By Jianzhong Wang
"Geometric constitution of High-Dimensional info and Dimensionality aid" adopts facts geometry as a framework to handle quite a few tools of dimensionality aid. as well as the advent to famous linear equipment, the booklet additionally stresses the lately constructed nonlinear equipment and introduces the purposes of dimensionality relief in lots of components, akin to face acceptance, picture segmentation, facts category, information visualization, and hyperspectral imagery information research. a variety of tables and graphs are incorporated to demonstrate the information, results, and shortcomings of the tools. MATLAB code of all dimensionality relief algorithms is equipped to help the readers with the implementations on desktops.
The ebook might be helpful for mathematicians, statisticians, computing device scientists, and knowledge analysts. it's also a worthwhile guide for different practitioners who've a uncomplicated historical past in arithmetic, records and/or laptop algorithms, like web 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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Additional resources for Geometric Structure of High-Dimensional Data and Dimensionality Reduction
Example text
The weights then derive a sparse DR kernel. LTSA finds an embedding that aligns the tangent spaces by using the eigenvector optimization of the DR kernel. Laplacian Eigenmaps (Leigs) Laplacian eigenmaps (Leigs) [30] use spectral technique in DR processing. In Leigs, the high-dimensional data is still assumed to be laid on a lowdimensional manifold. The method first builds a graph that defines a neighborhood system on the data set, then constructs a kernel that approximates the Laplace-Beltrami operator on the manifold.
Data compression or source coding is the process of encoding data down to size smaller than their normal presentation. Because of the huge size of high-dimensional data, it is necessary to transmit the compressed data instead. There are two compression schemes: lossless and lossy. Lossy compression scheme can reach a higher compression rate than the lossless one, but, as a trade-off, will lose some information. Hence, the quality control for a lossy compression scheme is crucial. There is a close relation between DR and compression: a high-dimensional data can be first reduced to a low-dimensional data, and then compressed further.
It is often represented in the matrix form X = D,n [x1 · · · xn ] = [xij ]i,j=1 . The data set X spans a subspace of RD denoted by S = span {x1 , · · · , xn }, which is the column space of the data matrix X. 30 Chapter 2 Preliminary Calculus on Manifolds We denote the column space of a matrix M by Col(M ) and the row space of M by Row(M ). Hence, span {x1 , · · · , xn } = Col(X). 1. In this book, the notation xj denotes the jth component of the vector x, and the notation xj denotes the jth vector in the data set X , whose ith component is xi,j .