Showing papers on "QR decomposition published in 2005"

A two-stage linear 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 involving high-dimensional data, such as image and text classification. An intrinsic limitation of classical LDA is the so-called singularity problems; that is, it fails when all scatter matrices are singular. Many LDA extensions were proposed in the past to overcome the singularity problems. Among these extensions, PCA+LDA, a two-stage method, received relatively more attention. In PCA+LDA, the LDA stage is preceded by an intermediate dimension reduction stage using principal component analysis (PCA). Most previous LDA extensions are computationally expensive, and not scalable, due to the use of singular value decomposition or generalized singular value decomposition. In this paper, we propose a two-stage LDA method, namely LDA/QR, which aims to overcome the singularity problems of classical LDA, while achieving efficiency and scalability simultaneously. The key difference between LDA/QR and PCA+LDA lies in the first stage, where LDA/QR applies QR decomposition to a small matrix involving the class centroids, while PCA+LDA applies PCA to the total scatter matrix involving all training data points. We further justify the proposed algorithm by showing the relationship among LDA/QR and previous LDA methods. Extensive experiments on face images and text documents are presented to show the effectiveness of the proposed algorithm.

Equal-diagonal QR decomposition and its application to precoder design for successive-cancellation detection

Abstract: In multiple-input multiple-output (MIMO) multiuser detection theory, the QR decomposition of the channel matrix H can be used to form the back-cancellation detector. In this paper, we propose an optimal QR decomposition, which we call the equal-diagonal QR decomposition, or briefly the QRS decomposition. We apply the decomposition to precoded successive-cancellation detection, where we assume that both the transmitter and the receiver have perfect channel knowledge. We show that, for any channel matrix H, there exists a unitary precoder matrix S, such that HS=QR, where the nonzero diagonal entries of the upper triangular matrix R in the QR decomposition of HS are all equal to each other. The precoder and the resulting successive-cancellation detector have the following properties. a) The minimum Euclidean distance between two signal points at the channel output is equal to the minimum Euclidean distance between two constellation points at the precoder input up to a multiplicative factor that equals the diagonal entry in the R-factor. b) The superchannel HS naturally exhibits an optimally ordered column permutation, i.e., the optimal detection order for the vertical Bell Labs layered space-time (V-BLAST) detector is the natural order. c) The precoder S minimizes the block error probability of the QR successive cancellation detector. d) A lower and an upper bound for the free distance at the channel output is expressible in terms of the diagonal entries of the R-factor in the QR decomposition of a channel matrix. e) The precoder S maximizes the lower bound of the channel's free distance subject to a power constraint. f) For the optimal precoder S, the performance of the QR detector is asymptotically (at large signal-to-noise ratios (SNRs)) equivalent to that of the maximum-likelihood detector (MLD) that uses the same precoder. Further, We consider two multiplexing schemes: time-division multiple access (TDMA) and orthogonal frequency-division multiplexing (OFDM). We d

FPGA Implementation of Matrix Inversion Using QRD-RLS Algorithm

Abstract: This paper presents a novel architecture for matrix inversion by generalizing the QR decomposition-based recursive least square (RLS) algorithm. The use of Squared Givens rotations and a folded systolic array makes this architecture very suitable for FPGA implementation. Input is a 4 × 4 matrix of complex, floating point values. The matrix inversion design can achieve throughput of 0.13M updates per second on a state of the art Xilinx Virtex4 FPGA running at 115 MHz. Due to the modular partitioning and interfacing between multiple Boundary and Internal processing units, this architecture is easily extendable for other matrix sizes.

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

Abstract: Dimension reduction is a critical data preprocessing step 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 algorithm 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 has been little work on designing incremental LDA algorithms that can efficiently incorporate new data items as they become available. 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 error rate on the reduced dimensional space. Our experiments on several real-world data sets reveal that the classification error rate achieved by the IDR/QR algorithm is very close to the best possible one achieved by other LDA-based algorithms. However, the IDR/QR algorithm has much less computational cost, especially when new data items are inserted dynamically.

Parallel out-of-core computation and updating of the QR factorization

Abstract: This article discusses the high-performance parallel implementation of the computation and updating of QR factorizations of dense matrices, including problems large enough to require out-of-core computation, where the matrix is stored on disk. The algorithms presented here are scalable both in problem size and as the number of processors increases. Implementation using the Parallel Linear Algebra Package (PLAPACK) and the Parallel Out-of-Core Linear Algebra Package (POOCLAPACK) is discussed. The methods are shown to attain excellent performance, in some cases attaining roughly 80% of the “realizable” peak of the architectures on which the experiments were performed.

Fast and stable QR eigenvalue algorithms for generalized companion matrices and secular equations

Abstract: We introduce a class ** of n×n structured matrices which includes three well-known classes of generalized companion matrices: tridiagonal plus rank-one matrices (comrade matrices), diagonal plus rank-one matrices and arrowhead matrices. Relying on the structure properties of **, we show that if A ∈ ** then A′=RQ ∈ **, where A=QR is the QR decomposition of A. This allows one to implement the QR iteration for computing the eigenvalues and the eigenvectors of any A ∈ ** with O(n) arithmetic operations per iteration and with O(n) memory storage. This iteration, applied to generalized companion matrices, provides new O(n2) flops algorithms for computing polynomial zeros and for solving the associated (rational) secular equations. Numerical experiments confirm the effectiveness and the robustness of our approach.

A scalable pipelined complex valued matrix inversion architecture

Abstract: This paper presents a fast, pipelined and scalable hardware architecture for inverting complex valued matrices. The matrix inversion algorithm involves, a QR-factorization based on the squared Givens rotations algorithm, the application of a recurrence algorithm for inversion of an upper triangular matrix R, and a matrix multiplication of R/sup -1/ with Q. We show that traditional triangular array architectures employing O(n/sup 2/) communicating processors can be mapped onto a scalable linear array architecture with only O(n) processors. The linear array architecture avoids drawbacks such as non-scalability, large area consumption and low throughput rate. The architecture is implemented using arithmetic operations with 12 bit fixed-point representation. The hardware implementation will be used as a core processor in a real-time smart antenna system.

A Comparative Study of QRD-M Detection and Sphere Decoding for MIMO-OFDM Systems

Abstract: We present a comparative study of two tree search based detection algorithms, namely, the M-algorithm combined with QR decomposition (QRD-M) and the sphere decoding (SD) algorithms, for multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) systems First, we show that nodes ordering before and during the tree search are important for both algorithms With appropriate ordering, QRD-M can improve detection performance significantly and SD can reduce decoding complexity substantially Then we compare the implementation complexity of the two algorithms, in terms of the number of nodes required to search or the required number of multiplications to achieve maximum likelihood detection performance It is interesting to show that the average complexity of SD is lower than that of QRD-M, whereas the worst case complexity of SD is much higher than that of QRD-M

A Formal Two-Phase Method for Decomposition of Complex Design Problems

Abstract: This paper presents a formal two-phase decomposition method for complex design problems that are represented in an attribute-component incidence matrix. Unlike the conventional approaches, this method decouples the overall decomposition process into two separate, autonomous function components: dependency analysis and matrix partitioning, which are algorithmically achieved by an extended Hierarchical Cluster Analysis (HCA) and a Partition Point Analysis (PPA), respectively. The extended HCA (Phase I) is applied to convert the (input) incidence matrix, which is originally unorganized, into a banded diagonal matrix. The PPA (Phase 2) is applied to further transform this matrix into a block-angular matrix according to a given set of decomposition criteria. This method provides both flexibility in the choice of the different settings on the decomposition criteria, and diversity in the generation of the decomposition solutions, both taking place in Phase 2 without resort to Phase I. These features essentially make this decomposition method effective, especially in its application to re-decomposition. A powertrain design example is employed for illustration and discussion.

Efficient solution of EFIE via low-rank compression of multilevel predetermined interactions

Abstract: This paper describes the predetermined interaction list oct-tree (PILOT) algorithm and its application in expediting the solution of full-wave electric field integral equation (EFIE)-based scattering problems for three-dimensional arbitrarily shaped conductors. PILOT combines features of the fast multipole method (FMM) and QR decomposition-based matrix compression techniques to optimize setup times, solve times, and memory requirements. The method is kernel independent and stable for electrically small structures unlike traditional FMM. The novel features of the algorithm, namely the mixed potential compression scheme and the hierarchical multilevel predetermined matrix structure are explained in detail. A complexity estimate is presented to demonstrate the scaling in time and memory requirements. Examples exhibiting the accuracy and the time and memory performances are also presented. Finally, a quantitative study is included to address the expected but gradual degradation of QR-based compression techniques for electrically large structures.

Complexity Analysis of MMSE Detector Architectures for MIMO OFDM Systems

Abstract: In this paper, a field programmable gate array (FPGA) implementation of a linear minimum mean square error (LMMSE) detector is considered for MIMO-OFDM systems. Two square root free algorithms based on QR decomposition (QRD) are introduced for the implementation of LMMSE detector. Both algorithms are based on QRD via Givens rotations, namely coordinate rotation digital computer (CORDIC) and squared Givens rotation (SGR) algorithms. Linear and triangular shaped array architectures are considered to exploit (he parallelism in the computations. An FPGA hardware implementation is presented and computational complexity of each implementation is evaluated and compared

An Interior-Point Approach to Trajectory Optimization

Abstract: This paper presents an interior-point approach for solving optimal control problems. Combining a flexible refinement scheme with dedicated linear algebra solvers, we obtain an efficient algorithm. Numerical results are displayed for various problems, among them several variants of atmospheric reentry.

Application-specific instruction set processor for SoC implementation of modern signal processing algorithms

Abstract: A novel application-specific instruction set processor (ASIP) for use in the construction of modern signal processing systems is presented. This is a flexible device that can be used in the construction of array processor systems for the real-time implementation of functions such as singular-value decomposition (SVD) and QR decomposition (QRD), as well as other important matrix computations. It uses a coordinate rotation digital computer (CORDIC) module to perform arithmetic operations and several approaches are adopted to achieve high performance including pipelining of the micro-rotations, the use of parallel instructions and a dual-bus architecture. In addition, a novel method for scale factor correction is presented which only needs to be applied once at the end of the computation. This also reduces computation time and enhances performance. Methods are described which allow this processor to be used in reduced dimension (i.e., folded) array processor structures that allow tradeoffs between hardware and performance. The net result is a flexible matrix computational processing element (PE) whose functionality can be changed under program control for use in a wider range of scenarios than previous work. Details are presented of the results of a design study, which considers the application of this decomposition PE architecture in a combined SVD/QRD system and demonstrates that a combination of high performance and efficient silicon implementation are achievable.

Communication system and method using a relay node

Abstract: A communication node that relays signals between a source node and a destination node includes (a) a first unitary matrix calculation unit configured to calculate a first unitary matrix based on a first channel between the source node and the relay node, (b) a second unitary matrix calculation unit configured to calculate a second unitary matrix based on a second channel between the relay node and the destination node, (c) a transformation matrix estimation unit configured to estimate a transformation matrix based on a triangular matrix derived from QR decomposition of the first and/or second channel matrix, (d) a relaying signal generator configured to generates a relaying signal by multiplying a received signal by at least one of the first unitary matrix, the second unitary matrix, and the transformation matrix, and (e) a transmission unit configured to transmit the relaying signal to the destination node.

Power system state estimation via globally convergent methods

Abstract: This paper introduces backtracking and trust region methods into power system state estimation. The traditional Newton (Gauss-Newton) method is not always reliable particularly in the presence of bad data, topological or parameter errors. The motivation was to enhance convergence properties of the state estimator under those conditions, and together with QR factorization to make a globally convergent and reliable algorithm. The trust region formulation shows that such a model is more robust than the traditional Newton (Gauss-Newton) or Backtracking (line search) algorithm. Both algorithms have been programmed and applied to representative power networks, and the computational requirement has been found.

Interpolation-based QR decomposition in MIMO-OFDM systems

Abstract: The extension of multiple-input multiple-output (MIMO) sphere decoding from the narrowband case to wideband systems based on orthogonal frequency division multiplexing (OFDM) requires the computation of a QR decomposition for each of the data-carrying OFDM tones. Since the number of data-carrying tones ranges from 48 (as in the IEEE 802.11a/g standards) to 6817 (as in the DVB-T standard), the corresponding computational complexity will in general be significant. This paper presents two algorithms for interpolation-based QR decomposition in MIMO-OFDM systems. An in-depth computational complexity analysis shows that the proposed algorithms, for a sufficiently high number of data-carrying tones and small channel order exhibit significantly smaller complexity than brute-force per-tone QR decomposition.

A Class of Incomplete Orthogonal Factorization Methods. II: Implementation and Results

Abstract: We present, implement and test several incomplete QR factorization methods based on Givens rotations for sparse square and rectangular matrices. For square systems, the approximate QR factors are used as right-preconditioners for GMRES, and their performance is compared to standard ILU techniques. For rectangular matrices corresponding to linear least-squares problems, the approximate R factor is used as a right-preconditioner for CGLS. A comprehensive discussion is given about the uses, advantages and shortcomings of the preconditioners.

A multifrontal QR factorization approach to distributed inference applied to multirobot localization and mapping

Abstract: QR factorization is most often used as a "black box" algorithm, but is in fact an elegant computation on a factor graph. By computing a rooted clique tree on this graph, the computation can be parallelized across subtrees, which forms the basis of so-called multifrontal QR methods. By judiciously choosing the order in which variables are eliminated in the clique tree computation, we show that one straightforwardly obtains a method for performing inference in distributed sensor networks. One obvious application is distributed localization and mapping with a team of robots. We phrase the problem as inference on a large-scale Gaussian Markov Random Field induced by the measurement factor graph, and show how multifrontal QR on this graph solves for the global map and all the robot poses in a distributed fashion. The method is illustrated using both small and large-scale simulations, and validated in practice through actual robot experiments.

Communication system and technique using QR decomposition with a triangular systolic array

Abstract: An apparatus, system, and method to perform QR decomposition of an input complex matrix are described. The apparatus may include a triangular systolic array to load the input complex matrix and an identity matrix, to perform a unitary complex matrix transformation requiring three rotation angles, and to produce a complex unitary matrix and an upper triangular matrix. The upper triangular matrix may include real diagonal elements. Other embodiments are described and claimed.

Bi-iterative least-square method for subspace tracking

Abstract: Subspace tracking is an adaptive signal processing technique useful for a variety of applications. In this paper, we introduce a simple bi-iterative least-square (Bi-LS) method, which is in contrast to the bi-iterative singular value decomposition (Bi-SVD) method. We show that for subspace tracking, the Bi-LS method is easier to simplify than the Bi-SVD method. The linear complexity algorithms based on Bi-LS are computationally more efficient than the existing linear complexity algorithms based on Bi-SVD, although both have the same performance for subspace tracking. A number of other existing subspace tracking algorithms of similar complexity are also compared with the Bi-LS algorithms.

Low-complexity maximum-likelihood decoder for four-transmit-antenna quasi-orthogonal space-time block code

Abstract: This letter proposes a very low-complexity maximum-likelihood (ML) detection algorithm based on QR decomposition for the quasi-orthogonal space-time block code (QSTBC) with four transmit antennas, called the LC-ML decoder. The proposed algorithm enables the QSTBC to achieve ML performance with significant reduction in computational load for any high-level modulation scheme.

Rational Krylov matrices and QR steps on Hermitian diagonal-plus-semiseparable matrices

Abstract: SUMMARY We prove that the unitary factor appearing in the QR factorization of a suitably dened rational Krylov matrix transforms a Hermitian matrix having pairwise distinct eigenvalues into a diagonal-plus- semiseparable form with prescribed diagonal term. This transformation is essentially uniquely dened by itsrst column. Furthermore, we prove that the set of Hermitian diagonal-plus-semiseparable matrices is invariant under QR iteration. These and other results are shown to be the rational counterpart of known facts involving structured matrices related to polynomial computations. Copyright ? 2005 John Wiley & Sons, Ltd.

Low complexity signal detection algorithm for MIMO-OFDM systems

Abstract: A very low complexity algorithm for ordering based on the MMSE criterion and detecting based on the QR decomposition of sorted channel matrix in BLAST architectures is presented. The algorithm needs only a fraction of computational effort compared to the standard ordering MMSE algorithm and achieves suboptimal performance.

Blind channel equalization with algebraic optimal step size

Abstract: The constant modulus algorithm (CMA) is arguably the most widespread iterative method for blind equalization of digital communication channels. The present contribution studies a recently proposed technique aiming at avoiding the shortcomings of conventional gradient-descent implementations. This technique is based on the computation of the step size leading to the absolute minimum of the CM criterion along the search direction. For the CM as well as other equalization criteria, this optimal step size can be calculated algebraically at each iteration by finding the roots of a low-degree polynomial. After developing the resulting optimal step-size CMA (OS-CMA), the algorithm is compared to its conventional constant step-size counterpart and more recent alternative CM-based methods. The optimal step size seems to improve the conditioning of the equalization problem as in prewhitening (e.g., via a prior QR decomposition of the data matrix), although it becomes more costly for long equalizers. The additional exploitation of the i.i.d. assumption through prewhitening can further improve performance, an outcome that had not been clearly interpreted in former works.

Efficient strategies for deriving the subset VAR models

Abstract: Algorithms for computing the subset Vector Autoregressive (VAR) models are proposed. These algorithms can be used to choose a subset of the most statistically-significant variables of a VAR model. In such cases, the selection criteria are based on the residual sum of squares or the estimated residual covariance matrix. The VAR model with zero coefficient restrictions is formulated as a Seemingly Unrelated Regressions (SUR) model. Furthermore, the SUR model is transformed into one of smaller size, where the exogenous matrices comprise columns of a triangular matrix. Efficient algorithms which exploit the common columns of the exogenous matrices, sparse structure of the variance-covariance of the disturbances and special properties of the SUR models are investigated. The main computational tool of the selection strategies is the generalized QR decomposition and its modification.