主曲线研究综述

主曲线（principal curves)是第一主成分的非线性推广，第一主成分是对数据集的一维线性最优描述，主轴线强调寻找通过数据分布的“中间（middle)并满足“自耦合”的光滑一维曲线，其理论基础是寻找嵌入高维空间的非欧氏低维流形，该文着重介绍了主曲线发展的动机，理论基础，典型的主曲线方法和算法实现及其不同领域的应用，并对存在的问题进行了分析。

An Overview of Principal Curves

Principal Curves are nonlinear generalizations of the first linear principal component which can be thought of as 'optimal' linear 1-d summarization of the data. They emphasize for finding 'self-consistent' smooth one dimensional curves that pass through the middle of a multidimensional data set, and the theoretical foundation is to seek lower-dimensional non-euclidean manifolds embedded multidimensional data space. This paper lays stress on the introduction of the developed motivations and the foundation of the theories, alternative approaches and corresponding algorithms, explores applications for various fields using principal curves. Finally, we analyze existing problems about principal curves.