Efficient Non-Parametric Methods for Dimensionality Reduction in High-Dimensional Non-Linear Multivariate Data

Abstract

Analysis of high-dimensional non-linear multivariate datasets frequently violates the assumptions underlying classical parametric techniques. This study conducts a systematic comparative evaluation of five prominent non-parametric dimensionality-reduction methods: Isomap, Uniform Manifold Approximation and Projection (UMAP), Locally Linear Embedding (LLE), Kernel Principal Component Analysis (KPCA), and t-Distributed Stochastic Neighbor Embedding (t-SNE), using one synthetic dataset and three real-world datasets drawn from genomics, macroeconomics, and social-media analytics. Performances of the methods were assessed with respect to their computational efficiency and their ability to preserve the local and global structure of the data.  This was measured through computational time, trustworthiness, mean square error, reconstruction error, and Spearman rank correlation. Across the empirical datasets, UMAP consistently exhibited superior speed and fidelity in structure preservation, with KPCA emerging as a strong second performer. Friedman rank tests indicated significant differences among the performances of the methods. However, the simulated data did not yield any notable significant difference.