t-SNE preserves local structure

the Johnson-Lindenstrauss lemma says

Random projection reduces dimensionality while approximately preserving pairwise distances

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

PCA vs t-SNE: PCA preserves global variance linearly, t-SNE preserves local structure nonlinearly

PCA: Linear variance preservation, t-SNE: Nonlinear local structure preservation

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

UMAP is faster than t-SNE

UMAP is faster due to approximate nearest neighbors and cross-entropy optimization

Convex optimization

Convex functions have only one global minimum