What is the difference between L1 and L2 distance in pgvector?

pgvector supports both L1 (Manhattan) distance and L2 (Euclidean) distance. They measure different things and have different properties that make each better suited for certain data types.

-- L2 distance (most common):
SELECT id, embedding <-> '[1,2,3]' AS l2_dist
FROM items ORDER BY l2_dist LIMIT 5;

-- L1 (Manhattan) distance:
SELECT id, embedding <+> '[1,2,3]' AS l1_dist
FROM items ORDER BY l1_dist LIMIT 5;

-- Comparison on same data:
SELECT
    id,
    embedding <-> '[1,2,3]' AS l2_distance,
    embedding <+> '[1,2,3]' AS l1_distance,
    embedding <=> '[1,2,3]' AS cosine_distance
FROM items
ORDER BY l2_distance
LIMIT 10;

-- L1 index:
CREATE INDEX ON items USING hnsw (embedding vector_l1_ops);

-- Quick rule of thumb:
-- Text embeddings: cosine distance <=> (magnitude-independent)
-- General purpose: L2 <->
-- Sparse or outlier-prone data: L1 <+>

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