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

Why do we need a special way to measure similarity in high-dimensional spaces?

Euclidean geometry

Euclidean distance measures absolute position in space

Manifold hypothesis

High-dimensional data lies on lower-dimensional manifolds

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

Can you shrink a 3D object into 2D without losing its shape?

the dot product measures alignment: it equals |a||b|cos(θ)

Why do vectors sometimes "agree" with each other?