Word2vec Conjecture and A Limitative Result

Being inspired by the success of textttword2vec citepmikolov2013distributed in capturing analogies, we study the conjecture that analogical relations can be represented by vector spaces. Unlike many previous works that focus on the distributional semantic aspect of textttword2vec, we study the purely representational question: can all semantic word-word relations be represented by differences (or directions) of vectors? We call this the word2vec conjecture and point out some of its desirable implications. However, we will exhibit a class of relations that cannot be represented in this way, thus falsifying the conjecture and establishing a limitative result for the representability of semantic relations by vector spaces over fields of characteristic 0, e.g., real or complex numbers.