Many objects in our world can be electronically represented with high-dimensional data- speech signals, images, videos, electrical text documents.We often need to analyze a large amount of data and process them. However,due to the high dimension of these data, directly processing them using reg-ular systems may be too complicated and unstable to be feasible. In order toprocess high-dimensional data, dimensionality reduction technique becomescrucial. Dimensionality reduction is a method to represent high-dimensionaldata by their low-dimensional embeddings so that the low-dimensional data can be effectively used either in processing systems, or for better understand-ing. This technique has proved an important tool and has been widely used in many fields of data analysis, data mining, data visualization, and machine learning.
- 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
