Does cosine similarity actually measure embedding similarity?

Cosine similarity is the default tool for quantifying semantic similarity between learned embeddings, on the intuition that direction matters more than norm. This paper shows that intuition is unsafe. Using regularized linear (matrix-factorization) models where closed-form solutions allow analysis, it derives that cosine similarities can be arbitrary and therefore meaningless: for some models they are not even unique, and for others they are implicitly controlled by the regularization applied during training. Since deep models combine multiple regularizations with implicit and unintended effects, taking cosine similarities of their embeddings can render results opaque and possibly arbitrary.

The keeper is a methodological caution with teeth: the same embeddings can produce different "similarities" depending on regularization the practitioner never explicitly chose for similarity, so a cosine score is not a stable, model-independent measure of semantic closeness. The paper outlines alternatives and urges not using cosine blindly.

This sharpens the vault's embedding-geometry caveats. It is the regularization-dependence complement to Why can't cosine space retrievers distinguish word order? (geometry-dependence) and underwrites the production-RAG warning in Do vector embeddings actually measure task relevance?: cosine over learned embeddings is doubly unreliable — wrong target (association) and unstable measure (regularization-controlled).