基于放大因子和延伸方向研究流形学习算法

流形学习是一种新的非监督学习方法，可以有效地发现高维非线性数据集的内在维数和进行维数约简，近年来越来越受到机器学习和认知科学领域研究者的重视．虽然目前已经出现了很多有效的流形学习算法，如等度规映射（ISOMAP）、局部线性嵌套（Locally Linear Embedding，LLE）等，然而，对观测空间的高维数据与降维后的低维数据之间的定量关系，尚难以直观地进行分析．这一方面不利于对数据内在规律的深入探察，一方面也不利于对不同流形学习算法的降维效果进行直观比较．文中提出了一种方法，可以从放大因子和延伸方向这两个方面显示出观测空间的高维数据与降维后的低维数据之间的联系；比较了两种著名的流形学习算法（ISOMAP和LLE）的性能，得出了一些有意义的结论；提出了相应的算法从而实现了以上理论．对几组数据的实验表明了研究的有效性和意义．

Investigating Manifold Learning Algorithms Based on Magnification Factors and Principal Spread Directions

As a new unsupervised learning technique, manifold learning has captured the attention of many researchers in the field of machine learning and cognitive sciences. The major algorithms include Isometric mapping (ISOMAP) and Locally Linear Embedding (LLE). The approaches can be used for discovering the intrinsic dimensions of nonlinear high-dimensional data effectively and aim researchers to analyze the data better. How to quantitatively analyze the relationship between the intrinsic dimensions and the observation space, however, has fewer reports. And thus further works in manifold learning may have suffered some difficulties. The paper focuses on two kinds of manifold learning algorithms (ISOMAP, LLE), and discusses magnification factors and principal spread directions from the observation space to the intrinsic low-dimensional space. Also the corresponding algorithm is proposed. Experiments show the effectiveness and advantages of the research.