Why Use Cosine Similarity Over Euclidean Distance? Insights & Limits

This is the fifth article in the Hulu machine‑learning interview series, focusing on cosine distance.

Cosine Distance Overview

In machine learning, features are often represented as vectors. Cosine similarity measures the angle between two vectors, ranging from -1 to 1, with identical vectors having a similarity of 1. Cosine distance is defined as 1 minus cosine similarity, yielding values in the range [0, 2]; identical vectors have a distance of 0.

Question 1: When to Prefer Cosine Similarity?

Cosine similarity considers only the angular relationship between vectors and ignores their magnitudes, making it suitable for comparing texts or other high‑dimensional data where vector lengths differ but content is similar. Euclidean distance can be large for such cases because it is magnitude‑sensitive. In high‑dimensional spaces (e.g., word2vec embeddings), cosine similarity remains stable (1 for identical direction, 0 for orthogonal, -1 for opposite), whereas Euclidean distance varies with dimensionality. When vectors are normalized to unit length, Euclidean distance and cosine similarity become monotonically related, so selecting the nearest neighbor by either metric yields the same result.

Question 2: Is Cosine Distance a Proper Metric?

Cosine distance does not satisfy all three metric axioms. It fulfills positive definiteness and symmetry but violates the triangle inequality. The article presents a counterexample with points A = (1,0), B = (1,1), C = (0,1) to demonstrate the failure of the triangle inequality.

The article also notes that other commonly used “distances” such as KL divergence are not true metrics because they lack symmetry and the triangle inequality.