Geometric Structure of High-Dimensional Data and by Jianzhong Wang

By Jianzhong Wang

"Geometric constitution of High-Dimensional information and Dimensionality aid" adopts facts geometry as a framework to handle numerous equipment of dimensionality relief. as well as the advent to famous linear tools, the ebook additionally stresses the lately built nonlinear tools and introduces the purposes of dimensionality relief in lots of components, comparable to face acceptance, picture segmentation, information type, info visualization, and hyperspectral imagery info research. a number of tables and graphs are incorporated to demonstrate the tips, results, and shortcomings of the equipment. MATLAB code of all dimensionality relief algorithms is supplied to assist the readers with the implementations on pcs.

The booklet could be worthy for mathematicians, statisticians, machine scientists, and knowledge analysts. it's also a invaluable guide for different practitioners who've a uncomplicated heritage in arithmetic, statistics and/or computing device algorithms, like net seek engine designers, physicists, geologists, digital engineers, and economists.

Jianzhong Wang is a Professor of arithmetic at Sam Houston country college, U.S.A.

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The geometric approaches that exploit the intrinsic dimension of a data set are often based on fractal dimensions or the nearest neighbor (NN) distances. Although the geometric methods also utilize the geometry of data neighborhood, they do not require the parameters as input. Most of these methods estimate the intrinsic dimension by counting volume growth rate of an expanding sphere on the manifold. 4 Intrinsic and Extrinsic Dimensions 15 introduced in the following. 2 Correlation Dimension The correlation dimension [14, 15] perhaps is the most popular fractal dimension.

When is “nearest neighbor” meaningful? In: Proceeedings of the Seventh International Conference on Database Theory, Lecture Notes in Computer Science, vol. 1540, pp. 217 – 235. SpringerVerlag, Jerusalem, Israel (1999). : Low bounds for high dimensional nearest neighbor search and relaterd problems. In: Proceeedings of the thirtyfirst annual ACM symposium on Theory of Computing, Atlanta, GA, pp. 312 – 321. ACM Press, New York (1999). : High-dimensional data analysis: optional metrics and feature selection.

A brief review of manifold comes first, followed by a preliminary discussion of the spectral graph theory. These are the main tools to reveal the geometry of data. At the end of the first part, we briefly discuss the data models and the formats of the input data (the observed data) and output data (the DR data). In Part II, linear DR methods are introduced. Besides introducing the deterministic linear DR methods — principal component analysis (PCA) and classical multidimensional scaling (CMDS), we also present the random projection method.

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