showed that relations between words are reflected to a large extent in the offsets between their vector embeddings (queen − king ≈ woman − man), and thus the vector of the hidden word b * will be similar to the vector b − a + a * , suggesting that the analogy question can be solved by optimizing: where V is the vocabulary excluding the question words b, a and a * , and sim is a similarity measure.
they used the cosine similarity measure, defined as: resulting in: Since cosine is inverse to the angle, high cosine similarity (close to 1) means that the vectors share a very similar direction.