1 Spectral Methods for Dimensionality Reduction
How can we search for low dimensional structure in high dimensional data? If the data is mainly confined to a low dimensional subspace, then simple linear methods can be used to discover the subspace and estimate its dimensionality. More generally, though, if the data lies on (or near) a low dimensional submanifold, then its structure may be highly nonlinear, and linear methods are bound to fail. Spectral methods have recently emerged as a powerful tool for nonlinear dimensionality reduction and manifold learning. These methods are able to reveal low dimensional structure in high dimensional data from the top or bottom eigenvectors of specially constructed matrices. To analyze data that lies on a low dimensional submanifold, the matrices are constructed from sparse weighted graphs whose vertices represent input patterns and whose edges indicate neighborhood relations. The main computations for manifold learning are based on tractable, polynomial-time optimizations, such as shortest path problems, least squares fits, semidefinite programming, and matrix diagonalization. This chapter provides an overview of unsupervised learning algorithms that can be viewed as spectral methods for linear and nonlinear dimensionality reduction.