Silhouette-Based Evaluation of PCA, Isomap, and t-SNE on Linear and Nonlinear Data Structures

Dimensionality reduction is fundamental for analyzing high-dimensional data, supporting visualization, denoising, and structure discovery. We present a systematic, large-scale benchmark of three widely used methods—Principal Component Analysis (PCA), Isometric Mapping (Isomap), and t-Distributed Stochastic Neighbor Embedding (t-SNE)—evaluated by average silhouette scores to quantify cluster preservation after embedding. Our full factorial simulation varies sample size $n\in \{100,200,300,400,500\}$, noise variance ${\sigma }^2\in \{0.25,0.5,0.75,1,1.5,2\}$, and feature count $p\in \{20,50,100,200,300,400\}$ under four generative regimes: (1) a linear Gaussian mixture, (2) a linear Student-t mixture with heavy tails, (3) a nonlinear Swiss-roll manifold, and (4) a nonlinear concentric-spheres manifold, each replicated 1000 times per condition. Beyond empirical comparisons, we provide mathematical results that explain the observed rankings: under standard separation and sampling assumptions, PCA maximizes silhouettes for linear, low-rank structure, whereas Isomap dominates on smooth curved manifolds; t-SNE prioritizes local neighborhoods, yielding strong local separation but less reliable global geometry. Empirically, PCA consistently achieves the highest silhouettes for linear structure (Isomap second, t-SNE third); on manifolds the ordering reverses (Isomap > t-SNE > PCA). Increasing ${\sigma }^2$ and adding uninformative dimensions (larger p) degrade all methods, while larger n improves levels and stability. To our knowledge, this is the first integrated study combining a comprehensive factorial simulation across linear and nonlinear regimes with distribution-based summaries (density and violin plots) and supporting theory that predicts method orderings. The results offer clear, practice-oriented guidance: prefer PCA when structure is approximately linear; favor manifold learning—especially Isomap—when curvature is present; and use t-SNE for the exploratory visualization of local neighborhoods. Complete tables and replication materials are provided to facilitate method selection and reproducibility.