beyer-1999-nn-distance-ratio-converges-to-1
IN premise — summaries/2026/08/24/wiki-Curse_of_dimensionality-chunk-3.md
Created 2026-08-24T17:11:09+00:00
Beyer, Goldstein, Ramakrishnan & Shaft (1999) proved that in sufficiently high dimensions with i.i.d. features, the ratio of farthest-to-nearest-neighbor distances converges to 1, rendering nearest-neighbor search meaningless.
Summary
When data lives in a very high number of independent dimensions, every point ends up roughly equidistant from any query, so the distinction between "nearest" and "farthest" neighbor vanishes. This means any system that relies on distance-based similarity, k-neighborhood reasoning, or nearest-neighbor search silently breaks down as feature count grows, forcing a shift toward dimensionality reduction, learned embeddings, or non-distance similarity measures.