基于Laplacian的离散性约简，包括谱聚类、Laplacian特征映射、局部保持投影、图嵌入和扩散映射：研究和综述
Laplacian-Based Dimensionality Reduction Including Spectral Clustering, Laplacian Eigenmap, Locality Preserving Projection, Graph Embedding, and Diffusion Map: Tutorial and Survey
摘要：
这是一个教程和调查文件的非线性维数和特征提取方法的基础上，拉普拉斯的图形数据。首先介绍了邻接矩阵、拉普拉斯矩阵的定义以及拉普拉斯算子的解释。然后，我们覆盖了图的切割和谱聚类，它应用于数据的子空间中的聚类。解释了拉普拉斯特征映射及其样本外扩展的不同优化变体。之后，我们引入了保局投影及其核变体作为拉普拉斯特征映射的线性特例。图嵌入的版本，然后解释这是广义版本的拉普拉斯特征映射和局部保持投影。最后介绍了一种基于数据的拉普拉斯算子和数据图上的随机游动的扩散映射方法。
英文摘要：
This is a tutorial and survey paper for nonlinear dimensionality and feature extraction methods which are based on the Laplacian of graph of data. We first introduce adjacency matrix, definition of Laplacian matrix, and the interpretation of Laplacian. Then, we cover the cuts of graph and spectral clustering which applies clustering in a subspace of data. Different optimization variants of Laplacian eigenmap and its out-of-sample extension are explained. Thereafter, we introduce the locality preserving projection and its kernel variant as linear special cases of Laplacian eigenmap. Versions of graph embedding are then explained which are generalized versions of Laplacian eigenmap and locality preserving projection. Finally, diffusion map is introduced which is a method based on Laplacian of data and random walks on the data graph.
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