| 英文摘要 | Feature extraction (i.e., dimensionality reduction) and classification are two fundamental issues in data modeling and data mining, and they are also the key parts of pattern recognition. Currently, manifold learning and semi-supervised classification are two hottest issues in feature extraction and classification. Manifold learning and semi-supervised classification are effective methods that can alleviate the problem of low labeled sample data size in high-dimensional space, and deserve deep research in both theory and application. First, manifold learning and semi-supervised classification are interdisciplinary and frontier fields of mathematics, computer science, information science and cognitive science. Second, manifold learning and semi-supervised classification have advanced the intersection of several disciplines of mathematics. Finally, manifold learning and semi-supervised classification have many important applications in machine learning, data mining and pattern recognition. Although manifold learning and semi-supervised classification have achieved great success in both theory and application, there are still many unaddressed issues. In this thesis, we focus on several key problems, and make a series of achievements, as introduced in the following: 1. For the unified interpretation and overview problem of manifold learning algorithms, we propose to utilize the Manifold Regularization (MR) framework. In order to gain the dimensionality reduction mapping f, MR tries to maintain the prior low-dimensional representation, and meanwhile, considers the complexity of f and the ability of f that can preserve certain intrinsic structure of data. With this framework, we connect various manifold learning methods from linear to nonlinear, unsupervised to supervised, single class to multi-class approaches. We utilize the MR framework to give a unified perspective to interpret them, present an overview of them, and investigate the common properties and intrinsic differences among them. 2. Most traditional manifold learning algorithms have three limitations when applied to semi-supervised classification: they are not applicable for data on multiple manifolds; it is difficult to introduce class labels; there is no explicit dimensionality reduction mapping. To address these problems, we adopt the strategy of "fitting the prior low-dimensional representation random generated according to the class labels of labeled points, and meanwhile preserving the spa... |
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