On the regularized Laplacian eigenmaps

To find an appropriate low-dimensional representation for complex data is one of the central problems in machine learning and data analysis. In this paper, a nonlinear dimensionality reduction algorithm called regularized Laplacian eigenmaps (RLEM) is proposed, motivated by the method for regularized spectral clustering. This algorithm provides a natural out-of-sample extension for dealing with points not in the original data set. The consistency of the RLEM algorithm is investigated. Moreover, a convergence rate is established depending on the approximation property and the capacity of the reproducing kernel Hilbert space measured by covering numbers. Experiments are given to illustrate our algorithm.