Withdrawn Paper Argues Cosine Similarity Validity Depends on Embedding Normalization
A preprint posted to arXiv argued that cosine similarity is geometrically valid when embeddings are constrained to the unit sphere, directly rebutting a prior paper that cautioned against its use. The author contended that the so-called 'gauge' ambiguity identified by Steck et al. (arXiv:2403.05440) arises from an incompatible dot-product training objective, not from cosine similarity itself. The paper was subsequently withdrawn by its author, who noted it originated as a blog post and requires revision before serving as a proper preprint.
The preprint, submitted by Taha Bouhsine, responded to a result by Steck, Ekanadham, and Kallus showing that cosine similarity of learned embeddings from matrix factorization models can be made arbitrary through a diagonal 'gauge' matrix D. Bouhsine accepted that result as correct but argued it applies only when embeddings are trained with dot-product objectives without normalization. The paper's central claim is that when embeddings are constrained to the unit sphere S^(d-1)—either during training or via post-hoc normalization with an appropriate objective—the D-matrix ambiguity disappears entirely. A key mathematical point offered is that on normalized embeddings, cosine distance reduces to exactly half the squared Euclidean distance, making cosine-based and Euclidean-based nearest-neighbor rankings identical. The author concluded that the practical problem is not cosine similarity per se, but the failure to normalize embeddings. The submission was withdrawn at version 2, with the author acknowledging it was drafted as a blog post companion and needs to be reformatted as a proper preprint before wider dissemination.
What's missing
Because the paper has been withdrawn and no PDF is available, the full mathematical proofs and any experimental validation cannot be assessed. It is also unclear whether the argument holds for all embedding training objectives beyond matrix factorization, or whether the claim about post-hoc normalization with 'an appropriate objective' introduces its own constraints that limit generality.
What different sources said
- arXiv cs.LGCenter
In Defense of Cosine Similarity: Normalization Eliminates the Gauge Freedom
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