Book Description

In many real world applications, data samples lying in a high dimensional ambient space can be modeled by very low dimensional nonlinear manifolds. Manifold learning, as a new framework of machine learning, discovers this low dimensional structure from the collection of the high dimensional data. In this thesis, some novel manifold learning methods are proposed, including conical dimension, semi-supervised nonlinear dimensionality reduction, active learning for the semi-supervised manifold learning, and mesh-free manifold learning. {it Conical dimension} is proposed as a novel local intrinsic dimension estimator, for estimating the intrinsic dimension of a data set consisting of points lying in the proximity of a manifold. It can also be applied to intersection and boundary detection. The accuracy and robustness of the algorithm are illustrated by both synthetic and real-world data experiments. Both synthetic and real life examples are shown. We propose the {it semi-supervised nonlinear dimensionality reduction} by introducing the prior information into basic nonlinear dimensionality reduction method, such as LLE and LTSA. The sensitivity analysis of our algorithms shows that prior information will improve the stability of the solution. We demonstrate the usefulness of our algorithm by synthetic and real life examples. A principled approach for selecting the data points for labeling used in semi-supervised manifold learning is proposed as {it active learning} method. Experiments on both synthetic and real-world problems show that our proposed methods can substantially improve the accuracy of the computed global parameterizations over several alternative methods. In the last section, we propose an alternative dimensionality reduction method, namely mesh-free manifold learning, which introduce the phase field models into dimensionality reduction problem to track the data movement during the time step of the dimensionality reduction procedure.