Dimensionality Reduction

Dimensionality reduction turns data with many variables into a smaller representation that is easier to visualize, analyze, store, or use in a predictive model. Linear methods such as PCA and supervised discriminant projections select directions of variation, while t-SNE, UMAP, graph and spectral methods, optimal transport, and autoencoders capture nonlinear structure. Researchers apply these approaches to images, text embeddings, biological data, and dynamical systems, evaluating them through reconstruction, classification, neighborhood, density, and topology preservation. A central tension is that preserving variance or local neighborhoods may discard task-relevant signals or distort global geometry, so methods and evaluations increasingly account for the intended use, uncertainty, and out-of-sample behavior.