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IEEE Global Telecommunications Conference (GLOBECOM 2010)

作者： Senaratne, Damith Tellambura, Chintha Univ Alberta Dept Elect & Comp Engn Edmonton AB Canada

ISBN: (纸本)9781424456383

In this paper we examine the use of generalized singular value decomposition (GSVD) for coordinated beamforming in MIMO systems. GSVD facilitates joint decomposition of a class of matrices arising inherently in source-to-2 destination MIMO broadcast scenarios. GSVD allows two channels of suitable dimensionality to be jointly diagonalized, i.e. to be reduced to non-interfering virtual broadcast channels, through the use of jointly determined transmit precoding and receiver reconstruction matrices. Potential applications for GSVD-based beamforming can be found in MIMO broadcasting, as well as in MIMO relaying under all amplify-and-forward, decode-and-forward, and code-and-forward relay processing schemes. Several of them are highlighted here. We also present simulation-based performance analysis results to justify the use of GSVD for coordinated beamforming.

关键词： generalized singular value decomposition beamforming MIMO relaying MIMO broadcasting network coding

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generalized singular value decomposition AND ITS APPLICATIONS IN MODEL ANALYSIS

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INTERNATIONAL JOURNAL OF THEORETICAL AND APPLIED FINANCE 2006年第2期9卷 171-184页

作者： Dulov, Eugene V. Zapata, Humberto Sarria Andrianova, Natalia A. Univ Nacl Colombia Fac Ciencias Dept Math Bogota Colombia Ulyanovsk State Univ Dept Math & Mech Ulyanovsk 432700 Russia

For a variety of processes we can observe and register their characteristics, making up a sequence of measurement vectors or matrices (rectangular in general). Our goal is to extract some model dependent information using the available information. Such approaches are typical in technology (for a neat chemistry example, see [7, 9]) and model analysis like parameter identification of linear stochastic dynamic systems. Since a stochastic nature of financial and economic data is evident, we can extend this data analysis technique to a number of new applications. If we are successful, some kind of adaptive filter can be further constructed (similar to the classic Kalman's one, for example). Inspired with formal model parameters, we can apply this filter to process financial data like stock information to predict and verify how close is a mathematical model to a real-time data. Namely, when provided with a set measurements represented by matrices A(i) is an element of M-m,M-n(R), we have to estimate a problem dependent characteristic matrices [GRAPHIC] with P, Q being orthonormal matrices, B-i is an element of Mr(R), r <= min{m, n}. Formulated as above, the problem is usually called a generalized singular value decomposition (GSVD) problem and could be solved numerically [1, 2]. These matrices provide some basic information applicable for higher level automated problem solver or human interpretation.

关键词： generalized singular value decomposition adaptive filtering identification data prediction

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Joint generalized singular value decomposition and tensor decomposition for image super-resolution

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SIGNAL IMAGE AND VIDEO PROCESSING 2022年第3期16卷 849-856页

作者： Fang, Ying Ling, Bingo Wing-Kuen Lin, Yuxin Huang, Ziyin Chan, Yui-Lam Guangdong Univ Technol Fac Informat Engn Guangzhou 510006 Peoples R China Hong Kong Polytech Univ Dept Elect & Informat Engn Hong Kong Peoples R China

The existing methods for performing the super-resolution of the three-dimensional images are mainly based on the simple learning algorithms with the low computational powers and the complex deep learning neural network-based learning algorithms with the high computational powers. However, these methods are based on the prior knowledge of the images and require a large database of the pairs of the low-resolution images and the corresponding high-resolution images. To address this difficulty, this paper proposes a method based on the joint generalized singular value decomposition and tensor decomposition for performing the super-resolution. Here, it is not required to know the prior knowledge of the pairs of the low-resolution images and the corresponding high-resolution images. First, an image is represented as a tensor. Compared to the three-dimensional singular spectrum analysis, the spatial structure of the local adjacent pixels of the image is retained. Second, both the generalized singular value decomposition and the Tucker decomposition are applied to the tensor to obtain two low-resolution tensors. It is worth noting that the correlation between these two low-resolution tensors is preserved. Also, these two decompositions achieve the exact perfect reconstruction. Finally, the high-resolution image is reconstructed. Compared to the de-Hankelization of the three-dimensional singular spectrum analysis, the required computational complexity of the reconstruction of our proposed method is much lower. The computer numerical simulation results show that our proposed method achieves a higher peak signal-to-noise ratio than the existing methods.

关键词： Super-resolution Tensor generalized singular value decomposition Tucker decomposition

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A new approach to generalized singular value decomposition

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2005年第2期27卷 434-444页

作者： Friedland, S Univ Illinois Dept Math Stat & Comp Sci Chicago IL 60607 USA Univ Minnesota Inst Math & Applicat Minneapolis MN 55455 USA

We propose a new algorithm to find the generalized singular value decompositions of two matrices with the same number of columns. We discuss in detail the sensitivity of our algorithm to errors in the entries of the m... 详细信息

关键词： gene expression matrix singular value decomposition extended singular value decomposition generalized singular value decomposition

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Nonlinear discriminant analysis using kernel functions and the generalized singular value decomposition

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2005年第1期27卷 87-102页

作者： Park, CH Park, H Chungnam Natl Univ Dept Comp Sci & Engn Taejon 305764 South Korea Univ Minnesota Dept Comp Sci & Engn Minneapolis MN 55455 USA Natl Sci Fdn Arlington VA 22230 USA

Linear discriminant analysis ( LDA) has been widely used for linear dimension reduction. However, LDA has limitations in that one of the scatter matrices is required to be nonsingular and the nonlinearly clustered structure is not easily captured. In order to overcome the problems caused by the singularity of the scatter matrices, a generalization of LDA based on the generalized singular value decomposition ( GSVD) was recently developed. In this paper, we propose a nonlinear discriminant analysis based on the kernel method and the GSVD. The GSVD is applied to solve the generalized eigenvalue problem which is formulated in the feature space defined by a nonlinear mapping through kernel functions. Our GSVD-based kernel discriminant analysis is theoretically compared with other kernel-based nonlinear discriminant analysis algorithms. The experimental results show that our method is an effective nonlinear dimension reduction method.

关键词： dimension reduction generalized singular value decomposition kernel functions linear discriminant analysis nonlinear discriminant analysis

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Multilevel information cryptosystem using generalized singular value decomposition, optical interference, and devil's vortex Fresnel lens encoding

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OPTICS AND LASERS IN ENGINEERING 2024年 181卷

作者： Abuturab, Muhammad Rafiq Maulana Azad Natl Urdu Univ Sch Sci Dept Phys Hyderabad 500032 India

A novel multilevel information cryptosystem based on generalized singular value decomposition (GSVD), optical interference, and devil's vortex Fresnel lens (DVFL) encoding, is proposed. The proposed cryptosystem exploits a set of four fused LL sub-bands from four RGB images, which are converted into a CMYK image and split into C, M, Y, and K channels. The GSVD operation is used to produce five matrices from C and M channels, and five matrices from Y and K channels independently. A single-channel image of the first set formed by fusion of the two unitary matrices from each pair is gyrator transformed. In a similar fashion, the transformed images of a number of sets are combined into a complex image, which is then inverse gyrator transformed. The proposed system exploits optical interference of one phase-only mask (POM) as DVFL and two analytically produced POMs. The parameters of DVFL are used as remarkably sensitive decryption keys. Moreover, the individual keys and two POMs are used as decryption keys to advance the security against potential attacks. To avoid the strict alignment of the three POMs in different arms during the experiment, the summation of the three POMs is displayed on a single spatial light modulator (SLM). Further, gyrator transform does not require axial movements. Therefore, the proposed method avoids problems that result from misalignment. The proposed method can be implemented by using a hybrid optoelectronic system. Numerical simulation results demonstrate the practicability and effectiveness of the proposed system.

关键词： generalized singular value decomposition devil 's vortex Fresnel lens phase mask Optical interference Optical gyrator transform

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A tangent algorithm for computing the generalized singular value decomposition

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SIAM JOURNAL ON NUMERICAL ANALYSIS 1998年第5期35卷 1804-1832页

作者： Drmac, Z Univ Colorado Dept Comp Sci Boulder CO 80309 USA

We present two new algorithms for floating-point computation of the generalized singular values of a real pair (A, B) of full column rank matrices and for floating-point solution of the generalized eigenvalue problem Hx = lambda Mx with symmetric, positive definite matrices H and M. The pair (A, B) is replaced with an equivalent pair (A', B'), and the generalized singular values are computed as the singular values of the explicitly computed matrix F = A'B'(?1). The singular values of F are computed using the Jacobi method. The relative accuracy of the computed singular value approximations does not depend on column scalings of A and B;that is, the accuracy is nearly the same for all pairs (AD(1);BD2), with D-1, D-2 arbitrary diagonal, nonsingular matrices. Similarly, the pencil H - lambda M is replaced with an equivalent pencil H' - lambda M', and the eigenvalues of H - lambda M are computed as the squares of the singular values of G = L-H L-M(-1), where L-H, L-M are the Cholesky factors of H', M', respectively, and the matrix G is explicitly computed as the solution of a linear system of equations. For the computed approximation lambda + delta lambda of any exact eigenvalue lambda, the relative error \delta lambda\/lambda is of order p(n)epsilon max{min(Delta is an element of D) (kappa 2)(Delta H Delta);min(Delta is an element of D) (kappa 2) (Delta M Delta)}, where p(n) is a modestly growing polynomial of the dimension of the problem, epsilon is the round-off unit of floating-point arithmetic, D denotes the set of diagonal nonsingular matrices, and kappa(2)(.) is the spectral condition number. Furthermore, ?oating-point computation corresponds to an exact computation with H + delta H, M + delta M, where, for all i, j, \delta H-ij\/root HiiHjj and \delta M-ij\/root MiiMjj are of order of epsilon times a modest function of n.

关键词： generalized singular value decomposition generalized eigenvalue problem Jacobi method relative accuracy singular value decomposition

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Generalizing discriminant analysis using the generalized singular value decomposition

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IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 2004年第8期26卷 995-1006页

作者： Howland, P Park, H Univ Minnesota Dept Comp Sci & Engn Minneapolis MN 55455 USA

Discriminant analysis has been used for decades to extract features that preserve class separability. It is commonly defined as an optimization problem involving covariance matrices that represent the scatter within and between clusters. The requirement that one of these matrices be nonsingular limits its application to data sets with certain relative dimensions. We examine a number of optimization criteria, and extend their applicability by using the generalized singular value decomposition to circumvent the nonsingularity requirement. The result is a generalization of discriminant analysis that can be applied even when the sample size is smaller than the dimension of the sample data. We use classification results from the reduced representation to compare the effectiveness of this approach with some alternatives, and conclude with a discussion of their relative merits.

关键词： linear discriminant analysis latent semantic indexing principal component analysis generalized singular value decomposition QR decomposition trace optimization

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The generalized singular value decomposition and the method of particular solutions

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2008年第3期30卷 1278-1295页

作者： Betcke, Timo Univ Manchester Sch Math Manchester M13 9PL Lancs England

A powerful method for solving planar eigenvalue problems is the method of particular solutions (MPS), which is also well known under the name "point matching method." The implementation of this method usually depends on the solution of one of three types of linear algebra problems: singular value decomposition, generalized eigenvalue decomposition, or generalized singular value decomposition. We compare and give geometric interpretations of these different variants of the MPS. It turns out that the most stable and accurate of them is based on the generalized singular value decomposition. We present results to this effect and demonstrate the behavior of the generalized singular value decomposition in the presence of a highly ill-conditioned basis of particular solutions.

关键词： eigenvalues method of particular solutions point matching subspace angles generalized singular value decomposition

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Sensitivity analysis for the generalized singular value decomposition

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NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS 2013年第1期20卷 138-149页

作者： Chen, Xiao Shan Li, Wen S China Normal Univ Sch Math Sci Guangzhou 510631 Guangdong Peoples R China

In this paper, we discuss the sensitivity of multiple nonzero finite generalized singular values and the corresponding generalized singular matrix set of a real matrix pair analytically dependent on several parameters. From our results, the partial derivatives of multiple nonzero singular values and their left and right singular vector matrices are *** (c) 2012 John Wiley & Sons, Ltd.

关键词： generalized singular value decomposition generalized singular value sensitivity analysis

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