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中文摘要：
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滋方法对环状流形数据进行降维具有显著优势．
英文摘要：
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.
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