QR Decomposition

The problem of Canonical Polyadic (CP) decomposition of semi-nonnegative semi-symmetric three-way arrays is often encountered in Independent Component Analysis (ICA), where the cumulant of a nonnegative mixing process is frequently involved, such as the Magnetic Resonance Spectroscopy (MRS). We propose a new method, called JD + QR , to solve such a problem. The nonnegativity constraint is imposed by means of a square change of variable. Then the high-dimensional optimization problem is decomposed into several sequential rational subproblems using QR matrix factorization. A numerical experiment on simulated arrays emphasizes its good performance. A BSS application on MRS data confirms the validity and improvement of the proposed method.

LU decomposition is a fundamental in linear algebra. Numerous tools exists that provide this important factorization. The authors present the conditions for a matrix to have none, one, or infinitely many LU factorizations. In the case where no factorization exists, the authors illustrate how to approximate an LU decomposition by considering LU factorization of nearby matrices.

LU decomposition is a fundamental in linear algebra. Numerous tools exists that provide this important factorization. The authors present the conditions for a matrix to have none, one, or infinitely many LU factorizations. In the case where no factorization exists, the authors illustrate how to approximate an LU decomposition by considering LU factorization of nearby matrices.

We consider efficient techniques for the design of a transceiver for a matrix channel with minimum mean square error (MMSE) decision-feedback (DF) detection when perfect channel information is available at both the transmitter and receiver. By combining the canonical property of the MMSE-DF detector and our recently developed equal-diagonal QRS decomposition of a matrix we obtain a uniform decomposition of mutual information in which each of the synthesized scalar subchannels has the same mutual information (under the assumption of error-free feedback). To assist our analysis of this uniform decomposition, we provide a new QR interpretation of the MMSE-DF receiver. This enables us to show that the natural detection order is optimal (in an SINR sense), and that for the proposed transmitter, the MMSE-DF detector is asymptotically equivalent to the maximum likelihood detector when the SNR is high. We also derive a low-complexity quadratic recursive algorithm for the characterization of all eligible S-factors in the QRS decomposition. When coupled with our QR interpretation of the MMSE-DF detector, this enables us to efficiently design the optimal transmitter and to efficiently implement the MMSE-DF receiver.

In this paper, a parallel group detection (PGD) algorithm is proposed in order to address the degradation in the bit error rate (BER) performance of linear detectors when they are used in high-load massive MIMO systems. The algorithm is constructed by converting the equivalent extended massive MIMO system into two subsystems, which can be simultaneously detected by the classical detection procedures. Then, using the PGD and the classical ZF as well as the QR-decomposition- (QRD-) based detectors, we proposed two new detectors, called ZF-based PGD (ZF-PGD) and QRD-based PGD (QRD-PGD). The PGD is further combined with the sorted longest basis (SLB) algorithm to make the signal recovery more accurate, thereby resulting in two new detectors, namely, the ZF-PGD-SLB and the QRD-PGD-SLB. Various complexity evaluations and simulations prove that the proposed detectors can significantly improve the BER performance compared to their classical linear and QRD counterparts with the practical com...

We propose Dijkstra's algorithm with bounded list size after QR decomposition for decreasing the computational complexity of near maximumlikelihood (ML) detection of signals over multipleinput-multiple-output (MIMO) channels. After that, we compare the performances of proposed algorithm, QR decomposition M-algorithm (QRD-MLD), and its improvement. When the list size is set to achieve the almost same symbol error rate (SER) as the QRD-MLD, the proposed algorithm has smaller average computational complexity.

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.

We present theory and algorithms for the equality constrained indefinite least squares problem, which requires minimization of an indefinite quadratic form subject to a linear equality constraint. A generalized hyperbolic QR factorization is introduced and used in the derivation of perturbation bounds and to construct a numerical method. An alternative method is obtained by employing a generalized QR factorization in combination with a Cholesky factorization. Rounding error analysis is given to show that both methods have satisfactory numerical stability properties and numerical experiments are given for illustration. This work builds on recent work on the unconstrained indefinite least squares problem by Chandrasekaran, Gu, and Sayed and by the present authors.

In this paper, a modified learning algorithm for the multilayer neural network with the multi-valued neurons (MLMVN) is presented. The MLMVN, which is a member of complex-valued neural networks family, has already demonstrated a number of important advantages over other techniques. A modified learning algorithm for this network is based on the introduction of an acceleration step, performing by means of the complex QR decomposition and on the new approach to calculation of the output neurons errors: they are calculated as the differences between the corresponding desired outputs and actual values of the weighted sums. These modifications improve significantly the existing derivative-free backpropagation learning algorithm for the MLMVN in terms of learning speed. A modified learning algorithm requires two orders of magnitude lower number of training epochs and less time for its convergence when compared to the existing learning algorithm. Good performance is confirmed not only by the much quicker convergence of the learning algorithm, but also by the compatible or even higher classification/prediction accuracy, which is obtained by testing over some benchmarks (Mackey-Glass and Jenkins-Box time series) and over some satellite spectral data examined in a comparison test.

The combination of multiple-input multiple-output (MIMO) signal processing with orthogonal frequency division multiplexing (OFDM) using Cyclic Prefix (CP) is regarded as a highly promising solution to achieving the data rates of next-generation wireless communication systems operating in frequency selective fading environments. In this paper, we propose a novel detection scheme for MIMO based transmission that combines the adaptive MMSE detection scheme with a priori information and the serial interference cancelation (SIC) in the spatial domain. Principle is to include adaptive MMSE detection in each SIC iteration. With the proposed method, both complexity and performance can be balanced with the iterative scheme. Simulation results show interesting gain in term of performance for QPSK modulation.

Internet simplified digital data transferring. This data needs to be secured; so securing digital data becomes an important concern. Steganography provides security for data by inserting it into a cover and concealing it. In this paper, a steganography algorithm was introduced. This algorithm used the stationary wavelet transform (SWT) and اhybrid-matrix decomposition techniques, singular value decomposition (SVD) and QR factorization to conceal a video in another video. The algorithm performance was measured with reference to the peak signal to noise ratio (PSNR) and structural similarity index measure (SSIM) for the cover and the secret videos. The algorithm successfully hid a video in another cover video and both are of the same size; the hiding capacity was 100%. The algorithm achieved a SSIM that reached 0.97 and a high PSNR value that reached 68.8 proving that the imperceptibility of the proposed algorithm is very high. The comparative analysis shows that the suggested algorit...

In this paper, a new method for dynamically estimating and updating the coefficients of a digital predistortion (DPD) linearizer is presented. By means of the partial least squares (PLS) algorithm, the basis matrix used in the DPD estimation/adaptation is dynamically updated at every iteration to minimize the linearization error. Moreover, only the minimum necessary DPD coefficients being required to meet a target estimation error are computed. The proposed estimation technique is compared with the standard least squares (LS) estimation solved by using QR decomposition. Experimental results show the similar linearization performance obtained with both estimation methods, but in the case of the dynamic PLS, using less coefficients at every iteration. Finally, the proposed algorithm allows a high level of parallelization, which makes it suitable for FPGA implementation.

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. v Abstract .

Iterative orthogonalization is aimed to ensure small deviation from orthogonality in the Gram-Schmidt process. Former applications of this technique are restricted to classical Gram-Schmidt (CGS) and column-oriented modified Gram-Schmidt (MGS). The major aim of this paper is to explain how iterative orthogonalization is incorporated into row-oriented MGS. The interest that we have in a row-oriented iterative MGS comes from the observation that this method is capable of performing column pivoting. The use of column pivoting delays the deteriorating effects of rounding errors and helps to handle rank-deficient least-squares problems. A second modification proposed in this paper considers the use of Gram-Schmidt QR factorization for solving linear least-squares problems. The standard solution method is based on one orthogonalization of the r.h.s. vector b against the columns of Q. The outcome of this process is the residual vector, r * , and the solution vector, x * . The modified scheme is a natural extension of the standard solution method that allows it to apply iterative orthogonalization. This feature ensures accurate computation of small residuals and helps in cases when Q has some deviation from orthogonality.

This paper proposes a new computational method for solving structured least squares problems that arise in the process of identification of coherent structures in fluid flows. It is deployed in combination with dynamic mode decomposition (DMD) which provides a nonorthogonal set of modes -corresponding to particular temporal frequencies -a subset of which is used to represent time snapshots of the underlying dynamics. The coefficients of the representation are determined from a solution of a structured linear least squares problem with the matrix that involves the Khatri-Rao product of a triangular and a Vandermonde matrix. Such a structure allows a very efficient normal equation based least squares solution, which is used in state of the art CFD tools such as the sparsity promoting DMD (DMDSP). A new numerical analysis of the normal equations approach provides insights about its applicability and its limitations. Relevant condition numbers that determine numerical robustness are identified and discussed. Further, the paper offers a corrected semi-normal solution and QR factorization based algorithms. It is shown how to use the Vandermonde-Khatri-Rao structure to efficiently compute the QR factorization of the least squares coefficient matrix, thus providing a new computational tool for the ill-conditioned cases where the normal equations may fail to compute a sufficiently accurate solution. Altogether, the presented material provides a firm numerical linear algebra framework for a class of structured least squares problems arising in a variety of applications.

This note reports an unexpected and rather erratic behavior of the LAPACK software implementation of the QR factorization with Businger-Golub column pivoting. It is shown that, due to finite precision arithmetic, software implementation of the factorization can catastrophically fail to produce triangular factor with the structure characteristic to the Businger-Golub pivot strategy. The failure of current state of the art software, and a proposed alternative implementations are analyzed in detail.

The m-th root of the diagonal of the upper triangular matrix R m in the QR decomposition of AX m B = Q m R m converges and the limit is given by the moduli of the eigenvalues of X with some ordering, where A, B, X ∈ C n×n are nonsingular. The asymptotic behavior of the strictly upper triangular part of R m is discussed. Some computational experiments are discussed.

In this paper, we propose two novel indices for type-2 fuzzy rule ranking to identify the most influential fuzzy rules in designing type-2 fuzzy logic systems, and name them as R-values and c-values of fuzzy rules separately. The R-values of type-2 fuzzy rules are obtained by applying QR decomposition in which there is no need to estimate a rank as required in the SVD-QR with column pivoting algorithm. The c-values of type-2 fuzzy rules are suggested to rank rules based on the effects of rule consequents. Experimental results on a signal recovery problem have shown that by using the proposed indices the most influential type-2 fuzzy rules can be effectively selected to construct parsimonious type-2 fuzzy models while the system performances are kept at a satisfied level. I. INTRODUCTION Type-2 fuzzy sets initially suggested by Zadeh in 1975 [1] offer an opportunity to model higher level uncertainty in the human decision making process than the type-1 fuzzy sets [2][3]. As one of the major research topics in fuzzy system community, type-2 fuzzy sets and type-2 fuzzy logic system (T2FLS) have achieved many successful applications in various areas where uncertainties occur such as in decision making [4][5], diagnostic medicine [6], signal processing [7][8], traffic forecasting [9], mobile robot control [10], pattern recognition [11][12], intelligent control [13][14]. However, one challenge in type-1 fuzzy systems remains in T2FLSs, that is, the curse of dimensionality: the number of fuzzy rules required increases exponentially with the input space dimension in grid partition. Additional challenge in T2FLS modelling is that there is a higher computational overhead than type-1 FLS modelling . Hence, complexity reduction techniques are urgently needed for T2FLS modelling. As a matter of fact, even in type-1 fuzzy logic system (FLS) modelling, developing parsimonious fuzzy modelling technique with as few fuzzy rules as possible is a very important research topic [15][16][17][20]. Interestingly, Liang and Mendel in the first place suggested to design parsimonious interval T2FLS (IT2FLS) by using SVD-QR decomposition method to perform rule reduction . However, one issue arising in applying the SVD-QR with column pivoting algorithm to fuzzy rule reduction is the estimation of an effective rank. In order to avoid the estimation of the rank for the SVD-QR with column pivoting algorithm, this paper proposes to

The QR decomposition in linear algebra calculation is widely applied. Despite that the algorithms for the decomposition are available, it lacks explanations how to implement them for non-square matrices. The research gives descriptions about the modifications, needed to apply for rectangular form of matrices. A software implementation of the algorithm is presented. It can be used for client or server side programming tools, performing linear algebra calculations and optimization.

Evaluation of the land surface albedos by employing the bidirectional reflectance distribution function (BRDF) models is one of the important problems in remote sensing. As is known, the retrieval process is an inverse problem. In Proposition 3 of , the authors consider that the number of independent observations should be greater than the number of the unknown parameters to describe the physical model as an overdetermined system, then the inverse process can be solved. However as Li et al (1998) pointed out that such a requirement can be hardly satisfied even in the coming EOS era, the inversion procedure is always underdetermined in some sense. Therefore, in order to solve the BRDF inversion problem, some new technique must be developed. Generally speaking, the inverse problems are ill-posed. Therefore, some regularization technique should be applied to suppress the ill-posedness. One kind of way to alleviate the ill-posedness is incorporating with some apriori knowledge which has been developed in . This is actually a constrained least squares error (LSE) method since the apriori knowledge can be considered as some kind of constraints to the solution. Another kind of way is by numerical truncated singular value decomposition by employing the hotspot remote sensing data . In this paper, we consider a new solution method, i.e., the l 1 norm solution method, which iteratively solves the kernel-driven bidirectional reflectance distribution function (BRDF) models for retrieval of land surface albedos. This method, is based on searching for an interior point solution for the problem in the feasible solution set. This method can always find a set of suitable BRDF coefficients for poor sampled data. Numerical performance is given for the widely used 18 data sets among the 73 data sets ].

A least-squares method with a direct minimization algorithm is introduced to solve the non-linear population balance equation that consists of both breakage and coalescence terms. The least-squares solver, direct minimization solver together with a finite difference solver are implemented for comparisons. It is shown that the coalescence term introduces a strong non-linear behavior which can affect the robustness of the numerical solvers. In the comparison with the least-squares method, the direct minimization method is proved to be capable of producing equally accurate results, while its formulation is better conditioned. In the case of a non-linear population balance equation system, the direct minimization method converges faster than the standard least-squares method.

This paper presents a unified derivation of rotation-based recursive least squares (RLS) algorithms based on QR decomposition. They solve the adaptive least squares problems of the linear combiner without a desired signal, the single and multichannel linear prediction and transversal filtering. All algorithms derived in this paper are based on Givens rotations. They offer superior numerical properties as shown by computer simulations. They are computationally efficient and highly concurrent. Parallel implementation of the resulting methods in the form of systolic array and its application to system identification is discussed in this paper.

This paper revisits the comrade matrix approach in finding the greatest common divisor (GCD) of two orthogonal polynomials. The present work investigates on the applications of the QR decomposition with iterative refinement (QRIR) to solve certain systems of linear equations which is generated from the comrade matrix. Besides iterative refinement, an alternative approach of improving the conditioning behavior of the coefficient matrix by normalizing its columns is also considered. As expected the results reveal that QRIR is able to improve the solutions given by QR decomposition while the normalization of the matrix entries do improves the conditioning behavior of the coefficient matrix leading to a good approximate solutions of the GCD.

Non-orthogonal multiple access (NOMA) with power-domain user multiplexing has been considered as one of the potential candidates of the fifth-Generation (5G) systems. In this paper, a two-stage user selection algorithm is proposed based on proportional fairness for downlink NOMA with zeroforcing beamforming (PF-NOMA-ZFBF) in order to improve the throughput-fairness trade-off for NOMA system. We focus on short term fairness, where short term refers to the minimum time window in which a specified fairness is guaranteed and evaluated using Jain's index. A transmit power allocation then applied to enhance the throughput of NOMA system. Simulation results show that the proposed PF-NOMA-ZFBF significantly improves user fairness index with at least 38.96% over conventional NOMA-ZFBF while maintaining the total system sum-rate (near maximum).

Stable distributions are characterized by four parameters which can be estimated via a number of methods, and although approximate maximum likelihood estimation techniques have been proposed, they are computationally intensive and difficult to implement. This article describes a fast, wavelet-based, regression-type method for estimating the parameters of a stable distribution. Fourier domain representations, combined with a wavelet multiresolution approach, are shown to be effective and highly efficient tools for inference in stable law families. Our procedures are illustrated and compared with other estimation methods using simulated data from stable distributions and an application to a real data example is explored. One novel aspect of this work is that here wavelets are being used to solve a parametric problem rather than a nonparametric one, which is the more typical context in wavelet applications.

Multiple antennas are used in both the transmitter and receiver ends of MIMO systems to increase spectrum efficiency. The implementation of two methods of decoding algorithms, the Viterbo-Boutros (VB) as well as Schnorr-Euchner (SE), is proposed in this method. The decoding algorithm is divided into a single FPGA device using a software/hardware co-design methodology. To maximize the decoding efficiency, three methods of parallelism are being adopted: using a concurrent channel matrix, using preprocessing and parallel decoding of imaginary or real selection, and concurrent execution of many phases in the closest lattice point search. With a Xilinx FPGA chip, the decoders for a 4x4 MIMO system with 16-QAM modulation have been prototyped. The VB and SE algorithms have hardware prototypes that relate to the support data rates of up to 84.4 and 38.2 Mb/s at a 22 dB SNR, which is faster than their respective implementations in a DSP.

Decomposition Approach for Inverse Matrix Calculation 113 of a corresponding inverse of a modified matrix A *-1 are derived in (Strassen, 1969). The components of the inverse matrix can be evaluated analytically. Finding the inverse matrix is related with a lot of calculations. Instead of direct finding an inverse matrix, it is worth to find analytical relations where lower dimensions inverse matrices components are available. Here analytical relations for inverse matrix calculation are derived and the corresponding MATLAB codes are illustrated.

This paper presents an improved version of incremental condition estimation, a technique for tracking the extremal singular values of a triangular matrix as it is being constructed one column at a time. We present a new motivation for this estimation technique using orthogonal projections. The paper focuses on an implementation of this estimation scheme in an accurate and consistent fashion. In particular, we address the subtle numerical issues arising in the computation of the eigensystem of a symmetric rank-one perturbed diagonal 22 matrix. Experimental results show that the resulting scheme does a good job in estimating the extremal singular values of triangular matrices, independent of matrix size and matrix condition number, and that it performs qualitatively in the same fashion as some of the commonly used nonincremental condition estimation schemes.

In this paper, a new method for dynamically estimating and updating the coefficients of a digital predistortion (DPD) linearizer is presented. By means of the partial least squares (PLS) algorithm, the basis matrix used in the DPD estimation/adaptation is dynamically updated at every iteration to minimize the linearization error. Moreover, only the minimum necessary DPD coefficients being required to meet a target estimation error are computed. The proposed estimation technique is compared with the standard least squares (LS) estimation solved by using QR decomposition. Experimental results show the similar linearization performance obtained with both estimation methods, but in the case of the dynamic PLS, using less coefficients at every iteration. Finally, the proposed algorithm allows a high level of parallelization, which makes it suitable for FPGA implementation.

QR decomposition and M-algorithm based near maximum likelihood block detection (QRM-MLBD) significantly improves the single-carrier (SC) multiple-input multipleoutput (SC-MIMO) transmission performance in a frequency-selective fading channel. In the conventional QRM-MLBD, the cyclic prefix (CP) is inserted in order to avoid the inter-block interference (IBI). However, CP insertion reduces the transmission efficiency. In this paper, an iterative overlap QRM-MLBD is proposed for SC-MIMO transmission with no CP insertion. It is confirmed by computer simulation that the iterative overlap QRM-MLBD with no CP insertion provides improved throughput performance while reducing the computational complexity over the conventional QRM-MLBD with CP insertion.

Recently, we proposed a pseudo block coded singlecarrier (PBC-SC) transmission using minimum mean square error (MMSE) based frequency-domain equalization and decoding (FDED) and show that MMSE based FDED achieves BER performance superior to 2-step decoding (which performs frequency-domain equalization and hard decision decoding separately). However, there still exists a large BER performance gap between MMSE based FDED and MF bound due to the presence of residual inter-symbol interference (ISI) after MMSE based FDED. In this paper, in order to narrow the performance gap, we propose frequency-domain block signal detection (FDBD) aided FDEDs for PBC-SC transmission: frequency-domain iterative interference cancellation (FDIC), maximum likelihood block signal detection employing QR decomposition and M algorithm (QRM-MLBD) and belief propagation (BP). We compare, by computer simulation, the BER performances and the computational complexities of three FDBD aided FDED schemes. It is shown that QRM-MLBD and BP can achieve almost the same BER performance as MF bound with much lower computational complexity than MMSE based FDED.

Near maximum likelihood block signal detection using QR decomposition and M-algorithm (QRM-MLBD) can improve a bit error rate (BER) performance of cyclic prefix inserted single-carrier (CP-SC) transmissions. However, it requires a fairly large number M of surviving paths in the M-algorithm and leads to very high computational complexity. Replacing the CP by training sequence (TS) was shown to reduce the number of M. Another approach to reduce the complexity of QRM-MLBD is to modify the tree structure constructed by QR decomposition for ML detection. Recently, we proposed a 2-step QRM-MLBD which prunes unreliable symbol candidates before tree search by using the minimum mean square error based frequency-domain equalization (MMSE-FDE) output. In this paper, we apply the 2step QRM-MLBD to TS inserted SC (TS-SC) transmission in order to further reduce the complexity of QRM-MLBD. We show by computer simulation that 2-step QRM-MLBD can reduce the complexity compared to conventional QRM-MLBD while keeping almost the same BER performance.

In this paper, we propose an adaptive single-carrier (SC) transmission suitable for near maximum likelihood (ML) block signal detection using QR decomposition and M-algorithm (QRM-MLBD). QRM-MLBD can significantly improve the bit error rate (BER) performance of SC block transmission. However, in order to achieve a close-to-ML performance, the use of a fairly large number M of surviving paths in the M-algorithm is required and hence its computational complexity is still high. In this paper, to reduce the required number of surviving paths, we introduce an unequal bit loading in an SC transmission block. In QRM-MLBD, the signal-to-noise power ratio (SNR) degrades for symbols to be detected at early stages in the M-algorithm. By noting this fact, we propose to load less number of bits (i.e., lower modulation level) on symbols close to the end of block. We show that the introduction of an unequal bit loading to SC block transmission using QRM-MLBD can achieve almost the same BER performance as the conventional QRM-MLBD while significantly reducing the required number M of surviving path in the M-algorithm.

Recently, a frequency-domain block signal detection (FDBD) using maximum likelihood detection (MLD) employing QR decomposition and M-algorithm (QRM-MLD) was proposed for the reception of the single-carrier (SC) signals transmitted over a frequency-selective fading channel. SC-FDBD with QRM-MLD can significantly improve the bit error rate (BER) performance of SC transmission while reducing significantly the computational complexity compared to the MLD. However, its computational complexity is still high. In this paper, we propose a computationally efficient 2-step QRM-MLD SC-FDBD. Compared to conventional QRM-MLD, the number of symbol candidates can be reduced by using the decision made by minimum mean square error based frequency-domain equalization (MMSE-FDE). We evaluate the BER performance achievable by 2-step QRM-MLD and show that it can significantly reduce the computational complexity while keeping the BER performance almost the same as the conventional QRM-MLD.

QR decomposition and M-algorithm based near maximum likelihood block detection (QRM-MLBD) significantly improves the single-carrier (SC) multiple-input multipleoutput (SC-MIMO) transmission performance in a frequency-selective fading channel. In the conventional QRM-MLBD, the cyclic prefix (CP) is inserted in order to avoid the inter-block interference (IBI). However, CP insertion reduces the transmission efficiency. In this paper, an iterative overlap QRM-MLBD is proposed for SC-MIMO transmission with no CP insertion. It is confirmed by computer simulation that the iterative overlap QRM-MLBD with no CP insertion provides improved throughput performance while reducing the computational complexity over the conventional QRM-MLBD with CP insertion.

The Cholesky decomposition plays an important role in finding the inverse of the correlation matrices. As it is a fast and numerically stable for linear system solving, inversion, and factorization compared to singular valued decomposition (SVD), QR factorization and LU decomposition. As different methods exist to find the Cholesky decomposition of a given matrix, this paper presents the comparative study of a proposed RChol algorithm with the conventional methods. The RChol algorithm is an explicit way to estimate the modified Cholesky factors of a dynamic correlation matrix. Cholesky decomposition is a fast and numerically stable for linear system solving, inversion, and factorization compared to singular valued decomposition (SVD), QR factorization and LU decomposition [1]. The wireless communication system is highly dependent on matrix inversion of the correlation matrix. Such system consists of a huge matrix inversion. An outdoor wireless communication has a time-varying channel which changes dynamically for mobile user. In case of narrowband channel, the channel is considered constant for a symbol duration, whereas for broadband, it is changing within a symbol period. Such time-varying channel forms the special structure of channel matrix and correlation matrix. To exploit such special structure, a novel modified recessive Cholesky (RChol) algorithm is introduced in [2]. Our proposed (RChol) algorithm is a computational efficient algorithm to compute the modified Cholesky factors of known as well as an unknown covariance matrix. In this paper, we present the comparative study of conventional Cholesky algorithm and the RChol algorithm to manifest the importance of the proposed algorithm in a highly dynamic wireless communication.

Low resolution analog-to-digital converters (ADCs) can be employed to improve the energy efficiency (EE) of a wireless receiver since the power consumption of each ADC is exponentially related to its sampling resolution and the hardware complexity. In this paper, we aim to jointly optimize the sampling resolution, i.e., the number of ADC bits, and analog/digital hybrid combiner matrices which provides highly energy efficient solutions for millimeter wave multiple-input multiple-output systems. A novel decomposition of the hybrid combiner to three parts is introduced: the analog combiner matrix, the bit resolution matrix and the baseband combiner matrix. The unknown matrices are computed as the solution to a matrix factorization problem where the optimal, fully digital combiner is approximated by the product of these matrices. An efficient solution based on the alternating direction method of multipliers is proposed to solve this problem. The simulation results show that the proposed solution achieves high EE performance when compared with existing benchmark techniques that use fixed ADC resolutions. Index Terms-energy efficient design, optimal bit resolution and hybrid combining, mmWave MIMO.

In this paper, we propose a new kernel discriminant analysis called kernel relevance weighted discriminant analysis (KRWDA) which has several interesting characteristics. First, it can effectively deal with the small sample size problem by using a QR decomposition on scatter matrices. Second, by incorporating a weighting function into discriminant criterion, it overcomes overemphasis on well-separated classes and hence can work under more realistic situations. Finally, using kernel theory, it handle non linearity efficiently. In order to improve performance of the proposed algorithm, we introduce two novel kernel functions and compare them with some commonly used kernels on face recognition field. We have performed multiple face recognition experiments to compare KRWDA with other dimensionality reduction methods showing that KRWDA consistently gives the best results. Keywords Kernel discriminant analysis Á RWLDA Á Kernel functions Á Small sample Á Size problem Á Face recognition 1 Originality and contribution This paper proposes a new kernel-based method called kernel relevance weighted discriminant analysis (KRWDA). It produces a nonlinear discriminant features for face recognition task and also avoids the SSS problem using QR decomposition. Furthermore, two novel kernel functions namely, logarithmic and power distance kernels, have been introduced. Based on our tests, we have noted that these kernel functions deliver the highest face recognition accuracy compared to the conventional kernels, such as polynomial kernel and gaussian RBF kernel.

In this paper, we propose a new kernel discriminant analysis called kernel relevance weighted discriminant analysis (KRWDA) which has several interesting characteristics. First, it can effectively deal with the small sample size problem by using a QR decomposition on scatter matrices. Second, by incorporating a weighting function into discriminant criterion, it overcomes overemphasis on well-separated classes and hence can work under more realistic situations. Finally, using kernel theory, it handle non linearity efficiently. In order to improve performance of the proposed algorithm, we introduce two novel kernel functions and compare them with some commonly used kernels on face recognition field. We have performed multiple face recognition experiments to compare KRWDA with other dimensionality reduction methods showing that KRWDA consistently gives the best results. Keywords Kernel discriminant analysis Á RWLDA Á Kernel functions Á Small sample Á Size problem Á Face recognition 1 Originality and contribution This paper proposes a new kernel-based method called kernel relevance weighted discriminant analysis (KRWDA). It produces a nonlinear discriminant features for face recognition task and also avoids the SSS problem using QR decomposition. Furthermore, two novel kernel functions namely, logarithmic and power distance kernels, have been introduced. Based on our tests, we have noted that these kernel functions deliver the highest face recognition accuracy compared to the conventional kernels, such as polynomial kernel and gaussian RBF kernel.

As the general purpose graphics processing units (GPGPU) are increasingly deployed for scientific computing for its raw performance advantages compared to CPUs, the fault tolerance issue has started to become more of a concern than before when they were exclusively used for graphics applications. The pairing of GPUs with CPUs to form a hybrid computing systems for better flexibility and performance creates a massive amounts of computations that have a higher possibility to be affected by transient error-a soft error that silently modifies data causing errors to pass unnoticed. This is despite the fact that the newest Fermi generation of GPUs from NVIDIA are equipped with error correcting units to protect their memories. This problem is particularly serious for applications that employ numerical linear algebra since large sections of data are often modified between steps, and therefore even a single error could eventually propagate into a large area of result. In order to give protection to dense linear algebra computations on such hybrid systems, we developed an algorithm that is resilient to soft errors. We chose the right-looking Householder QR factorization as a demonstration of our algorithm for a hybrid system that features both GPUs and CPUs. Algorithm based fault tolerance (ABFT) is used to protect from errors in the trailing matrix and the right factor, while a checkpointing method is used to ensure the left factor is error-free. This work is based on a previous study of fault tolerance in matrix factorizations. Our contribution includes (1) a stable multiple-error checkpointing and recovery mechanism for the left-factor, which is also scalable in performance in the hybrid execution environment and does not cause severe performance degradation. (2) optimized Givens rotation utilities on the GPU to efficiently reduce an upper Hessenberg matrix to upper triangular form, and (3) a recovery algorithm based on QR update inside a hybrid system. Experimental results show that, our fault tolerant QR factorization can successfully detect and correct data altered by soft errors in both the left and right factors and we observe a decreasing percentage of overhead as the matrix size grows.

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Watermark algorithms for large language models (LLMs) have achieved extremely high accuracy in detecting text generated by LLMs. Such algorithms typically involve adding extra watermark logits to the LLM's logits at each generation step. However, prior algorithms face a trade-off between attack robustness and security robustness. This is because the watermark logits for a token are determined by a certain number of preceding tokens; a small number leads to low security robustness, while a large number results in insufficient attack robustness. In this work, we propose a semantic invariant watermarking method for LLMs that provides both attack robustness and security robustness. The watermark logits in our work are determined by the semantics of all preceding tokens. Specifically, we utilize another embedding LLM to generate semantic embeddings for all preceding tokens, and then these semantic embeddings are transformed into the watermark logits through our trained watermark model. Subsequent analyses and experiments demonstrated the attack robustness of our method in semantically invariant settings: synonym substitution and text paraphrasing settings. Finally, we also show that our watermark possesses adequate security robustness. Our code and data are available at https://github.com/THU-BPM/Robust_Watermark.

Hereby, I, Ta Minh THANH, consciously assure that for the manuscript "Consideration of A Robust Watermarking Algorithm for Color Image Using Improved QR Decomposition" the following is fulfilled: 1) This material is the authors' own original work, which has not been previously published elsewhere. 2) The paper is not currently being considered for publication elsewhere. 3) The paper reflects the authors' own research and analysis in a truthful and complete manner. 4) The paper properly credits the meaningful contributions of co-authors and co-researchers. 5) The results are appropriately placed in the context of prior and existing research. 6) All sources used are properly disclosed (correct citation). Literally copying of text must be indicated as such by using quotation marks and giving proper reference. 7) All authors have been personally and actively involved in substantial work leading to the paper, and will take public responsibility for its content. I agree with the above statements and declare that this submission follows the policies of Solid State Ionics as outlined in the Guide for Authors and in the Ethical Statement.

We present a new algorithm for solving an eigenvalue problem for a real symmetric matrix which is a rank-one modification of a diagonal matrix. The algorithm computes each eigenvalue and all components of the corresponding eigenvector with high relative accuracy in O(n) operations. The algorithm is based on a shift-and-invert approach. Only a single element of the inverse of the shifted matrix eventually needs to be computed with double the working precision. Each eigenvalue and the corresponding eigenvector can be computed separately, which makes the algorithm adaptable for parallel computing. Our results extend to the complex Hermitian case. The algorithm is similar to the algorithm for solving the eigenvalue problem for real symmetric arrowhead matrices from: