How does PCA behave when data has strong nonlinear structure?

PCA performs poorly on nonlinear manifolds because it only captures linear variance directions.

PCA assumes that the data lies in a linear subspace and tries to find orthogonal axes that maximize variance. When data lies on a nonlinear manifold (e.g., spirals, circles), PCA cannot unfold the structure, leading to overlapping projections and loss of separability. This limitation motivates nonlinear methods like Kernel PCA, t-SNE, or UMAP.