一种基于Biharmonic样条插值的流形学习算法  

A Manifold Learning Algorithm Based on Biharmonic Spline Interpolation Technique

作　　者：顾艳春[1,2] 马争鸣[2] 梁宇滔[2] 

机构地区：[1]佛山科学技术学院电子与信息工程学院,广东佛山528000 [2]中山大学信息科学与技术学院,广东广州510220

出　　处：《中山大学学报（自然科学版）》2013年第5期82-90,96,共10页Acta Scientiarum Naturalium Universitatis Sunyatseni

基　　金：广东省自然科学基金资助项目(8452800001001086);佛山科学技术学院资助项目(2010X063)

摘　　要：作为一种有效的非线性降维方法,流形学习在众多领域吸引了广泛的关注并取得了长足的发展。但当样本点较为稀疏时,样本点的局部邻域很难满足流形学习局部同胚的前提条件,此时流形学习算法往往效果变差甚至失效。一种有效的解决方法是增加一些新的插值点。但已有的插值方法选取的插值点与原样本点均存在线性关系。从线性代数的理论来说,由插值点和原有邻域点张成的线性子空间与原有邻域点张成的子空间是一样的,因此,不会改善线性逼近的误差。而且,插值点没有反应出流形的本质结构和特征,从理论上背离了数据降维的目的。为此,提出了一种基于Biharmonic非线性插值技术的流形学习算法BbMLA。由于是从高维曲面逼近的角度非线性的选择插值点,插值出的样本点不会被原有邻域点线性表示,从而能更好的重构原样本点。将BbMLA应用到多个数据集后,图示说明了插值点能够有效的改善邻域内的样本点结构,同时插值后的流形学习算法具有较好的有效性和稳定性。As an effective non-linear dimension reduction method, manifold learning has attracted wide- spread attention and made great progress. But when sample points are not dense, these algorithms often become worse or even failed just because the points in some neighborhoods do not meet the requirement of local homeomorphism. An effective solution to this question is to increase some new interpolation points. Unfortunately, the points selected by existing interpolation methods nowadays are all linear with the origi- nal sample points. From the theory of linear algebra, the subspace spanned by the interpolation points and the original neighbors is the same as the subspace spanned by the original ones; therefore, the inter- polation points will not improve the linear approximation error either. Moreover, the interpolation points have no consideration to the native structure and characteristics of the manifold, which deviates from the purpose of data dimensionality reduction. To this end, a new manifold learning algorithm based on a non- linear interpolation method called Biharmonic is proposed. Experimental results demonstrate the improve- ment of the neighborhood structure. The effectiveness and stability of this algorithm are further confirmed by applying it to the classical manifold learning algorithms.

关 键 词：流形学习 数据降维 曲面拟合 插值 

分 类 号：TP391.4[自动化与计算机技术—计算机应用技术]