On the Emergence of Linear Analogies in Word Embeddings

Models such as Word2Vec and GloVe construct word embeddings based on the co-occurrence probability P (i, j) of words i and j in text corpora. The resulting vectors W i not only group semantically similar words but also exhibit a striking linear analogy structure-for example, W king − W man + W woman ≈ W queen whose theoretical origin remains unclear. Previous observations indicate that this analogy structure: (i) already emerges in the top eigenvectors of the matrix M (i, j) = P (i, j)/P (i)P (j), (ii) strengthens and then saturates as more eigenvectors of M (i, j), which controls the dimension of the embeddings, are included, (iii) is enhanced when using log M (i, j) rather than M (i, j), and (iv) persists even when all word pairs involved in a specific analogy relation (e.g., king-queen, man-woman) are removed from the corpus. To explain these phenomena, we introduce a theoretical generative model in which words are defined by binary semantic attributes, and co-occurrence probabilities are derived from attribute-based interactions. This model analytically reproduces the emergence of linear analogy structure and naturally accounts for properties (i)-(iv). It can be viewed as giving fine-grained resolution into the role of each additional embedding dimension. It is robust to various forms of noise and agrees well with co-occurrence statistics measured on Wikipedia and the analogy benchmark introduced by Mikolov et al.