Brief

This paper, originally submitted to arXiv, has been withdrawn by its author. It aimed to defend cosine similarity in learned embeddings, arguing that a previously identified ambiguity related to a "gauge" matrix is only an issue when embeddings are not normalized. The authors proposed that by constraining embeddings to the unit sphere, either during or after training, the ambiguity disappears. They demonstrated that on normalized embeddings, cosine similarity is monotonically equivalent to half the squared Euclidean distance, suggesting that the perceived problem lies in a failure to normalize rather than cosine similarity itself. AI