orthogonal matrices preserve distances: O^T O = I means no stretching or squashing

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

t-SNE preserves local structure

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

the Johnson-Lindenstrauss lemma says

Random projection reduces dimensionality while approximately preserving pairwise distances

Computational complexity of matrix multiplication

O(n³) naive matrix multiplication

the Gram-Schmidt process does: orthogonalizes a set of vectors

Orthogonalizes a set of vectors using Gram-Schmidt

Cholesky decomposition

Cholesky decomposition factors A = LL^T for symmetric positive definite matrices