Dimension Reduction：A Guided Tour

We give a tutorial overview of several geometric methods for dimension reduction. We divide the methods into projective methods and methods that model the manifold on which the data lies. For projective methods, we review projection pursuit, principal component analysis (PCA), kernel PCA, probabilistic PCA, canonical correlation analysis, oriented PCA, and several techniques for sufficient dimension reduction. For the manifold methods, we review multidimensional scaling (MDS), landmark MDS, Isomap, locally linear embedding, Lapl acian eigenmaps and spectral clustering. The Nystr¨om method, which links several of the manifold algorithms, is also reviewed. The goal is to provide a self-contained overview of key concepts underlying many of these algorithms, and to give pointers for further reading.