Laplacian eigenmaps (Leigs) method is based on the idea of manifold unsupervised learning. Let H be the observed high-dimensional data, which reside on a low-dimentional manifold M. Let h be the coordinate mapping on M so that Y = h(H)is a DR of H. Each component of the coordinate mapping h is a linear function on M. Hence, all components of h nearly reside on the numerically null space of the Laplace-Beltrsmi operator on M. In Leigs method, a Laplace-Beltrami operator is constructed on the data graph. Then the DR data set is derived from the eigenvectors of the operator corresponding to several smallest eigenvalues. The chapter is organized as follows. In Section 12.1, we describe the Laplacian eigenmaps method and focus the discussion on the construction of the Laplace-Beltrami operator on the data set. In Section 12.2, the Leigs DR algorithm is developed. In Section 12.3, some experiments of the algorithm are presented.
KeywordsWeight Matrix Coordinate Mapping Graph Neighborhood Noise Standard Deviation Laplacian Eigenmaps
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