Principal Component Analysis

PCA represents data with orthogonal directions that capture as much variance as possible, usually by decomposing a covariance matrix or data matrix. Sparse and structured PCA make components easier to interpret, while robust and streaming methods address outliers, changing data, and limited memory; kernel and manifold-based variants model nonlinear structure. Researchers apply these methods to settings such as images, sensor streams, genomics, and learned representations, and assess both reconstruction and downstream tasks like clustering or classification. A central tension is that preserving variance does not necessarily preserve rare or decision-relevant information, while adding robustness, interpretability, or nonlinear flexibility can increase computational and modeling complexity.