And therein lies this massive change. This new approach allows languages to be treated like vector spaces with precise mathematical properties. Now the study of language is becoming a problem of vector space mathematics.

Today, Timothy Baldwin at the University of Melbourne in Australia and a few pals explore one of the curious mathematical properties of this vector space: that adding and subtracting vectors produces another vector in the same space.

The question they address is this: what do these composite vectors mean? And in exploring this question they find that the difference between vectors is a powerful tool for studying language and the relationship between words.

Baldwin and co ask how reliable this approach can be and how far it can be taken. To do this, they compare how vector relationships change according to the corpus of words studied. For example, do the same vector relationships work in the corpus of words from Wikipedia as in the corpus of words from Google News or Reuters English newswire?

However, there are some interesting outliers where words have more than one meaning and so have ambiguous representations in these vectors spaces. Examples in the third person plural cluster include study and studies, run and runs, increase and increases, all words that can be nouns and verbs, which distorts their vectors in these spaces.

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But why? Why do arithmetic operators apply to vectors generated by non-linear models such as word2vec? What conditions have to be satisfied by the training corpus for these analogies to hold in the vector space?

We can interpret the inner product of a word and context vector because even though SGNS and GloVe learn vectors iteratively in practice, they are implicitly factorizing a word-context matrix containing a co-occurrence statistic. When the factorized matrix can be perfectly reconstructed, where $\vec{x}, \vec{y}$ are word vectors and $\vec{x}_c, \vec{y}_c$ their corresponding context vectors:

This means that we can now use the SGNS or GloVe identity above to rewrite these conditions in terms of statistics over the training corpus. Regardless of which identity we choose, we arrive at the Co-occurrence Shifted PMI Theorem (or csPMI Theorem, for short). Remarkably, even though SGNS and GloVe are completely different embedding models, the conditions under which analogies manifest in their vector spaces is the same!

Let $\text{csPMI}(x,y) = \text{PMI}(x,y) + \log p(x,y)$, $W$ be an SGNS or GloVe word vector space with no reconstruction error, $M$ be the word-context matrix that is implicitly factorized by SGNS or GloVe, and $S$ be a set of ordered pairs such that $|S| > 1$ and all words in $S$ have a vector in $W$.

In a noiseless SGNS or GloVe space, a linear analogy holds exactly over a set of word pairs iff the co-occurrence shifted PMI is the same for every word pair and across any two word pairs, provided the row vectors of those words in the factorized word-context matrix are coplanar.

This, in turn, reaffirms the long-standing intuition on why word analogies hold, helps explain why vector addition is a decent means of composing words, and provides a new interpretation of Euclidean distance in word vector spaces. Unlike past theories of word analogy arithmetic, there is plenty of empirical evidence to support the csPMI Theorem, making it much more tenable.

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