Note: t-SNE and UMAP, nonlinear dimensionality reduction for visualization

The problem, from first principles

03-pca-projection reduces dimensionality by finding directions of maximum variance — a single global linear transformation applied identically everywhere. This works well when the data genuinely lies near a linear subspace, but many real datasets have structure PCA can't see: clusters that curve, spiral, or fold through high-dimensional space in ways no single set of straight-line directions captures.

From theory to code

Implement gaussian_affinities(input, sigma), which converts pairwise distances into a row-normalized similarity distribution — the core building block t-SNE uses to describe "how likely is point j to be point i's neighbor" before it ever attempts any low-dimensional placement. Reuse 01-knn's pairwise_distances.

Constraints

• input: shape (n_samples, n_features). Returns shape (n_samples, n_samples).
• Each row sums to 1 (a genuine probability distribution per point).
• The diagonal is always exactly 0 — a point is never its own neighbor.
• sigma controls how sharply affinity falls off with distance.

Hints

Open one at a time. Each gives away a little more than the last.