Rotate King to get Queen: Word Relationships as Orthogonal Transformations in Embedding Space
This provides an alternative representation for word analogies in NLP, but it is incremental as it builds on known linear substructures without fundamentally changing the field.
The paper tackles the problem of how word relationships are represented in embedding spaces, showing that orthogonal and linear transformations can model relationships like gender as accurately as traditional vector arithmetic, with linear transformations achieving higher accuracy.
A notable property of word embeddings is that word relationships can exist as linear substructures in the embedding space. For example, $\textit{gender}$ corresponds to $\vec{\textit{woman}} - \vec{\textit{man}}$ and $\vec{\textit{queen}} - \vec{\textit{king}}$. This, in turn, allows word analogies to be solved arithmetically: $\vec{\textit{king}} - \vec{\textit{man}} + \vec{\textit{woman}} \approx \vec{\textit{queen}}$. This property is notable because it suggests that models trained on word embeddings can easily learn such relationships as geometric translations. However, there is no evidence that models $\textit{exclusively}$ represent relationships in this manner. We document an alternative way in which downstream models might learn these relationships: orthogonal and linear transformations. For example, given a translation vector for $\textit{gender}$, we can find an orthogonal matrix $R$, representing a rotation and reflection, such that $R(\vec{\textit{king}}) \approx \vec{\textit{queen}}$ and $R(\vec{\textit{man}}) \approx \vec{\textit{woman}}$. Analogical reasoning using orthogonal transformations is almost as accurate as using vector arithmetic; using linear transformations is more accurate than both. Our findings suggest that these transformations can be as good a representation of word relationships as translation vectors.