Similarity Search

Every recommendation engine, every semantic search system, every content-based filter, every duplicate detector, and every RAG pipeline has the same operation at its core: given this item, find the most similar items.

Spotify finds songs similar to the one you're playing. Pinterest finds pins visually similar to the one you liked. Stripe detects transactions that look similar to known fraud patterns. Your email client finds messages similar to the one you flagged as spam.

All of these are similarity search problems. The item is represented as a vector (an embedding), and "similar" means "close in vector space." The database's job is to find the K closest vectors to your query vector, the K-nearest neighbors.

It sounds simple. It's not. The challenge is doing this fast at scale. With 1 billion vectors in 1,536 dimensions, brute-force comparison takes seconds per query. Users expect results in milliseconds. Bridging that gap, from seconds to milliseconds, is what separates a toy demo from a production system.

The choice of distance metric fundamentally changes what "similar" means.

Cosine Similarity

Measures the angle between two vectors. Ignores magnitude, only cares about direction.

cosine_sim(A, B) = (A . B) / (||A|| * ||B||)

Range: -1 (opposite) to 1 (identical direction)

Use when: Comparing text embeddings. A long document and a short document about the same topic should be similar, even though the long document's vector might have larger magnitudes. Cosine similarity treats [0.1, 0.2, 0.3] and [0.2, 0.4, 0.6] as identical, same direction, different scale.

Euclidean Distance (L2)

Straight-line distance between two points in vector space.

L2(A, B) = sqrt(sum((A_i - B_i)^2))

Range: 0 (identical) to infinity

Use when: The magnitude of the vector carries meaning. For image embeddings where brighter/darker images produce different magnitudes, L2 distance captures those differences. Also used for spatial data and clustering.

Dot Product (Inner Product)

dot(A, B) = sum(A_i * B_i)

Range: -infinity to infinity

Use when: Vectors are normalized (unit length). In that case, dot product equals cosine similarity but is faster to compute (no division by magnitudes). Many embedding models produce normalized vectors specifically so dot product can be used.

Manhattan Distance (L1)

Sum of absolute differences. Less common for embeddings, but used in specific domains.

The practical choice: If you're using OpenAI, Cohere, or Sentence Transformer embeddings, cosine similarity (or dot product on normalized vectors) is almost always correct. The model was trained to produce vectors where cosine similarity reflects semantic similarity.