Vector Distance Metrics & L2 Normalization

Vector distance metrics are mathematical functions measuring geometric similarity in high-dimensional embedding spaces, primarily Cosine Similarity $\cos(u, v) = \frac{u \cdot v}{\|u\| \|v\|}$, Euclidean Distance (L2) $d(u, v) = \|u - v\|_2$, and Dot Product $\langle u, v \rangle = u \cdot v$; when vectors are L2-normalized ($\|u\|_2 = 1$), L2 distance is monotonically equivalent to cosine similarity via $d^2(u, v) = 2 - 2\cos(u, v)$, and dot product directly equals cosine similarity.

Configuring distance metric types in vector databases (Milvus, Qdrant), computing semantic similarity in RAG retrieval, and unifying tensor dimensions for hardware-accelerated GEMM operations.

Raw cosine similarity requires computing norms $\|u\|$ and $\|v\|$ with square roots and divisions during runtime search, creating severe SIMD/GPU throughput bottlenecks; offline L2 normalization enables pure dot product GEMM execution with zero norm calculation overhead.

Derive the mathematical equivalence formula between L2 distance and Cosine Similarity under L2 normalization?

Why is Dot Product execution on GPU/SIMD much faster than raw unnormalized Cosine Similarity?

In unnormalized vectors, why does dot product retrieval suffer severe bias towards text length and vector magnitude?

How to choose between Cosine Similarity and Manhattan Distance (L1) for high-dimensional sparse vectors (e.g., BM25/SPLADE)?

When modifying Metric Type (IP vs COSINE vs L2) in vector DB collection indexes, is full index rebuild required?