  Methodologies for multidimensionality reduction aim at discovering
low-dimensional manifolds where data ranges. Principal Component Analysis (PCA)
is very effective if data have linear structure. But fails in identifying a
possible dimensionality reduction if data belong to a nonlinear low-dimensional
manifold. For nonlinear dimensionality reduction, kernel Principal Component
Analysis (kPCA) is appreciated because of its simplicity and ease
implementation. The paper provides a concise review of PCA and kPCA main ideas,
trying to collect in a single document aspects that are often dispersed.
Moreover, a strategy to map back the reduced dimension into the original high
dimensional space is also devised, based on the minimization of a discrepancy
functional.
