| 王建忠，1967年毕业于北京大学数学力学系，1981年获浙江大学应用数学系硕士学位，后执教于武汉大学并于1988年晋升为教授，现任美国得克萨斯州山姆休斯顿大学教授。主要研究方向包括：样条和逼近理论、小波分析、图像处理以及高维数据分析等。曾主持多项美国自然科学基金及其他基金课题的研究。发表学术论文70余篇，专利1项。 |
| Chapter 1 Introduction 1.1 0verview of Dimensionality R,eduction 1.2 High Dimension Data Acquisition 1.2.1 Collection of Images in Face Recognition 1.2.2 Handwriting Letters and Digits 1.2.3 Text Documents 1.2.4 Hyperspectral Images 1.3 Curse of the Dimensionality 1.3.1 Volume of Cubes and Spheres 1.3.2 Volume of a Thin Spherical Shell 1.3.3 Tail Probability of the Multivariate Gaussian Distributions 1.3.4 Diagonals of Cube 1.3.5 Concentration of Norms and Distances 1.4 Intrinsic and Extrinsic Dimensions 1.4.1 Intrinsic Dimension Estimation 1.4.2 Correlation Dimension 1.4.3 Capacity Dimension 1.4.4 Multiscale Estimation 1.5 0utline of the Book 1.5.1 Categories of DR Problems 1.5.2 Scope of This Book 1.5.3 0ther Topics Related to This Book 1.5.4 Artificial Surfaces for Testing DR Algorithms Part I Data Geometry Chapter 2 Preliminary Calculus on Manifolds 2.1 Linear Manifold 2.1.1 Subspace and Projection 2.1.2 Functions on Euclidean Spaces 2.1.3 Laplace Operator and Heat Diffusion Kernel 2.2 Differentiable Manifolds 2.2.1 Coordinate Systems and Parameterization 2.2.2 Tangent Spaces and Tangent Vectors 2.2.3 Riemannian Metrics 2.2.4 Geodesic Distance 2.3 Functions and Operators on Manifolds 2.3.1 Functions on Manifolds 2.3.2 0perators on Manifolds Chapter 3 Geometric Structure of High-Dirnensional 3.1 Similarity and Dissimilarity of Data 3.1.1 Neighborhood Definition 3.1.2 Algorithms for Construction of Neighborhood 3.2 Graphs on Data Sets 3.2.1 Undirected Graphs 3.2.2 Directed Graphs 3.2.3 Neighborhood and Data Graphs 3.3 Spectral Analysis of Graphs 3.3.1 Laplacian of Graphs 3.3.2 Laplacian on Weighted Graphs 3.3.3 Contracting Operator on Weighted Graph Chapter 4 Data Models and Structures of Kernels of DR 4.1 Data Models in Dimensionality Reduction 4.1.1 Input Data of First Type 4.1.2 Input Data of Second Type 4.1.3 Constraints on Output Data 4.1.4 Consistence of Data Graph 4.1.5 Robust Graph Connection Algorithm 4.2 Constructions of DR Kernels 4.2.1 DR Kernels of Linear Methods …… Part Ⅱ Linear Dimensionality reduction Part Ⅲ Nonlinear Dimensionality Reduction |
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