Nonlinear Dimensionality Reduction for Data on Manifold with Rings

Abstract

近年来出现了多种新型的非线性降维方法,且在一些应用中体现出良好的效果.然而,当面对球体、柱体等环状流形产生的非线性流形数据时,这些方法往往会失效.针对这一问题,提出了针对环状流形数据的环结构检测算法与非线性降维方法.理论上,基于目前极受关注的Isomap降维方法的运行原理,给出了一个判断环状流形的充要条件;算法上利用所得的判断定理,制订了基于数据的环状流形检测算法;最后基于所找到的环结构,利用极坐标展开的思想设计了针对环状流形数据的非线性降维策略.针对一系列典型环状流形数据集的仿真实验结果表明,与其他流形学习降维方法相比,该方法对环状流形数据进行降维具有显著优势.;Isomap has attracted attentions recently due to its prominent performance on nonlinear dimensionality reduction. However, how to implement effective learning for data on manifold with rings is still a remaining problem. To solve this problem, a systemic strategy is presented in this study. Based on the intrinsic implementation principle of Isomap, a theorem is presented which gives a sufficient and necessary condition to judge whether a manifold is with rings. Besides, an algorithm for detecting ring structures in the manifold is constructed and a nonlinear dimensionality reduction strategy is developed through polar coordinates transformation. A series of simulation results implemented on a series of synthetic and real-world data sets generated by manifolds with or without rings verify the prominent performance of the new method.