Why does the Johnson–Lindenstrauss lemma work despite high dimensionality?

Random projections preserve pairwise distances with high probability even after reducing dimensions.

The Johnson–Lindenstrauss (JL) lemma states that a set of n points in high-dimensional space can be embedded into O(log n / ε²) dimensions while approximately preserving pairwise Euclidean distances within (1±ε). The key insight is that random linear projections distribute distortion evenly across all directions. Because of measure concentration, most random projections behave similarly, preventing extreme distortions. This is counterintuitive because it shows that structure is preserved not by careful design, but by randomness itself.