A Note On Laplacian Eigenmaps

The embedding provided by the Laplacian Eigenmap algorithm preserves local information optimally in a certain sense . In recovering a low dimensional parametrization of data lying on a low dimensional submanifold in high dimensional space, we need to understand and grasp the structure of image of the Laplacian Eigenmap. In particular ,for the path associated with k points, what is the structure of image of the Laplacian Eigenmap in 2-dimensional Euclidean space. They play an important role in image manifolds.In this note,we show that the image of Laplacian Eigenmap in 2-dimensional Euclidean space is lied in a parabola.