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Singular value decomposition

About: Singular value decomposition is a research topic. Over the lifetime, 14695 publications have been published within this topic receiving 339599 citations. The topic is also known as: SVD & Singular value decomposition; SVD.


Papers
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Journal ArticleDOI
TL;DR: A multilinear generalization of the best rank-R approximation problem for matrices, namely, the approximation of a given higher-order tensor, in an optimal least-squares sense, by a tensor that has prespecified column rank value, rowRank value, etc.
Abstract: In this paper we discuss a multilinear generalization of the best rank-R approximation problem for matrices, namely, the approximation of a given higher-order tensor, in an optimal least-squares sense, by a tensor that has prespecified column rank value, row rank value, etc For matrices, the solution is conceptually obtained by truncation of the singular value decomposition (SVD); however, this approach does not have a straightforward multilinear counterpart We discuss higher-order generalizations of the power method and the orthogonal iteration method

1,638 citations

Journal ArticleDOI
TL;DR: Identification algorithms based on the well-known linear least squares methods of gaussian elimination, Cholesky decomposition, classical Gram-Schmidt, modified Gram- Schmidt, Householder transformation, Givens method, and singular value decomposition are reviewed.
Abstract: Identification algorithms based on the well-known linear least squares methods of gaussian elimination, Cholesky decomposition, classical Gram-Schmidt, modified Gram-Schmidt, Householder transformation, Givens method, and singular value decomposition are reviewed. The classical Gram-Schmidt, modified Gram-Schmidt, and Householder transformation algorithms are then extended to combine structure determination, or which terms to include in the model, and parameter estimation in a very simple and efficient manner for a class of multivariate discrete-time non-linear stochastic systems which are linear in the parameters.

1,620 citations

Journal ArticleDOI
TL;DR: In this article, a singular value decomposition analysis of the TLS problem is presented, which provides a measure of the underlying problem's sensitivity and its relationship to ordinary least squares regression.
Abstract: Totla least squares (TLS) is a method of fitting that is appropriate when there are errors in both the observation vector $b (mxl)$ and in the data matrix $A (mxn)$. The technique has been discussed by several authors and amounts to fitting a "best" subspace to the points $(a^{T}_{i},b_{i}), i=1,\ldots,m,$ where $a^{T}_{i}$ is the $i$-th row of $A$. In this paper a singular value decomposition analysis of the TLS problem is presented. The sensitivity of the TLS problem as well as its relationship to ordinary least squares regression is explored. Aan algorithm for solving the TLS problem is proposed that utilizes the singular value decomposition and which provides a measure of the underlying problem''s sensitivity.

1,587 citations

Book
04 Aug 1999
TL;DR: This book discusses vector spaces, signal processing, and the theory of Constrained Optimization, as well as basic concepts and methods of Iterative Algorithms and Dynamic Programming.
Abstract: I. INTRODUCTION AND FOUNDATIONS. 1. Introduction and Foundations. II. VECTOR SPACES AND LINEAR ALGEBRA. 2. Signal Spaces. 3. Representation and Approximation in Vector Spaces. 4. Linear Operators and Matrix Inverses. 5. Some Important Matrix Factorizations. 6. Eigenvalues and Eigenvectors. 7. The Singular Value Decomposition. 8. Some Special Matrices and Their Applications. 9. Kronecker Products and the Vec Operator. III. DETECTION, ESTIMATION, AND OPTIMAL FILTERING. 10. Introduction to Detection and Estimation, and Mathematical Notation. 11. Detection Theory. 12. Estimation Theory. 13. The Kalman Filter. IV. ITERATIVE AND RECURSIVE METHODS IN SIGNAL PROCESSING. 14. Basic Concepts and Methods of Iterative Algorithms. 15. Iteration by Composition of Mappings. 16. Other Iterative Algorithms. 17. The EM Algorithm in Signal Processing. V. METHODS OF OPTIMIZATION. 18. Theory of Constrained Optimization. 19. Shortest-Path Algorithms and Dynamic Programming. 20. Linear Programming. APPENDIXES. A. Basic Concepts and Definitions. B. Completing the Square. C. Basic Matrix Concepts. D. Random Processes. E. Derivatives and Gradients. F. Conditional Expectations of Multinomial and Poisson r.v.s.

1,568 citations

Journal ArticleDOI
TL;DR: This work introduces a new dimensionality reduction technique which it is called Piecewise Aggregate Approximation (PAA), and theoretically and empirically compare it to the other techniques and demonstrate its superiority.
Abstract: The problem of similarity search in large time series databases has attracted much attention recently. It is a non-trivial problem because of the inherent high dimensionality of the data. The most promising solutions involve first performing dimensionality reduction on the data, and then indexing the reduced data with a spatial access method. Three major dimensionality reduction techniques have been proposed: Singular Value Decomposition (SVD), the Discrete Fourier transform (DFT), and more recently the Discrete Wavelet Transform (DWT). In this work we introduce a new dimensionality reduction technique which we call Piecewise Aggregate Approximation (PAA). We theoretically and empirically compare it to the other techniques and demonstrate its superiority. In addition to being competitive with or faster than the other methods, our approach has numerous other advantages. It is simple to understand and to implement, it allows more flexible distance measures, including weighted Euclidean queries, and the index can be built in linear time.

1,550 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023562
20221,182
2021628
2020786
2019899
2018817