流形学习及其在模式识别中的应用
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摘要
随着信息技术的飞速发展,数据的采集工作变得越来越容易。然而数据的海量性、高维性和分布的非线性特性却使人们感到越来越难以对其进行驾驭和处理。一方面我们可以获取的数据量变得越来越大;而另一方面,我们却难以找到所需的信息。在此背景下,流形学习应运而生,并为越来越多的研究者所关注。而其目标是解决高维数据分析中数据分布非线性所带来的难题,探索高维非线性数据集中的真实分布几何。
本论文面向模式识别来研究流形学习,其目的在于促进流形学习在模式识别中的成功应用。论文的主要工作大体上可以分为三个部分:构造非线性等距映射关系(即微分同胚),探讨数据集的内蕴几何(包括内蕴维数、非线性特性、内蕴几何模型),计算审美的初步探索。具体来讲,本文的主要创新性工作包括:
1、提出具有显式等距映射的ISOMAP算法。针对原ISOMAP算法缺少从高维空间到低维空间显式映射关系的不足,基于迭代优化设计出E-ISOMAP算法,并给出其监督版本SE-ISOMAP算法。由于显式等距映射的存在,E-ISOMAP和SE-ISOMAP可以用于基于测地线距离的非线性特征抽取。
2、提出采用“分两步走”的方式来解决ISOMAP算法中非线性等距映射的构造问题。在学习参数化的测地线距离函数和构造距离保持映射的基础上,实现了ISOMAP算法中从高维空间到低维空间的非线性等距映射的显式构造,可以用于基于测地线距离的非线性特征提取。
3、展开对非负局部线性重构系数的实验研究,探讨它在内蕴维数估计和在发掘数据集内精细类别子结构方面的可能应用。实验表明:在噪声较小、内蕴维数较低的情况下,显著非负局部线性重构系数的数目和分布可以指示出数据集的内蕴维数;非负局部线性重构系数的分布能够指示出数据集内的精细类别子结构,可以用于对邻域关系图的剪枝,以提高基于测地线距离的半监督分类的识别精度。
4、针对某些存在多个类别的数据集,提出主纤维丛(PrincipalFiber Bundle:PFB)模型假设。在主纤维丛假设下,提出基于双重邻域关系图的“丛流形学习”(Bundle Manifold Learning:BML)算法,用来发现数据集中潜在的精细子结构。在基准数据库上的实验表明:BML算法能够发现多类别数据集中的精细子结构,而现有的其他流形学习算法都不能。
5、提出计算审美的研究任务,结合HCL2000数据库完成美观度标注数据集,利用数据可视化技术给出对美观度标注结果的初步分析,为计算审美研究的深入开展提供依据。
With the fantastic development of information technology, data collection has become increasingly easy. However, the large amounts of data resources have puzzled people, due to their huge quantity, high dimensionality and nonlinearity. Although the data resource is sufficient, we are confronting with the embarrassment that the needed information cannot be discovered. And then, a new research direction of high dimensional data analysis, called, manifold learning, emerged as the times require, and attracts a surge of research interests. The goal of manifold learning is to solve the difficulties caused from the nonlinearity of data distribution in high dimensional dataset, and to explore the faithful intrinsic geometry hiding in high dimensional dataset.
In this dissertation, we expand manifold learning towards to pattern recognition and the motivation is to facilitate its applications in pattern recognition. The work of this dissertation consists of three parts: (1) constructing nonlinear isometric mapping (i.e. diffeomorphism); (2) investigating the intrinsic geometry (i.e. intrinsic dimensionality, nonlinearity and geometrical model) of high dimensional dataset; and (3) attempting towards computational aesthetics. Specifically speaking, the following innovative works are achieved in this thesis:
1. Proposed an E-ISOMAP algorithm, which aims at remedying the deficiency of lack of an explicit nonlinear isometric mapping in the original ISOMAP algorithm. Based on iterative majorization procedure, a version of ISOMAP algorithm with explicit nonlinear isometric mapping (E-ISOMAP) is presented and its supervised version (SE-ISOMAP) is also given. Owning to the existence of explicit isometric mapping, E-ISOMAP and SE-ISOMAP algorithms can be used for nonlinear feature extraction based on geodesic distance.
2. Proposed a "two-step" approach for constructing the nonlinear isometric mapping of ISOMAP. Based on learning of parameterized geodesic distance function and constructing of distance-preserving mapping (i.e. triangulation), the nonlinear isometric mapping which is from high dimensional Euclidean space into low dimensional Euclidean space is constructed in an explicit way and a framework for feature extraction with ISOMAP is also formulated.
3. Investigated the behavior of non-negative local linear reconstruction coefficients and discussed their possible applications for estimating the intrinsic dimensionality and discovering the subtle structure hiding in high dimensional datasets. Experimental results have shown that: (1) the number of the dominant non-negative local linear reconstruction coefficients indicates the intrinsic dimensionality of dataset provided the noisy level is low and the intrinsic dimension is small; (2) the non-negative local linear reconstruction coefficients can discover the subtle intrinsic structure hiding in dataset and hence can be used for conducting the pruning operation to improve the accuracy of geodesic distance based semi-supervised classification.
4. Put forward a Principal Fiber Bundle (PFB) model assumption to formulate the intrinsic geometry of certain dataset, which consists of samples from multiple classes. Under PFB assumption, we presented a naive Bundle Manifold Learning (BML) algorithm, which utilizes the double neighborhood graphs, to discover the subtle structure hiding in dataset which lies on bundle manifold.
5. Brought forward a novel research task, termed as Computational Aesthetics, which judges the handsomeness of Chinese Handwriting Character by computer automatically. Primary datasets HCL2000-CA towards to computational aesthetics are prepared and some exploratory experiments on HCL2000-CA dataset are given.
本论文面向模式识别来研究流形学习,其目的在于促进流形学习在模式识别中的成功应用。论文的主要工作大体上可以分为三个部分:构造非线性等距映射关系(即微分同胚),探讨数据集的内蕴几何(包括内蕴维数、非线性特性、内蕴几何模型),计算审美的初步探索。具体来讲,本文的主要创新性工作包括:
1、提出具有显式等距映射的ISOMAP算法。针对原ISOMAP算法缺少从高维空间到低维空间显式映射关系的不足,基于迭代优化设计出E-ISOMAP算法,并给出其监督版本SE-ISOMAP算法。由于显式等距映射的存在,E-ISOMAP和SE-ISOMAP可以用于基于测地线距离的非线性特征抽取。
2、提出采用“分两步走”的方式来解决ISOMAP算法中非线性等距映射的构造问题。在学习参数化的测地线距离函数和构造距离保持映射的基础上,实现了ISOMAP算法中从高维空间到低维空间的非线性等距映射的显式构造,可以用于基于测地线距离的非线性特征提取。
3、展开对非负局部线性重构系数的实验研究,探讨它在内蕴维数估计和在发掘数据集内精细类别子结构方面的可能应用。实验表明:在噪声较小、内蕴维数较低的情况下,显著非负局部线性重构系数的数目和分布可以指示出数据集的内蕴维数;非负局部线性重构系数的分布能够指示出数据集内的精细类别子结构,可以用于对邻域关系图的剪枝,以提高基于测地线距离的半监督分类的识别精度。
4、针对某些存在多个类别的数据集,提出主纤维丛(PrincipalFiber Bundle:PFB)模型假设。在主纤维丛假设下,提出基于双重邻域关系图的“丛流形学习”(Bundle Manifold Learning:BML)算法,用来发现数据集中潜在的精细子结构。在基准数据库上的实验表明:BML算法能够发现多类别数据集中的精细子结构,而现有的其他流形学习算法都不能。
5、提出计算审美的研究任务,结合HCL2000数据库完成美观度标注数据集,利用数据可视化技术给出对美观度标注结果的初步分析,为计算审美研究的深入开展提供依据。
With the fantastic development of information technology, data collection has become increasingly easy. However, the large amounts of data resources have puzzled people, due to their huge quantity, high dimensionality and nonlinearity. Although the data resource is sufficient, we are confronting with the embarrassment that the needed information cannot be discovered. And then, a new research direction of high dimensional data analysis, called, manifold learning, emerged as the times require, and attracts a surge of research interests. The goal of manifold learning is to solve the difficulties caused from the nonlinearity of data distribution in high dimensional dataset, and to explore the faithful intrinsic geometry hiding in high dimensional dataset.
In this dissertation, we expand manifold learning towards to pattern recognition and the motivation is to facilitate its applications in pattern recognition. The work of this dissertation consists of three parts: (1) constructing nonlinear isometric mapping (i.e. diffeomorphism); (2) investigating the intrinsic geometry (i.e. intrinsic dimensionality, nonlinearity and geometrical model) of high dimensional dataset; and (3) attempting towards computational aesthetics. Specifically speaking, the following innovative works are achieved in this thesis:
1. Proposed an E-ISOMAP algorithm, which aims at remedying the deficiency of lack of an explicit nonlinear isometric mapping in the original ISOMAP algorithm. Based on iterative majorization procedure, a version of ISOMAP algorithm with explicit nonlinear isometric mapping (E-ISOMAP) is presented and its supervised version (SE-ISOMAP) is also given. Owning to the existence of explicit isometric mapping, E-ISOMAP and SE-ISOMAP algorithms can be used for nonlinear feature extraction based on geodesic distance.
2. Proposed a "two-step" approach for constructing the nonlinear isometric mapping of ISOMAP. Based on learning of parameterized geodesic distance function and constructing of distance-preserving mapping (i.e. triangulation), the nonlinear isometric mapping which is from high dimensional Euclidean space into low dimensional Euclidean space is constructed in an explicit way and a framework for feature extraction with ISOMAP is also formulated.
3. Investigated the behavior of non-negative local linear reconstruction coefficients and discussed their possible applications for estimating the intrinsic dimensionality and discovering the subtle structure hiding in high dimensional datasets. Experimental results have shown that: (1) the number of the dominant non-negative local linear reconstruction coefficients indicates the intrinsic dimensionality of dataset provided the noisy level is low and the intrinsic dimension is small; (2) the non-negative local linear reconstruction coefficients can discover the subtle intrinsic structure hiding in dataset and hence can be used for conducting the pruning operation to improve the accuracy of geodesic distance based semi-supervised classification.
4. Put forward a Principal Fiber Bundle (PFB) model assumption to formulate the intrinsic geometry of certain dataset, which consists of samples from multiple classes. Under PFB assumption, we presented a naive Bundle Manifold Learning (BML) algorithm, which utilizes the double neighborhood graphs, to discover the subtle structure hiding in dataset which lies on bundle manifold.
5. Brought forward a novel research task, termed as Computational Aesthetics, which judges the handsomeness of Chinese Handwriting Character by computer automatically. Primary datasets HCL2000-CA towards to computational aesthetics are prepared and some exploratory experiments on HCL2000-CA dataset are given.
引文
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