If a vector database supports multiple distance metrics, how might the index be stored or optimized differently for each (for example, an index optimized for inner product vs one for L2)?

When a vector database supports multiple distance metrics, the underlying index structures and optimization strategies must adapt to the mathematical properties of each metric. For example, an index optimized for inner product (dot product) will handle vector magnitude and angular relationships differently than one designed for Euclidean (L2) distance. These differences influence how data is organized, stored, and queried to maximize performance and accuracy.

For L2 distance, indexes often rely on spatial partitioning techniques like Voronoi diagrams or hierarchical navigable small world (HNSW) graphs, which prioritize geometric proximity. For instance, in a product quantization (PQ) index, vectors are split into subvectors, and codebooks are trained to minimize L2 reconstruction error. This allows approximate nearest neighbor (ANN) searches to efficiently compute distances by comparing quantized subvectors. In contrast, inner product optimization requires handling both vector direction and magnitude. Here, indexes might normalize vectors during preprocessing to convert inner product calculations into cosine similarity, simplifying comparisons. Alternatively, clustering algorithms like spherical k-means (which normalizes centroids) are used for inverted file (IVF) indexes to group vectors by angular similarity, ensuring clusters align with the inner product’s emphasis on vector direction.

The way distance computations are accelerated also varies. L2 benefits from precomputing squared terms to avoid redundant calculations (e.g., using the identity ||a-b||² = ||a||² + ||b||² - 2a·b). Indexes optimized for L2 might store precomputed squared magnitudes of vectors to speed up pairwise comparisons. For inner product, indexes might focus on optimizing matrix multiplication (e.g., using optimized BLAS libraries) or employ pruning strategies that discard vectors with magnitudes below a threshold when query vectors are normalized. Additionally, graph-based indexes like HNSW adjust their edge selection criteria: For L2, edges prioritize geometric neighbors, while for inner product, edges might prioritize vectors with high dot product values, even if they’re not spatially close.

Finally, hardware optimizations can differ. For example, L2 calculations map well to SIMD instructions for parallel arithmetic operations, while inner product might leverage fused multiply-add operations. Databases might also use metric-specific compression: PQ codebooks for L2 minimize Euclidean error, whereas inner product codebooks could maximize dot product preservation. These tailored approaches ensure that each metric’s unique requirements—whether geometric (L2) or angular/magnitude-sensitive (inner product)—are efficiently addressed in storage and query execution.