dimensionalityreducing

Dimensionality reducing, or dimensionality reduction, refers to techniques that reduce the number of random variables under consideration, often by constructing a lower-dimensional representation of data. The goal is to simplify data, reduce noise and storage requirements, mitigate the curse of dimensionality, and enable visualization or faster downstream processing.

Dimensionality reduction can be categorized as feature extraction or feature selection. Feature extraction builds new, lower-dimensional features from the original data, while feature selection retains a subset of existing features. Methods can be linear or nonlinear. Linear methods seek projections that preserve as much information as possible, while nonlinear methods aim to capture complex relationships and manifolds in the data.

Common methods include principal component analysis (PCA), which identifies directions of greatest variance; factor analysis and linear discriminant analysis (LDA), which incorporate probabilistic structure or class labels; independent component analysis (ICA) and non-negative matrix factorization (NMF) for decomposing data into interpretable components. Nonlinear approaches include kernel PCA, Isomap, locally linear embedding (LLE), diffusion maps, t-distributed stochastic neighbor embedding (t-SNE), and uniform manifold approximation and projection (UMAP). PCA serves as a standard baseline; t-SNE and UMAP are popular for visualizing high-dimensional data in 2D or 3D, while LDA is supervised for discriminative tasks.

Typical workflows involve standardizing data, choosing a target dimensionality, applying the method, and evaluating information preservation using explained variance or reconstruction error. For neighborhood preservation, metrics such as trustworthiness and continuity may be used. Considerations include the trade-off between information loss and simplification, interpretability of new components, computational cost, and the suitability of the method for the intended task. Dimensionality reduction is widely used across image processing, genomics, text mining, neuroscience, and marketing analytics.