Dimension reduction has long been associated with retinotopic mapping for understanding cortical maps. Multisensory information is processed, fused, fed and mapped to a 2-D cortex in a near-optimal information preserving manner. Data projection and visualization, inspired by this mechanism, are playing an increasingly important role in many computational applications such as cluster analysis, classification, data mining, knowledge management and retrieval, decision support, marketing, image processing and analysis. Such tasks involving either visual and spatial analysis or reduction of features or volume of the data are essential in many fields from biology, neuroscience, decision support, to management science. The topic has recently attracted a great deal of attention. There have been considerable advances in methodology and techniques for extracting nonlinear manifold as to reduce data dimensionality and a number of novel methods have been proposed from statistics, geometry theory and adaptive neural networks. Typical approaches include multidimensional scaling, nonlinear PCA and principal curve/surface. This paper provides an overview on this challenging and emerging topic. It discusses various recent methods such as self-organizing maps, kernel PCA, principal manifold, isomap, local linear embedding, Laplacian eigenmap and spectral clustering, and many of them can be seen as a combined, adaptive learning framework. Their usefulness and potentials will be presented and illustrated in various applications.
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