Showing papers on "QR decomposition published in 2004"

Near-maximum-likelihood detection of MIMO systems using MMSE-based lattice reduction

Abstract: In recent publications the use of lattice-reduction for signal detection in multiple antenna systems has been proposed. In this paper, we adopt these lattice-reduction-aided schemes to the MMSE criterion. We show that an obvious way to do this is infeasible and propose an alternative method based on an extended system model, which in conjunction with simple successive interference cancellation nearly reaches the performance of maximum-likelihood detection. Furthermore, we demonstrate that, a sorted QR decomposition can significantly reduce the computational effort associated with lattice-reduction. Thus, the new algorithm clearly outperforms existing methods with comparable complexity.

Fast antenna subset selection in MIMO systems

Abstract: Multiple antenna wireless communication systems have recently attracted significant attention due to their higher capacity and better immunity to fading as compared to systems that employ a single-sensor transceiver. Increasing the number of transmit and receive antennas enables to improve system performance at the price of higher hardware costs and computational burden. For systems with a large number of antennas, there is a strong motivation to develop techniques with reduced hardware and computational costs. An efficient approach to achieve this goal is the optimal antenna subset selection. In this paper, we propose a fast antenna selection algorithm for wireless multiple-input multiple-output (MIMO) systems. Our algorithm achieves almost the same outage capacity as the optimal selection technique while having lower computational complexity than the existing nearly optimal antenna selection methods. The optimality of the proposed technique is established in several important specific cases. A QR decomposition-based interpretation of our algorithm is provided that sheds a new light on the optimal antenna selection problem.

MMSE-based lattice-reduction for near-ML detection of MIMO systems

Abstract: Recently the use of lattice-reduction for signal detection in multiple antenna systems has been proposed. In this paper, we adopt these lattice-reduction aided schemes to the MMSE criterion. We show that an obvious way to do this is suboptimum and propose an alternative method based on an extended system model. In conjunction with simple successive interference cancellation this scheme almost reaches the performance of maximum-likelihood detection. Furthermore, we demonstrate that the application of sorted QR decomposition (SQRD) as a initialization step can significantly reduce the computational effort associated with lattice-reduction. Thus, the new algorithm clearly outperforms existing methods with comparable complexity.

Numerical Analysis and Scientific Computation

Abstract: 1. Nonlinear Equations. Biscetion and Inverse Linear Interpolation. Newton's Method. The Fixed Point Theorem. Quadratic Convergence of Newton's Method. Variants of Newton's Method. Brent's Method. Effects of Finite Precision Arithmetic. Newton's Method for Systems. Broyden's Method. 2. Linear Systems. Gaussian Elimination with Partial Pivoting. The LU Decomposition. The LU Decomposition with Pivoting. The Cholesky Decomposition. Condition Numbers. The QR Decomposition. Householder Triangularization and the QR Decomposition. Gram-Schmidt Orthogonalization and the QR Decomposition. The Singular Value Decomposition. 3. Iterative Methods. Jacobi and Gauss-Seidel Iteration. Sparsity. Iterative Refinement. Preconditioning. Krylov Space Methods. Numerical Eigenproblems. 4. Polynomial Interpolation. Lagrange Interpolating Polynomials. Piecewise Linear Interpolation. Cubic Splines. Computation of the Cubic Spline Coefficients. 5. Numerical Integration. Closed Newton-Cotes Formulas. Open Newton-Cotes Formulas and Undetermined Coeffients. Gaussian Quadrature. Gauss-Chebyshev Quadrature. Radau and Lobatto Quadrature. Adaptivity and Automatic Integration. Romberg Integration. 6. Differential Equations. Numerical Differentiation. Euler's Method. Improved Euler's Method. Analysis of Explicit One-Step Methods. Taylor and Runge-Kutta Methods. Adaptivity and Stiffness. Multi-Step Methods. 7. Nonlinear Optimization. One-Dimensional Searches. The Method of Steepest Descent. Newton Methods for Nonlinear Optimization. Multiple Random Start Methods. Direct Search Methods. The Nelder-Mead Method. Conjugate Direction Methods. 8. Approximation Methods. Linear and Nonlinear Least Squares. The Best Approximation Problem. Best Uniform Approximation. Applications of the Chebyshev Polynomials. Afterword. Bibliography. Answers. Index.

IDR/QR: an incremental dimension reduction algorithm via QR decomposition

Abstract: Dimension reduction is critical for many database and data mining applications, such as efficient storage and retrieval of high-dimensional data. In the literature, a well-known dimension reduction scheme is Linear Discriminant Analysis (LDA). The common aspect of previously proposed LDA based algorithms is the use of Singular Value Decomposition (SVD). Due to the difficulty of designing an incremental solution for the eigenvalue problem on the product of scatter matrices in LDA, there is little work on designing incremental LDA algorithms. In this paper, we propose an LDA based incremental dimension reduction algorithm, called IDR/QR, which applies QR Decomposition rather than SVD. Unlike other LDA based algorithms, this algorithm does not require the whole data matrix in main memory. This is desirable for large data sets. More importantly, with the insertion of new data items, the IDR/QR algorithm can constrain the computational cost by applying efficient QR-updating techniques. Finally, we evaluate the effectiveness of the IDR/QR algorithm in terms of classification accuracy on the reduced dimensional space. Our experiments on several real-world data sets reveal that the accuracy achieved by the IDR/QR algorithm is very close to the best possible accuracy achieved by other LDA based algorithms. However, the IDR/QR algorithm has much less computational cost, especially when new data items are dynamically inserted.

The calculation of linear least squares problems

Abstract: We first survey componentwise and normwise perturbation bounds for the standard least squares (LS) and minimum norm problems. Then some recent estimates of the optimal backward error for an alleged solution to an LS problem are presented. These results are particularly interesting when the algorithm used is not backward stable. The QR factorization and the singular value decomposition (SVD), developed in the 1960s and early 1970s, remain the basic tools for solving both the LS and the total least squares (TLS) problems. Current algorithms based on Householder or Gram-Schmidt QR factorizations are reviewed. The use of the SVD to determine the numerical rank of a matrix, as well as for computing a sequence of regularized solutions, is then discussed. The solution of the TLS problem in terms of the SVD of the compound matrix $(b\ A)$ is described. Some recent algorithmic developments are motivated by the need for the efficient implementation of the QR factorization on modern computer architectures. This includes blocked algorithms as well as newer recursive implementations. Other developments come from needs in different application areas. For example, in signal processing rank-revealing orthogonal decompositions need to be frequently updated. We review several classes of such decompositions, which can be more efficiently updated than the SVD. Two algorithms for the orthogonal bidiagonalization of an arbitrary matrix were given by Golub and Kahan in 1965, one using Householder transformations and the other a Lanczos process. If used to transform the matrix $(b\ A)$ to upper bidiagonal form, this becomes a powerful tool for solving various LS and TLS problems. This bidiagonal decomposition gives a core regular subproblem for the TLS problem. When implemented by the Lanczos process it forms the kernel in the iterative method LSQR. It is also the basis of the partial least squares (PLS) method, which has become a standard tool in statistics. We present some generalized QR factorizations which can be used to solve different generalized least squares problems. Many applications lead to LS problems where the solution is subject to constraints. This includes linear equality and inequality constraints. Quadratic constraints are used to regularize solutions to discrete ill-posed LS problems. We survey these classes of problems and discuss their solution. As in all scientific computing, there is a trend that the size and complexity of the problems being solved is steadily growing. Large problems are often sparse or structured. Algorithms for the efficient solution of banded and block-angular LS problems are given, followed by a brief discussion of the general sparse case. Iterative methods are attractive, in particular when matrix-vector multiplication is cheap.

A Comparison of Pruning Algorithms for Sparse Least Squares Support Vector Machines

Abstract: Least Squares Support Vector Machines (LS-SVM) is aproven method for classification and function approximation In comparison to the standard Support Vector Machines (SVM) it only requires solving a linear system, but it lacks sparseness in the number of solution terms Pruning can therefore be applied Standard ways of pruning the LS-SVM consist of recursively solving the approximation problem and subsequently omitting data that have a small error in the previous pass and are based on support values We suggest a slightly adapted variant that improves the performance significantly We assess the relative regression performance of these pruning schemes in a comparison with two (for pruning adapted) subset selection schemes, -one based on the QR decomposition (supervised), one that searches the most representative feature vector span (unsupervised)-, random omission and backward selection on independent test sets in some benchmark experiments

On the shifted qr iteration applied to companion matrices

Abstract: We show that the shifted QR iteration applied to a companion matrix F maintains the weakly semiseparable structure of F. More precisely, if Ai iI = QiRi, Ai+1 := RiQi + iI, i = 0; 1; : : :, where A0 = F , then we prove that Qi, Ri and Ai are semiseparable matrices having semiseparability rank at most 1, 4 and 3, respectively. This structural property is used to design an algorithm for performing a single step of the QR iteration in just O(n) flops. The robustness and reliability of this algorithm is discussed. Applications to approximating polynomial roots are shown.

Efficient Kernel Discriminant Analysis via QR Decomposition

Abstract: Linear Discriminant Analysis (LDA) is a well-known method for feature extraction and dimension reduction. It has been used widely in many applications such as face recognition. Recently, a novel LDA algorithm based on QR Decomposition, namely LDA/QR, has been proposed, which is competitive in terms of classification accuracy with other LDA algorithms, but it has much lower costs in time and space. However, LDA/QR is based on linear projection, which may not be suitable for data with nonlinear structure. This paper first proposes an algorithm called KDA/QR, which extends the LDA/QR algorithm to deal with nonlinear data by using the kernel operator. Then an efficient approximation of KDA/QR called AKDA/QR is proposed. Experiments on face image data show that the classification accuracy of both KDA/QR and AKDA/QR are competitive with Generalized Discriminant Analysis (GDA), a general kernel discriminant analysis algorithm, while AKDA/QR has much lower time and space costs.

A low-rank IE-QR algorithm for matrix compression in volume integral equations

Abstract: A single-level matrix compression algorithm based on pivoted QR factorization with partial matrix assembling, which exploits the rank deficiency of matrix blocks for physically separated groups of basis functions, is presented for the volume integral equation solution of electromagnetic scattering from arbitrarily shaped dielectric bodies. For a system of N equations, an amount of work of the order O(N/sup 2/) has traditionally been required by the method of moments (MoM). The algorithm of the present paper reduces both computational complexity and storage requirement to O(N/sup 1.5/) with relatively less dependence on the integral equation kernel. Hence, the proposed algorithm is more practical for large-scale problems and can be implemented in a wide range of applications with few or no modifications.

A single-level dual rank IE-QR algorithm to model large microstrip antenna arrays

Abstract: A single-level dual rank IE-QR algorithm is introduced so that the resulting dense method of moments (MOM) matrix is efficiently compressed. For a system of N equations, an amount of work of the order O(N/sup 2/) has traditionally been required for both matrix assembly and matrix-vector multiplication in an iterative matrix solver. The algorithm of the present paper reduces the memory requirement and CPU time for both matrix assembly and matrix-vector multiplication to O(N/sup 3/2/) making it practical to solve for large antenna arrays with full wave approach. In conjunction with a "geometric-neighboring" preconditioner for matrix solution using GMRES, the current approach solves problems involving large antenna arrays using only a fraction of what are needed by conventional MOM both in term of memory and total CPU time.

FPGA based embedded processing architecture for the QRD-RLS algorithm

Abstract: A novel implementation of the QR decomposition based recursive least squares (RLS) algorithm on Altera Stratix FPGAs is presented. CORDIC (coordinate rotation by digital computer) operators are efficiently time-shared to perform the QR decomposition while consuming minimal resources. Back substitution is then performed on the embedded soft Nios processor by utilizing custom instructions to yield the final weight vectors. Analytical resource estimates along with actual implementation results illustrating the weight calculation delays are also presented.

Spectral Clustering for Robust Motion Segmentation

Abstract: In this paper, we propose a robust motion segmentation method using the techniques of matrix factorization and subspace separation. We first show that the shape interaction matrix can be derived using QR decomposition rather than Singular Value Decomposition(SVD) which also leads to a simple proof of the shape subspace separation theorem. Using the shape interaction matrix, we solve the motion segmentation problems by the spectral clustering techniques. We exploit multi-way Min-Max cut clustering method and provide a novel approach for cluster membership assignment. We further show that we can combine a cluster refinement method based on subspace separation with the graph clustering method to improve its robustness in the presence of noise. The proposed method yields very good performance for both synthetic and real image sequences.

Oct-tree-based multilevel low-rank decomposition algorithm for rapid 3-D parasitic extraction

Abstract: Fast parasitic extraction is an integral part of high-speed microelectronic simulation at the package and on-chip level. Integral equation methods and related fast solvers for the iterative solution of the resulting dense matrix systems have enabled linear time complexity and memory usage. However, these methods tend to have large disparities between setup and matrix-vector product times that affect their efficiency when applied to multiple excitation problems, i.e., problems with a large number of nets. For example, FastCap, which is based on the fast multipole method, has a significantly faster setup time than the multilevel QR decomposition-based IES/sup 3/, but relatively slow matrix-vector products. In this paper, we present a novel oct-tree-based QR compression technique for fast iterative solution. The regular cube structure of the fast multipole method and the QR compression scheme for interaction submatrices as in IES/sup 3/ are combined to achieve a predetermined compressible matrix-block structure and, consequently, superior memory, setup, and solve time efficiencies.

Two methods for decreasing the computational complexity of the MIMO ML decoder

Abstract: SUMMARY We propose use of QR factorization with sort and Dijkstra’s algorithm for decreasing the computational complexity of the sphere decoder that is used for ML detection of signals on the multi-antenna fading channel. QR factorization with sort decreases the complexity of searching part of the decoder with small increase in the complexity required for preprocessing part of the decoder. Dijkstra’s algorithm decreases the complexity of searching part of the decoder with increase in the storage complexity. The computer simulation demonstrates that the complexity of the decoder is reduced by the proposed methods significantly.

Robust partial pole assignment for vibrating systems with aerodynamic effects

Abstract: This paper proposes a novel algorithm for robust partial eigenvalue assignment (RPEVA) problem for a cubic matrix pencil arising from modeling of vibrating systems with aerodynamic effects. The RPEVA problem for a cubic pencil is the one of choosing suitable feedback matrices to reassign a few (say k < 3n) unwanted eigenvalues while leaving the remaining large number (3n - k) of them unchanged, in such a way that the the eigenvalues of the closed-loop matrix are as insensitive as possible to small perturbation of the data. The latter amounts to minimizing the condition number of the closed-loop eigenvector matrix. The problem is solved directly in the cubic matrix polynomial setting without making any transformation to a standard first-order state-space system. This allows us to take advantage of the exploitable structures such as the sparsity, definiteness, bandness, etc., very often offered by large practical problems. The major computational requirements are: (i)solutions of a small Sylvester equation, (ii) QR factorizations, and (iii) solutions or some standard least squares problems. The least-squares problems result from matrix rank-one and rank-two update techniques used in the algorithm for reassigning, respectively, simple and complex eigenvalues. The practical effectiveness of the method is demonstrated by implementational results on simulated data provided by the Boeing company.

PILOT: A fast algorithm for enhanced 3D parasitic capacitance extraction efficiency

Abstract: Integral-equation methodologies applied to extract parasitics for board, package, and on-chip structures involve solving a dense system of equations. In this paper, we present an improved matrix-compression technique for fast iterative solution of such dense systems, which applies QR decomposition on multilevel oct-tree-based interaction sub-matrices. The regular-tree structure of the fast-multipole method and the rank-revealing QR-based matrix-compression scheme are combined in order to achieve superior time and memory efficiency. As is demonstrated by the numerical-simulation results presented herein, the new algorithm is found to be faster and more memory efficient than both existing QR-based methods and FastCap. © 2004 Wiley Periodicals, Inc. Microwave Opt Technol Lett 41: 169–173, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20083

Linear projection methods in face recognition under unconstrained illuminations: a comparative study

Abstract: Face recognition under unconstrained illuminations (FR/I) received extensive study because of the existence of illumination subspace. P. Belhumer et al. (1996) presented a study on the comparison between principal component analysis (PCA) and subspace linear discriminant analysis (LDA) for this problem. PCA and subspace LDA are two well-known linear projection methods that can be characterized as trace optimization on scatter matrices. Generally, a linear projection method can be derived by applying a specific matrix analysis technique on specific scatter matrices under some optimization criterion. Several novel linear projection methods were proposed recently using generalized singular value decomposition or QR decomposition matrix analysis techniques [H. Park, et al., 2003], [J. Ye and Q. Li, 2004]. In this paper, we present a comparative study on these linear projection methods in FR/I. We further involve multiresolution analysis in the study. Our comparative study is expected to give a relatively comprehensive view on the performance of linear projection methods in FR/I problems.

Algorithms for Computing the QR Decomposition of a Set of Matrices with Common Columns

Abstract: The QR decomposition of a set of matrices which have common columns is investigated. The triangular factors of the QR decompositions are represented as nodes of a weighted directed graph. An edge between two nodes exists if and only if the columns of one of the matrices is a subset of the columns of the other. The weight of an edge denotes the computational complexity of deriving the triangular factor of the destination node from that of the source node. The problem is equivalent to constructing the graph and finding the minimum cost for visiting all the nodes. An algorithm which computes the QR decompositions by deriving the minimum spanning tree of the graph is proposed. Theoretical measures of complexity are derived and numerical results from the implementation of this and alternative heuristic algorithms are given.

Reduced complexity equalization schemes for zero padded OFDM systems

Abstract: Three reduced complexity equalization schemes for Zero-padded OFDM systems are described. These schemes guarantee Zero-Forcing (ZF) equalization irrespective of the channel nulls. Two of these schemes implement the minimum-norm ZF equalizer efficiently using QR decomposition. In the third scheme, the channel zeros are grouped as being inside or outside or on the unit circle. These groups are then equalized sequentially in a manner so as to tackle excess noise amplification. The three schemes are compared for their computational complexity and Bit Error Rate (BER) performance. It is shown that the attractive scheme depends on the system specifications. The BERComputations trade off occurring in the choice of the right algorithm is also highlighted.

A QR –method for computing the singular values via semiseparable matrices

Abstract: The standard procedure to compute the singular value decomposition of a dense matrix, first reduces it into a bidiagonal one by means of orthogonal transformations Once the bidiagonal matrix has been computed, the QR–method is applied to reduce the latter matrix into a diagonal one In this paper we propose a new method for computing the singular value decomposition of a real matrix In a first phase, an algorithm for reducing the matrix A into an upper triangular semiseparable matrix by means of orthogonal transformations is described A remarkable feature of this phase is that, depending on the distribution of the singular values, after few steps of the reduction, the largest singular values are already computed with a precision depending on the gaps between the singular values An implicit QR–method for upper triangular semiseparable matrices is derived and applied to the latter matrix for computing its singular values The numerical tests show that the proposed method can compete with the standard method (using an intermediate bidiagonal matrix) for computing the singular values of a matrix

Two resultant based methods computing the greatest common divisor of two polynomials

Abstract: In this paper we develop two resultant based methods for the computation of the Greatest Common Divisor (GCD) of two polynomials. Let S be the resultant Sylvester matrix of the two polynomials. We modified matrix S to S*, such that the rows with non-zero elements under the main diagonal, at every column, to be gathered together. We constructed modified versions of the LU and QR procedures which require only the of floating point operations than the operations performed in the general LU and QR algorithms. Finally, we give a bound for the error matrix which arises if we perform Gaussian elimination with partial pivoting to S*. Both methods are tested for several sets of polynomials and tables summarizing all the achieved results are given.

Reduced-complexity sphere decoding via detection ordering for linear multi-input multi-output channels

Abstract: Sphere decoding is a powerful approach for maximum-likelihood (ML) detection over Gaussian multi-input multi-output (MIMO) linear channels. We propose a new detection ordering approach, which minimizes the corresponding diagonal element of the upper-triangular matrix R over all possible column permutations in each step of the QR decomposition. Compared with the previously proposed V-BLAST ZF-DFE ordering approach, our approach has two major advantages: (1) it is efficiently embedded in the QR decomposition with a small computational overhead, rendering itself suitable for fast-varying channels, while the V-BLAST ZF-DFE ordering is not suitable for fast-varying channels since it incurs large computation overhead; (2) the sphere decoder with our proposed detection ordering achieves 17%-69% and 9%-59% reductions in the number of multiplications and the number of additions, respectively, in comparison to that with the V-BLAST ZF-DFE ordering.

Elimination of Redundant Cut Joint Constraints for Multibody System Models

Abstract: The problem of dependent cut joint constraints for kinematic loops in rigid multibody systems is addressed. The constraints are reduced taking into account the subalgebra generated by the screw system of the kinematic loop. The elimination of dependent constraint equations is based on constructing a basis matrix of the screw algebra generated by loop's screw system. This matrix is configuration independent and thus always valid. The determination of the sufficient constraints is achieved with a SVD or QR decomposition of this matrix. Unlike all other proposed approaches the presented method is singularity consistent because it is not the Jacobian which is decomposed, but instead a basis matrix for the loop algebra. Since this basis is obtained after a finite number of cross products the computational effort is negligible. Furthermore, because the elimination process is only necessary once in advance of the integration/simulation process, it proved valuable even if it does not remove all dependent constraints, as for paradoxical mechanisms.

Incremental and Decremental Least Squares Support Vector Machine and Its Application to Drug Design

Abstract: The least squares support vector machine (LS-SVM) has shown to exhibit excellent classification performance in many applications. In this paper, we propose an incremental and decremental LS-SVM based on updating and downdating the QR decomposition. It can efficiently compute the updated solution when data points are appended or removed. The experiment results illustrated that the proposed incremental algorithm efficiently produces the same solutions as those obtained by LS-SVM which recomputes the solution all over even for small changes in the data. For drug design, the results of each biochemistry laboratory test on a new compound can be iteratively included in the training set. This procedure can further improve precision in order to select the next best predicted organic compound. Instead of retraining entire data points, it is much efficient to update solution by incremental LS-SVM.