random projection to O(log n/ε²) dimensions preserves pairwise distances within 1±ε

the Johnson-Lindenstrauss lemma says

Random projection reduces dimensionality while approximately preserving pairwise distances

List of unsolved problems in mathematics

Random points in high dimensions are nearly equidistant due to the uniform distribution of volume in high-dimensional space

t-SNE preserves local structure

t-SNE preserves local structure by converting distances to probabilities and minimizing Kullback-Leibler divergence

cosine similarity works better than Euclidean distance in high dimensions

Cosine similarity measures orientation, not magnitude, making it more robust to irrelevant dimensions in high-dimensional spaces

non-convex loss landscapes are hard: many local minima and saddle points

Non-convex loss landscapes are hard due to many local minima and saddle points

the curse of dimensionality makes nearest neighbor search unreliable

High dimensionality dilutes data density, making nearest neighbors less distinct and search unreliable