Semantics at an Angle: When Cosine Similarity Works Until It Doesn't

Abstract

Cosine similarity is a standard comparison rule for learned representations in information retrieval, natural language processing, computer vision, and multimodal learning. Its popularity is well founded: it removes positive radial scale, is computationally convenient, and often matches objectives that train normalized embeddings. These same properties also delimit what cosine can express. Normalization discards radial variation; anisotropic representations can compress angular contrast; high-dimensional neighborhoods can develop hubs; and a symmetric, uncalibrated score may be mismatched to the relation of interest. This article offers a selective review organized around a simple principle: the usefulness of cosine similarity depends jointly on the learned representation, any normalization or post-processing, the scoring rule, and the downstream decision. We derive the main geometric identities, distinguish failure mechanisms that are often conflated, review representative evidence about embedding norms, and describe objective-matched, geometry-aware, hubness-aware, norm-aware, and learned alternatives. The central conclusion is conditional rather than adversarial: cosine's positive-scale invariance is justified when radial variation is nuisance or fixed by the representation contract, but its angular geometry and downstream decision must still be validated.