Nonlinear Dimensionality Reduction: t-SNE and Manifold Learning

PCA and ICA both assume the data's underlying structure is captured by a linear transformation. Many real datasets instead lie on a curved, low-dimensional manifold embedded in high-dimensional space — the canonical toy example is the "Swiss roll," a 2D sheet rolled up in 3D — where no linear projection can unroll the structure without distorting it. This note covers two families of tools for exactly this case: Locally Linear Embedding (LLE), which recovers manifold structure through purely local linear relationships, and t-SNE, the standard tool for producing 2D/3D visualisations of high-dimensional clusters.