Nonlinear Dimensionality Reduction for Data on Manifold with Rings

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.