5. Comparing the Metrics

Overview

Here you compare the three main metrics: Euclidean, Manhattan, cosine. You'll see when to use each one and why cosine dominates in semantic search.

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Comparison table

Aspect	Euclidean	Manhattan	Cosine
What it measures	Straight line	Grid path	Angle
Sensitive to magnitude	Yes	Yes	No
Values	[0, ∞)	[0, ∞)	[-1, 1]
High dimensions	⚠️ Problematic	⚠️ Problematic	✅ Robust
Use with embeddings	Uncommon	Rare	⭐ Standard
Normalization needed	No	No	Optional (it helps)

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When to use each metric

Euclidean:

• ✅ Physical distances (GPS, maps)
• ✅ When magnitude matters
• ❌ NOT optimal for high-dimensional embeddings

Manhattan:

• ✅ Movement on a grid
• ✅ Robust to outliers
• ❌ Rare with embeddings

Cosine:

• ✅ Text embeddings (OpenAI, BERT)
• ✅ Semantic search
• ✅ Any case where only direction matters

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Why cosine dominates in AI

The problem in high dimensions (e.g. 1536D):

All random vectors are ~equidistant
→ Euclidean distance loses its discriminative power

The solution:

Cosine measures the angle, not the distance
→ Angles are preserved better in high dimensions
→ More interpretable values (0.95 vs 0.45 is clear)

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A comparative example

A = [3, 4]
B = [6, 8]
C = [0, 5]

Euclidean:
- dist(A, B) = √((6-3)² + (8-4)²) = 5
- dist(A, C) = √((0-3)² + (5-4)²) ≈ 3.16

Cosine:
- cos(A, B) = 1.0 (same direction)
- cos(A, C) = 20 / (5 × 5) = 0.80 (different direction)

Interpretation: Euclidean says "C is closer to A". Cosine says "B is more similar to A (same direction)".

For embeddings: Cosine is correct (B is conceptually more similar).

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Next capsule: 06-metrics-in-semantic-search.md — Use in production.