List of unsolved problems in mathematics

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

Random projection reduces dimensionality while preserving pairwise distances within ε² due to the Johnson-Lindenstrauss lemma

the Johnson-Lindenstrauss lemma says

Random projection reduces dimensionality while approximately preserving pairwise distances

saddle points are more common than local minima in high dimensions

Saddle points arise due to mixed partial derivatives being zero, leading to more complex curvature in high dimensions

Surface tension

High surface tension in water

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

the curse of dimensionality makes nearest neighbor search unreliable

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