Introduction
Principal Component Analysis (PCA) is by far the most popular dimensionality-reduction method. It finds the flat surface (subspace) that lies closest to the data, then projects the data onto it.
Imagine photographing a long, flat surfboard to show its shape. From the end, it looks like a thin line — little information. From above, you see its full length and width — the most information in a single 2-D photo. PCA automatically finds the "best camera angle": the directions in which the data are most spread out.