子空间学习是机器学习的一个重要组成部分，具有重要的理论意义和实用价值。本文对子空间学习相关的一些关键问题进行了研究。主要内容如下：1，从一个新视角出发引出非监督学习的一个统一框架，并在此框架下提出了局部样条嵌入算法。给出了半监督学习一个一般性学习框架，并从三个不同的角度进行解释，提供了良好的理论基础。2，通过线性约束和线性回归这两种线性化方法，分别将非监督学习和半监督学习线性化，导出相应的子空间学习框架。同时提出了一种更加灵活的线性化方法，将前两种形式上不相关的线性化方法统一起来，且具有更好的性能。对于监督子空间方法，证明了线性鉴别分析和线性最小二乘回归的解集存在一个交集，从而揭示了这两个方法之间的本质联系，并提出了LDA分类器的概念。3，对子空间方法中常涉及到的迹比值优化问题提出了一种快速的迭代解法，并确保收敛到全局最优解。应用这种快速的优化算法，分别提出了一种基于迹比值的特征选择方法和一种基于迹比值的子空间降维方法。这种基于直接求解迹比值目标函数的方法比传统的方法具有更好的性能。4，针对张量子空间方法中维数确定这个实际问题，提出了一种可以自动确定最优维数的局部张量鉴别分析算法。针对传统的核化方法需要将解转换成内积形式这个问题，提出了一种更加方便的线性方法核化的统一框架，并基于这个框架提出了一种新的核化方法。总体来说，本文围绕子空间学习从高层的框架构建问题，到基本的理论，方法和应用中的一些关键问题展开了系统的研究，构成了对子空间学习的一个完整的研究体系。

Subspace learning is an important issue in machine learning, and isof great theoretical and practical significance. In this thesis, wefocus on the subspace learning related problems, and the maincontributions of our work are summarized as follows:1, We derive an unified unsupervised learning framework from a novelviewpoint, and propose the local spline embedding algorithm forunsupervised learning under this framework. For the semi-supervisedlearning, we propose a general framework and interpret it on graphfrom three different viewpoints, which provides a well theoreticalfoundation for the framework.2, Through the two linearization methods including linear constraintand linear regression, we linearize the unsupervised learning andsemi-supervised learning respectively and derive the correspondingsubspace learning frameworks. In addition, we propose a moreflexible linearization method, which can unify the former twoformally uncorrelated linearization methods. Experimental resultsshow that the subspace learning method derived from the newlinearization method outperforms the one derived from the former twolinearization methods. For the supervised methods, we prove that theintersection of the solution sets of Linear Discriminant Analysis(LDA) and Linear Squares Regression (LSR) is not empty, whichreveals that LDA and LSR are closed related. Based on this intrinsicrelationship, we propose the concept of LDA classifier.3, We propose a fast iterative algorithm to find the global optimumof a trace ratio optimization problem, which is frequentlyencountered in subspace learning methods. With the proposedalgorithm, we propose the trace ratio based feature selection andtrace ratio based feature extraction methods respectively.Experimental results demonstrate that the trace ratio based methodsoutperforms the traditional methods.4, We study the two important extensions of subspace methods,including the tensor methods and the kernel methods. In tensormethods, automatically determining the optimal dimensions is animportant issue. We propose a local tensor discriminant analysisalgorithm to automatically determine the optimal dimensions byitself. Traditional kernelization for a linear algorithm shouldtransform its solution into the inner form first. We propose ageneral kernelization framework to kernelize most of linearalgorithms, in which the transformation is not required. Based onthe framework, we also propose a new kernelization method.In summary, this thesis systematically studies the subspace learingrelated problems from the high-level framework to the foundationaltheory, method and application, and constructs a whole researchsystem on subspace learning.