Explore Dimensionality Reduction

Try a tilted plane: PCA can preserve its two-dimensional structure almost exactly. A Swiss roll has a curved surface: Isomap uses paths through neighbors to try to unfold it. Too many graph neighbors shortcut the folds; too few can disconnect the graph. Classical MDS fits Euclidean distances and is equivalent to PCA here, up to rotation and reflection. Compare their neighborhood scores, then try t-SNE. Its island sizes and distances between islands do not reliably represent the original geometry. Neighborhood retention is the average overlap of 8-neighbor sets, not an overall quality score.

Perplexity sets a target neighborhood scale. This small exact t-SNE implementation uses symmetric Gaussian affinities in 3D and Student-t affinities in 2D, gradient descent with momentum, a fixed learning rate of 10, and seeded random initialization. The KL value uses the unexaggerated objective, so it need not decrease during warm-up. These are synthetic 3D-to-2D studies; real high-dimensional datasets and UMAP are not included.