A review of dimension reduction techniques

Abstract

The problem of dimension reduction is introduced as a way to overcome the curse of the dimen- sionality when dealing with vector data in high-dimensional spaces and as a modelling tool for such data. It is de ned as the search for a low-dimensional manifold that embeds the high-dimensional data. A classi cation of dimension reduction problems is proposed. A survey of several techniques for dimension reduction is given, including principal component analysis, projection pursuit and projection pursuit regression, principal curves and methods based on topologically continuous maps, such as Kohonen's maps or the generalised topographic mapping. Neural network implementations for several of these techniques are also reviewed, such as the projec- tion pursuit learning network and the BCM neuron with an objective function. Several appendices complement the mathematical treatment of the main text.