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Dimensionality reduction, also known as manifold learning, is an area of machine learning used for extracting informative features from data for better representation of data or separation between classes. This book presents a cohesive review of linear and nonlinear dimensionality reduction and manifold learning. Three main aspects of dimensionality reduction are covered: spectral dimensionality reduction, probabilistic dimensionality reduction, and neural network-based dimensionality reduction, which have geometric, probabilistic, and information-theoretic points of view to dimensionality reduction, respectively. The necessary background and preliminaries on linear algebra, optimization, and kernels are also explained to ensure a comprehensive understanding of the algorithms. The tools introduced in this book can be applied to various applications involving feature extraction, image processing, computer vision, and signal processing. This book is applicable to a wide audience who would like to acquire a deep understanding of the various ways to extract, transform, and understand the structure of data. The intended audiences are academics, students, and industry professionals. Academic researchers and students can use this book as a textbook for machine learning and dimensionality reduction. Data scientists, machine learning scientists, computer vision scientists, and computer scientists can use this book as a reference. It can also be helpful to statisticians in the field of statistical learning and applied mathematicians in the fields of manifolds and subspace analysis. Industry professionals, including applied engineers, data engineers, and engineers in various fields of science dealing with machine learning, can use this as a guidebook for feature extraction from their data, as the raw data in industry often require preprocessing. The book is grounded in theory but provides thorough explanations and diverse examples to improve the reader’s comprehension of the advanced topics. Advanced methods are explained in a step-by-step manner so that readers of all levels can follow the reasoning and come to a deep understanding of the concepts. This book does not assume advanced theoretical background in machine learning and provides necessary background, although an undergraduate-level background in linear algebra and calculus is recommended.

Dimension reduction (Statistics) --- Machine learning. --- Learning, Machine --- Artificial intelligence --- Machine theory --- Dimensionality reduction (Statistics) --- Reduction, Dimension (Statistics) --- Reduction, Dimensionality (Statistics) --- Statistics

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This book describes for the first time a simulation method for the fast calculation of contact properties and friction between rough surfaces in a complete form. In contrast to existing simulation methods, the method of dimensionality reduction (MDR) is based on the exact mapping of various types of three-dimensional contact problems onto contacts of one-dimensional foundations. Within the confines of MDR, not only are three dimensional systems reduced to one-dimensional, but also the resulting degrees of freedom are independent from another. Therefore, MDR results in an enormous reduction of the development time for the numerical implementation of contact problems as well as the direct computation time and can ultimately assume a similar role in tribology as FEM has in structure mechanics or CFD methods, in hydrodynamics. Furthermore, it substantially simplifies analytical calculation and presents a sort of “pocket book edition” of the entirety contact mechanics. Measurements of the rheology of bodies in contact as well as their surface topography and adhesive properties are the inputs of the calculations. In particular, it is possible to capture the entire dynamics of a system – beginning with the macroscopic, dynamic contact calculation all the way down to the influence of roughness – in a single numerical simulation model. Accordingly, MDR allows for the unification of the methods of solving contact problems on different scales. The goals of this book are on the one hand, to prove the applicability and reliability of the method and on the other hand, to explain its extremely simple application to those interested.

Engineering. --- Continuum Mechanics and Mechanics of Materials. --- Operating Procedures, Materials Treatment. --- Engineering Design. --- Appl.Mathematics/Computational Methods of Engineering. --- Engineering mathematics. --- Materials. --- Engineering design. --- Structural control (Engineering). --- Ingénierie --- Mathématiques de l'ingénieur --- Matériaux --- Conception technique --- Contrôle des structures (Ingénierie) --- Contact mechanics. --- Dimension reduction (Statistics). --- Friction. --- Surface roughness -- Measurement -- Methodology. --- Chemical & Materials Engineering --- Engineering & Applied Sciences --- Applied Mathematics --- Materials Science --- Contact problems (Mechanics) --- Mechanics, Contact --- Applied mathematics. --- Continuum mechanics. --- Industrial engineering. --- Mechanics --- Bearings (Machinery) --- Tribology --- Mechanics, Applied --- Mechanics. --- Mechanics, Applied. --- Manufactures. --- Solid Mechanics. --- Manufacturing, Machines, Tools, Processes. --- Mathematical and Computational Engineering. --- Engineering --- Engineering analysis --- Mathematical analysis --- Design, Engineering --- Industrial design --- Strains and stresses --- Manufactured goods --- Manufactured products --- Products --- Products, Manufactured --- Commercial products --- Manufacturing industries --- Applied mechanics --- Engineering, Mechanical --- Engineering mathematics --- Classical mechanics --- Newtonian mechanics --- Physics --- Dynamics --- Quantum theory --- Mathematics --- Design

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"Geometric Structure of High-Dimensional Data and Dimensionality Reduction" adopts data geometry as a framework to address various methods of dimensionality reduction. In addition to the introduction to well-known linear methods, the book moreover stresses the recently developed nonlinear methods and introduces the applications of dimensionality reduction in many areas, such as face recognition, image segmentation, data classification, data visualization, and hyperspectral imagery data analysis. Numerous tables and graphs are included to illustrate the ideas, effects, and shortcomings of the methods. MATLAB code of all dimensionality reduction algorithms is provided to aid the readers with the implementations on computers.  The book will be useful for mathematicians, statisticians, computer scientists, and data analysts. It is also a valuable handbook for other practitioners who have a basic background in mathematics, statistics and/or computer algorithms, like internet search engine designers, physicists, geologists, electronic engineers, and economists. Jianzhong Wang is a Professor of Mathematics at Sam Houston State University, U.S.A.

Pattern perception. --- Dimension reduction (Statistics) --- Computational complexity. --- Visualization. --- Dimensionality reduction (Statistics) --- Reduction, Dimension (Statistics) --- Reduction, Dimensionality (Statistics) --- Computer science. --- Data structures (Computer science). --- Mathematical statistics. --- Data mining. --- Applied mathematics. --- Engineering mathematics. --- Computer Science. --- Data Mining and Knowledge Discovery. --- Probability and Statistics in Computer Science. --- Applications of Mathematics. --- Data Structures, Cryptology and Information Theory. --- Complexity, Computational --- Electronic data processing --- Machine theory --- Design perception --- Pattern recognition --- Form perception --- Perception --- Figure-ground perception --- Visualisation --- Imagery (Psychology) --- Imagination --- Visual perception --- Statistics --- Mathematics. --- Data structures (Computer scienc. --- Data Structures and Information Theory. --- Math --- Science --- Informatics --- Algorithmic knowledge discovery --- Factual data analysis --- KDD (Information retrieval) --- Knowledge discovery in data --- Knowledge discovery in databases --- Mining, Data --- Database searching --- Information structures (Computer science) --- Structures, Data (Computer science) --- Structures, Information (Computer science) --- File organization (Computer science) --- Abstract data types (Computer science) --- Engineering --- Engineering analysis --- Mathematical analysis --- Mathematics --- Statistical inference --- Statistics, Mathematical --- Probabilities --- Sampling (Statistics) --- Statistical methods