As a well-known nonlinear dimensionality reduction method, Laplacian Eigenmaps (LE) aims to find low dimensional representations of the original high dimensional data by preserving the local geometry between them. LE has attracted great attentions because of its capability of offering useful results on a broader range of manifolds. However, when applying it to some real-world data, several limitations have been exposed such as uneven data sampling, out-of-sample problem, small sample size, discriminant feature extraction and selection, etc. In order to overcome these problems, a large number of extensions to LE have been made. So in this paper, we make a systematical survey on these extended versions of LE. Firstly, we divide these LE based dimensionality reduction approaches into several subtypes according to different motivations to address the issues existed in the original LE. Then we successively discuss them from strategies, advantages or disadvantages to performance evaluations. At last, the future works are also suggested after some conclusions are drawn. (C) 2018 Elsevier B.V. All rights reserved.
