Some aspects of nonlinear dimensionality reduction

In this paper we discuss nonlinear dimensionality reduction within the framework of principal curves. We formulate dimensionality reduction as problems of estimating principal subspaces for both noiseless and noisy cases, and propose the corresponding iterative algorithms that modify existing principal curve algorithms. An R squared criterion is introduced to estimate the dimension of the principal subspace. In addition, we present new regression and density estimation strategies based on our dimensionality reduction algorithms. Theoretical analyses and numerical experiments show the effectiveness of the proposed methods.