What is the difference between linear and nonlinear dimensionality reduction?

Linear methods assume data lies on a linear subspace, while nonlinear methods capture curved manifolds.

Linear dimensionality reduction methods such as PCA assume that data variation can be captured through linear combinations of features. This works well when relationships are approximately linear. Nonlinear methods like t-SNE, UMAP, or kernel PCA assume that data lies on a nonlinear manifold embedded in higher-dimensional space. They attempt to preserve local neighborhoods or geodesic distances rather than global linear variance structure.