局部切空间对齐算法的核主成分分析解释

A Kernel PCA View of the Local Tangent Space Alignment Algorithm

基于核方法的降维技术和流形学习是两类有效而广泛应用的非线性降维技术,它们有着各自不同的出发点和理论基础,在以往的研究中很少有研究关注两者的联系。LTSA算法利用数据的局部结构构造一种特殊的核矩阵,然后利用该核矩阵进行核主成分分析。本文针对局部切空间对齐这种流形学习算法,重点研究了LTSA算法与核PCA的内在联系。研究表明,LTSA在本质上是一种基于核方法的主成分分析技术。

Recently, nonlinear dimensionality reduction has attracted extensive interests of researchers in the machine learning community. The techniques for nonlinear dimensionality reduction can be divided into two categories： kernel-based methods and manifold learning. These two methods have different motivations and derivations. This paper interprets the well-known manifold learning algorithm LTSA as a kernel method. We show that LTSA can be described as the kernel PCA, and LTSA utilizes the local neighborhood information to construct a special kernel matrix, and the global embedding obtained by LTSA with modified constraints is equivalent to principal coordinates by the kernel PCA with this special kernel.