cosine similarity works better than Euclidean distance in high dimensions

List of algorithms

Cosine similarity measures the angle between vectors, not their magnitude

cosine similarity is preferred over dot product for normalized embeddings

Cosine similarity measures orientation, not magnitude, making it ideal for normalized embeddings

Euclidean geometry

Euclidean distance measures absolute position in space

random projection to O(log n/ε²) dimensions preserves pairwise distances within 1±ε

Random projection reduces dimensionality while preserving pairwise distances within ε² due to the Johnson-Lindenstrauss lemma

the curse of dimensionality makes nearest neighbor search unreliable

High dimensionality dilutes data density, making nearest neighbors less distinct and search unreliable

Manifold hypothesis

High-dimensional data lies on lower-dimensional manifolds