Showing papers on "QR decomposition published in 2006"

Fast linear algebra is stable

Abstract: In an earlier paper, we showed that a large class of fast recursive matrix multiplication algorithms is stable in a normwise sense, and that in fact if multiplication of $n$-by-$n$ matrices can be done by any algorithm in $O(n^{\omega + \eta})$ operations for any $\eta > 0$, then it can be done stably in $O(n^{\omega + \eta})$ operations for any $\eta > 0$. Here we extend this result to show that essentially all standard linear algebra operations, including LU decomposition, QR decomposition, linear equation solving, matrix inversion, solving least squares problems, (generalized) eigenvalue problems and the singular value decomposition can also be done stably (in a normwise sense) in $O(n^{\omega + \eta})$ operations.

Triangular systolic array with reduced latency for QR-decomposition of complex matrices

Abstract: The novel CORDIC-based architecture of the triangular systolic array for QRD of large size complex matrices is presented. The proposed architecture relies on QRD using a three angle complex rotation approach that provides significant reduction of latency (systolic operation time) and makes the QRD in such a way that the upper triangular matrix R has only real diagonal elements.

Triangular Systolic Array withReduced Latency for QR-decomposition ofComplex Matrices

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Branch-and-Bound Algorithms for Computing the Best-Subset Regression Models

Abstract: An efficient branch-and-bound algorithm for computing the best-subset regression models is proposed. The algorithm avoids the computation of the whole regression tree that generates all possible subset models. It is formally shown that if the branch-and-bound test holds, then the current subtree together with its right-hand side subtrees are cut. This reduces significantly the computational burden of the proposed algorithm when compared to an existing leaps-and-bounds method which generates two trees. Specifically, the proposed algorithm, which is based on orthogonal transformations, outperforms by O(n3) the leaps-and-bounds strategy. The criteria used in identifying the best subsets are based on monotone functions of the residual sum of squares (RSS) such as R2, adjusted R2, mean square error of prediction, and Cp. Strategies and heuristics that improve the computational performance of the proposed algorithm are investigated. A computationally efficient heuristic version of the branch-and-bound strategy ...

Simple LU and QR based non-orthogonal matrix joint diagonalization

Abstract: A class of simple Jacobi-type algorithms for non-orthogonal matrix joint diagonalization based on the LU or QR factorization is introduced. By appropriate parametrization of the underlying manifolds, i.e. using triangular and orthogonal Jacobi matrices we replace a high dimensional minimization problem by a sequence of simple one dimensional minimization problems. In addition, a new scale-invariant cost function for non-orthogonal joint diagonalization is employed. These algorithms are step-size free. Numerical simulations demonstrate the efficiency of the methods.

Improved k-best sphere decoding algorithms for MIMO systems

Abstract: Multiple-input multiple-output (MIMO) technique is a key enabling technology for today's high-rate wireless communications. The sphere decoding algorithm (SDA) has been used for maximum likelihood (ML) detection in MIMO systems. However, it suffers from variable computation complexity and non-fixed throughput. Therefore, the k-best sphere decoding algorithm is proposed for MIMO detections for its less complexity and fixed throughput. The disadvantage for k-best SDA is that it has performance loss due to the reason that ML detection is not guaranteed. In this paper, we propose some improved k-best sphere decoding algorithms which improve the MIMO detection performance. Simulation results show that by applying sorted QR decomposition for the channel matrix, and/or introducing dynamic K values for different layers, our improved algorithms can achieve 1-2 dB detection performance gain for 4 /spl times/ 4 64QAM MIMO systems over the traditional k-best SDA without introducing extra computational complexity.

Pseudo-Inverse MMSE Based QRD-M Algorithm for MIMO OFDM

Abstract: We propose a pseudo-inverse minimum mean squared error (MMSE) based QR decomposition (QRD) M-algorithm (QRD-M) for multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) systems. The proposed algorithm can effectively mitigate the detrimental effect of MIMO channel ill-conditions on the bit error rate (BER) performance. As a result, the number of retained branches (M value) can be reduced to half of the original QRD-M algorithm in Kyeong Jin Kim and R.A. Iltis (2002) in order to achieve the same performance for both the uncoded and coded systems. The detection complexity is therefore greatly reduced

Soft-Output Sphere Decoding: Performance and Implementation Aspects

Abstract: Multiple-input multiple-output (MIMO) detection algorithms providing soft information for a subsequent channel decoder pose significant implementation challenges due to their high computational complexity. In this paper, we show how sphere decoding can be used as an efficient tool to implement soft-output MIMO detection with flexible trade-offs between computational complexity and (error rate) performance. In particular, we demonstrate that single tree search, ordered QR decomposition, channel matrix regularization, and log-likelihood ratio clipping are the key ingredients for realizing soft-output MIMO detectors with near max-log performance at a computational complexity that is reasonably close to that of hard-output sphere decoding.

Computational Workloads for Commonly Used Signal Processing Kernels

Abstract: : In the course of designing or evaluating signal processing algorithms, one often must determine the computational workload needed to implement the algorithms on a digital computer. The floating-point operation (flop) counts for real versions of the most common signal processing kernels are well documented. However, the flop counts for kernels operating on complex inputs are not as readily found. This report collects the flop count expressions for both real and complex kernels and also presents brief outlines of the derivations for the flop count expressions. Specifically, the following computational kernels are addressed: (1) the dimensions of the two multiplicands (m x n and n x p) for the matrix-matrix multiplication; (2) the length of the vector n for the fast Fourier transform; (3) the size of the triangular system n for forward and back substitutions; (4) the dimensions of the input matrix m x n for the Householder QR decomposition, eigenvalue decomposition, and singular value decomposition.

MIQR: A Multilevel Incomplete QR Preconditioner for Large Sparse Least-Squares Problems

Abstract: This paper describes a multilevel incomplete QR factorization for solving large sparse least-squares problems. The algorithm builds the factorization by exploiting structural orthogonality in general sparse matrices. At any given step, the algorithm finds an independent set of columns, i.e., a set of columns that have orthogonal patterns. The other columns are then block orthogonalized against columns of the independent set, and the process is repeated recursively for a certain number of levels on these remaining columns. The final level matrix is processed with a standard QR or incomplete QR factorization. Dropping strategies are employed throughout the levels in order to maintain a good level of sparsity. A few improvements to this basic scheme are explored. Among these is the relaxation of the requirement of independent sets of columns. Numerical tests are proposed which compare this scheme with the standard incomplete QR preconditioner, the robust incomplete factorization preconditioner, and the algebraic recursive multilevel solver (on normal equations).

Efficient pre-processing in the parallel block-Jacobi SVD algorithm

Abstract: One way, how to speed up the computation of the singular value decomposition of a given matrix [email protected]?C^m^x^n,m>=n, by the parallel two-sided block-Jacobi method, consists of applying some pre-processing steps that would concentrate the Frobenius norm near the diagonal. Such a concentration should hopefully lead to fewer outer parallel iteration steps needed for the convergence of the entire algorithm. It is shown experimentally, that the QR factorization with the complete column pivoting, optionally followed by the LQ factorization of the R-factor, can lead to a substantial decrease of the number of outer parallel iteration steps, whereby the details depend on the condition number and on the distribution of singular values including their multiplicity. A subset of ill-conditioned matrices has been identified, for which the dynamic ordering becomes inefficient. Best results in numerical experiments performed on the cluster of personal computers were achieved for well-conditioned matrices with a multiple minimal singular value, where the number of parallel iteration steps was reduced by two orders of magnitude. However, the gain in speed, as measured by the total parallel execution time, depends decisively on the implementation of the distributed QR and LQ factorizations on a given parallel architecture. In general, the reduction of the total parallel execution time up to one order of magnitude has been achieved.

A Fixed-Point Implementation for QR Decomposition

Abstract: Matrix triangularization and orthogonalization are prerequisites to solving least square problems and find applications in a wide variety of communication systems and signal processing applications such as MIMO systems and matrix inversion. QR decomposition using modified Gram-Schmidt (MGS) orthogonalization is one of the numerically stable techniques used in this regard. This paper presents a fixed point implementation of QR decomposition based on MGS algorithm using a novel LUT based approach. The proposed architecture is based on log-domain arithmetic operations. The error performance of various fixed-point arithmetic operations has been discussed and optimum LUT sizes are presented based on simulation results for various fractional-precisions. The proposed architecture also paves way for an efficient parallel VLSI implementation of QR decomposition using MGS

A fast identification algorithm for box-cox transformation based radial basis function neural network

Abstract: In this letter, a Box-Cox transformation-based radial basis function (RBF) neural network is introduced using the RBF neural network to represent the transformed system output. Initially a fixed and moderate sized RBF model base is derived based on a rank revealing orthogonal matrix triangularization (QR decomposition). Then a new fast identification algorithm is introduced using Gauss-Newton algorithm to derive the required Box-Cox transformation, based on a maximum likelihood estimator. The main contribution of this letter is to explore the special structure of the proposed RBF neural network for computational efficiency by utilizing the inverse of matrix block decomposition lemma. Finally, the Box-Cox transformation-based RBF neural network, with good generalization and sparsity, is identified based on the derived optimal Box-Cox transformation and a D-optimality-based orthogonal forward regression algorithm. The proposed algorithm and its efficacy are demonstrated with an illustrative example in comparison with support vector machine regression.

Singular value decomposition beamforming for a multiple-input-multiple-output communication system

Abstract: For one embodiment, a MIMO (Multiple-Input-Multiple-Output) communication system where a first sequence of beamformed signals is transmitted, a beamformed channel is observed in response to transmission of the first sequence of beamformed signals, a QR decomposition of the observed beamformed channel is performed to provide a unitary matrix, and a second sequence of beamformed signals is transmitted using as a beamformer the unitary matrix. Other embodiments are described and claimed.

When Fisher meets Fukunaga-Koontz: A New Look at Linear Discriminants

Abstract: The Fisher Linear Discriminant (FLD) is commonly used in pattern recognition. It finds a linear subspace that maximally separates class patterns according to Fisher’s Criterion. Several methods of computing the FLD have been proposed in the literature, most of which require the calculation of the so-called scatter matrices. In this paper, we bring a fresh perspective to FLD via the Fukunaga-Koontz Transform (FKT). We do this by decomposing the whole data space into four subspaces, and show where Fisher’s Criterion is maximally satisfied. We prove the relationship between FLD and FKT analytically, and propose a method of computing the most discriminative subspace. This method is based on the QR decomposition, which works even when the scatter matrices are singular, or too large to be formed. Our method is general and may be applied to different pattern recognition problems. We validate our method by experimenting on synthetic and real data.

Digital Hardware Aspects of Multiantenna Algorithms

Abstract: The field of wireless communication is growing rapidly, with new requirements for the next generation of mobile and wireless communications technology. In order to achieve the capacities needed for future wireless systems, the design and implementation of advanced communications techniques such as multiantenna systems is required. These systems are realized by computationally complex algorithms, requiring new digital hardware architectures to be developed. The development of efficient and scalable hardware building blocks for realization of multiantenna algorithms is the focus of this thesis. The first part of the thesis deals with the implementation of complex valued division. Two architectures implementing a numerically robust algorithm for computing complex valued division with standard arithmetic units are presented. The first architecture is based on a parallel computation scheme offering high throughput rate and low latency, while the second architecture is based on a resource conservative time-multiplexed computation scheme offering good throughput rate. The two implementations are compared to an implementation of a CORDIC based complex valued division. The second part of the thesis discusses implementation aspects of fundamental matrix operations found in many multiantenna algorithms. Four matrix operations were implemented; triangular matrix inversion, QR-decomposition, matrix inversion, and singular value decomposition. Matrix operations are usually implemented using large arrays of processors, which are difficult to scale and consume a lot of resources. In this thesis a method based on the data flow was applied to map the algorithms to scalable linear arrays. An even more resource conservative design based on a single processing element was also derived. All the architectures are capable of handling complex valued data necessary for the implementation of communication algorithms. In the third part of the thesis, developed building blocks are used to implement the Capon beamformer algorithm. Two architectures are presented; the first architecture is based on a linear data flow, while the second architecture utilizes the single processing element architecture. The Capon beamformer implementation is going to be used in a channel sounder to determine the direction-of-arrival of impinging signals. Therefore it was important to derive and implement flexible and scalable architectures to be able to adapt to different measuring scenarios. The linear flow architecture was implemented and tested with measured data from the channel sounder. By analyzing each block in the design, a minimum wordlength design could be derived. The fourth part of the thesis presents a design methodology for hardware implementation on FPGA. (Less)

Indefinite QR Factorization

Abstract: Let A be a Hermitian positive definite matrix given by its rectangular factor G such that A=G*G. It is well known that the Cholesky factorization of A is equivalent to the QR factorization of G. In this paper, an analogue of the QR factorization for Hermitian indefinite matrices is constructed. This problem has been considered by many authors, but the problem of zero diagonal elements has not been solved so far. Here we show how to overcome this difficulty.

On sampling algorithms in multilevel QR factorization method for magnetoquasistatic analysis of integrated circuits over multilayered lossy substrates

Abstract: This paper proposes improvements in the row and column samplings of the multilevel QR factorization method known as IES3, a fast integral equation solver previously proposed for efficient three-dimensional (3-D) parameter extraction. First, a rigorous Gram-Schmidt row sampling is developed to replace the row sampling algorithm in IES3, leading to a more stable algorithm. Second, to further enhance the efficiency of column sampling, a new scheme based on the idea of locating the interpolation points is presented. Error analyses indicate that the proposed schemes have higher accuracies than the original sampling in IES3, especially when the number of sampled points is small. The IES3 that uses one of these improved algorithms is called improved multilevel matrix QR factorization (IMLMQRF). These IMLMQRFs are applied in the magnetoquasistatic analysis of printed circuits on multilayered lossy medium for extractions of inductances and resistances. The frequency dependency of such parameters is also illustrated

Efficient detection algorithm for 2Nx2N MIMO systems using alamouti code and QR decomposition

Abstract: We propose an efficient 2Ntimes2N MIMO detection algorithm where the transmit signals are grouped in pairs and separately coded using the standard Alamouti space-time code. At the receiver, one or more QR decompositions are performed and the upper triangular property of the R matrices so obtained is exploited in order to successively decode the transmitted symbols starting with those interference-free symbols corresponding to the last two rows and columns of R. Bit-error-rate simulation results, for a 4times4 MIMO system and a bandwidth efficiency of 8 bits/s/Hz, show that the proposed technique, while less complex than ordered MMSE V-BLAST, outperforms the latter by 2-6 dB at a BER of 10-4

Wavelets Based Neural Network for Function Approximation

Abstract: In this paper, a new type of WNN is proposed to enhance the function approximation capability. In the proposed WNN, the nonlinear activation function is a linear combination of wavelets, that can be updated during the networks training process. As a result the approximate error is significantly decreased. The BP algorithm and the QR decomposition based training method for the proposed WNN is derived. The obtained results indicate that this new type of WNN exhibits excellent learning ability compared to the conventional ones.

Adaptive nonlinear discriminant analysis by regularized minimum squared errors

Abstract: Kernelized nonlinear extensions of Fisher's discriminant analysis, discriminant analysis based on generalized singular value decomposition (LDA/GSVD), and discriminant analysis based on the minimum squared error formulation (MSE) have recently been widely utilized for handling undersampled high-dimensional problems and nonlinearly separable data sets. As the data sets are modified from incorporating new data points and deleting obsolete data points, there is a need to develop efficient updating and downdating algorithms for these methods to avoid expensive recomputation of the solution from scratch. In this paper, an efficient algorithm for adaptive linear and nonlinear kernel discriminant analysis based on regularized MSE, called adaptive KDA/RMSE, is proposed. In adaptive KDA/RMSE, updating and downdating of the computationally expensive eigenvalue decomposition (EVD) or singular value decomposition (SVD) is approximated by updating and downdating of the QR decomposition achieving an order of magnitude speed up. This fast algorithm for adaptive kernelized discriminant analysis is designed by utilizing regularization techniques and the relationship between linear and nonlinear discriminant analysis and the MSE. In addition, an efficient algorithm to compute leave-one-out cross validation is also introduced by utilizing downdating of KDA/RMSE.

Method and apparatus for quasi maximum likelihood MIMO detection

Abstract: Techniques for MIMO detection that has performance approaching that of a maximum likelihood (ML) algorithm while having a reduced computational complexity are described. One method of an embodiment decomposes a MIMO channel into a 2×2 system. QR decomposition and maximum likelihood detection may be performed on the 2×2 system. Thereafter, certain metrics are combined and either a soft demodulation or decision and cancellation follows. Other embodiments are described and claimed.

Multiple-QR-Decomposition Assisted Group Detection for Reduced-Complexity-and-Latency MIMO-OFDM Receivers

Abstract: We propose a new multiple-QR-decomposition assisted group detection (Multi-QRD-GD) algorithm for the development of reduced-complexity-and-latency multi-input multi-output (MIMO) orthogonal frequency division multiplexing (OFDM) receivers. In addition, we investigate an effect of adaptive grouping (AG) based on received signal-to-noise power ratio for Multi-QRD-GD. In this paper, we describe the functions of the Multi-QRD-GD algorithm and AG scheme. Then, we compare bit error rate (BER) performances of the proposed Multi-QRD-GD and conventional detection algorithms by computer simulations. In addition, we compare the computational complexities of the algorithms.

Wavelets based neural network for function approximation

Abstract: In this paper, a new type of WNN is proposed to enhance the function approximation capability. In the proposed WNN, the nonlinear activation function is a linear combination of wavelets, that can be updated during the networks training process. As a result the approximate error is significantly decreased. The BP algorithm and the QR decomposition based training method for the proposed WNN is derived. The obtained results indicate that this new type of WNN exhibits excellent learning ability compared to the conventional ones.

A Novel Detection Algorithm Using the Sorted QR Decomposition Based on Log-Likelihood Ratio in V-BLAST Systems

Abstract: In this paper, we propose a novel detection algorithm using log-likelihood ratio (LLR) to detect signals in V-BLAST systems. This algorithm utilizes the sorted QR decomposition (SQRD) of the channel matrix, and applies LLR to determine the order of detection. Simulation results show that the proposed algorithm provides a better performance than the conventional SQRD with a few additional computations. Approximately, the average BER performance of our algorithm is better than that of the conventional SQRD algorithm by 6 dB for BPSK and by 2 dB for QPSK respectively at 10-3 target BER

Updating and downdating an upper trapezoidal sparse orthogonal factorization

Abstract: We describe how to update and downdate an upper trapezoidal sparse orthogonal factorization, namely the sparse QR factorization of A T k where A k is a 'tall and thin' full column rank matrix formed with a subset of the columns of a fixed matrix A. In order to do this, we have adapted Saunders' techniques of the early 1970s for square matrices, to rectangular matrices (with fewer columns than rows) by using the static data structure of George and Heath of the early 1980s but allowing row downdating on it. An implicitly determined column permutation allows us to dispense with computing a new ordering after each update/downdate; it fits well into the LINPACK downdating algorithm and ensures that the updated trapezoidal factor will remain sparse. We give all the necessary formulae even if the orthogonal factor is not available, and we comment on our implementation using the sparse toolbox of MATLAB 5.