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AbstractGaussian distributions are widely used in Bayesian variational inference to approximate intractable posterior densities, but the ability to accommodate skewness can improve approximation accuracy significantly, when data or prior information is scarce. We study the properties of a subclass of closed skew normals constructed using affine transformation of independent standardized univariate skew normals as the variational density, and illustrate how it provides increased flexibility and accuracy in approximating the joint posterior in various applications, by overcoming limitations in existing skew normal variational approximations. The evidence lower bound is optimized using stochastic gradient ascent, where analytic natural gradient updates are derived. We also demonstrate how problems in maximum likelihood estimation of skew normal parameters occur similarly in stochastic variational inference, and can be resolved using the centered parameterization. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

Summary: This study underscores the growing significance of multimodal transportation within the cargo sector and its consequential environmental impacts. We present a novel mathematical model for operation scheduling, incorporating variables such as resource availability, customer service benchmarks, and environmental considerations. Our objective is to mitigate transportation expenses and reduce delivery delays. The proposed approach advocates LU decomposition with a pivot strategy for rapid model resolution, adherence to convergence criteria, optimization of cost strategies, and efficient resource utilization. Leveraging the adaptive neural fuzzy inference system (ANFIS) and genetic algorithm (GA), our methodology facilitates learning from past decisions to enhance solutions, aligning supply, and demand efficiently. We evaluate financial and environmental implications across four scenarios, offering insights into the economic and environmental advantages of various transportation modes—trains, ships, airplanes—compared to truck transportation, with a specific focus on CO2 emission impacts. Implementing the ANFIS+GA model in multimodal scenarios yield impressive results: minimal MAPE transportation cost of 0.17%, R2 transportation cost of 0.996, MAPE CO2 emissions of 0.13%, and R2 CO2 emissions of 0.996. By identifying cost‐efficient routes and optimizing resource allocations, our approach enables informed decisions regarding vehicle distribution, supplier selection, and contract negotiations. Additionally, we use LU decomposition to establish the supplier risk threshold, crucial for comparing emission trade variances. Multimodal scenarios typically yield lower emissions, favoring buying emission allowances low and selling them high. Notably, the risk threshold affects low‐emission provider utilization, impacting transportation emissions. With a risk threshold of 0.12 and an emission price of 1.2, our ANFIS+GA‐based multimodal approach achieves a significant −20% deviation in CO2 emissions. [ABSTRACT FROM AUTHOR]

Due to the increasing applications of dual quaternion and their matrices in recent years, as well as the significance of LU decomposition as a matrix decomposition technique, in this paper, we propose dual quaternion Gaussian transformation and obtain dual quaternion LU decomposition by using Gaussian transformation. We also use the total order of dual numbers to obtain the partial pivoting dual quaternion LU decomposition. Based on the real structure-preserving algorithm of quaternion matrix, we propose the real structure-preserving algorithms of LU decomposition and partial pivoting LU decomposition for dual quaternion matrix. Numerical experiments have verified the effectiveness of the new real structure-preserving approaches. [ABSTRACT FROM AUTHOR]

The secure transmission of medical images and related digital patient records on untrusted channels has recently become a focus in healthcare industries. Data hiding and encryption are important tools for this goal. This paper proposes a data-hiding method for medical images in the context of invisibility, robustness, security and low time cost. A dual watermarking is introduced to accomplish chaos-based encryption to ensure medical images' copyright protection and content security. First, a local binary pattern based on neighbouring pixels is used to compute an optimal value, called an embedding factor, for embedding both marks. Second, the host medical image is marked using the lifting wavelet transform, the lower–upper (LU) decomposition and singular value decomposition with an embedding factor to protect ownership. Last, the marked image is encrypted by using a 3D-chaotic map. The method is tested on two standard datasets, which is convenient for medical applications. Our experimental results and performance analysis demonstrate that the proposed scheme produces a peak signal-to-noise ratio (PSNR) and NC wat 1 / NC wat 2 of 54.82 dB and 0.9916/0.9928, respectively. Furthermore, the key space analysis of our encryption technique is greater than 2 100 , which provides a high protection degree against brute-force attacks. The proposed scheme achieves better robustness, imperceptibility, and security performance than previous schemes. Our method can reduce time costs by 98% (91% on average), which results in lower time costs than other conventional encryption techniques. [ABSTRACT FROM AUTHOR]

Graphics processing units (GPUs) have become integral to embedded systems and supercomputing centres due to their large memory, cutting-edge technology and high performance per watt. However, their susceptibility to transient errors requires a comprehensive analysis of error sensitivity, as well as the development of error mitigation techniques and fault-tolerant algorithms. This study focuses on evaluating the soft-error sensitivity of two distinct versions of LU decomposition algorithms implemented on two very different GPUs—a low-power SoC embedded GPU and a high-performance massively parallel GPU. Through extensive fault injection campaigns on both GPUs, we examine the vulnerability of the algorithms, identify error causes, and determine critical code components requiring enhanced protection. The experiments reveal that most single bit flip fault injections in the instruction results lead to erroneous outcomes or unrecoverable errors. Notably, efficient GPU resource utilisation can increase the number of masked errors, thereby enhancing error resilience. Additionally, while different parts of the code exhibit similar error occurrence types and rates, the propagation of errors to elements within the result matrix differs significantly. [ABSTRACT FROM AUTHOR]

In this paper, we construct the dual-complex Gauss transformation and propose novel algorithms for LU decomposition and partial pivoting LU decomposition of dual-complex matrices. In addition, the Cholesky and LDL H decompositions of dual-complex matrices are also obtained as special cases. Strict authentication is designed to cater to applications that have a zero-tolerance policy for any modifications or alterations made to the protected object. To increase the feasibility, we first propose the dual-complex matrix representation of color images and design a scheme for strict color image authentication by means of the dual-complex LU decomposition. Numerical experiments are provided to demonstrate the efficiency of the algorithm. [ABSTRACT FROM AUTHOR]

As a kind of commutative and associative four-dimensional algebra, commutative quaternion has better applications in color image and signal processing than quaternion. Matrix decomposition is of great concern in the theoretical study and numerical calculation of commutative quaternion. Two kinds of disadvantages of commutative quaternion make the decomposition of commutative quaternion matrix extremely difficult. On one hand, commutative quaternion is not a kind of complete four dimensional division algebra because of the zero divisors. On the other hand, computing the inverse of commutative quaternion is very complicated. In this paper, the semi-tensor product (STP) of matrices will be used to overcome the above two kinds of shortcomings. And we will propose a real structure-preserving algorithm based on STP of matrices for commutative quaternion LU decomposition, which makes full use of high-level operations, relation of operations between commutative quaternion matrices and their L-representation matrices. Numerical experiments will be provided to demonstrate the efficiency of the real structure-preserving algorithm based on STP of matrices. Meanwhile, we will apply the structure-preserving algorithm for strict image authentication. [ABSTRACT FROM AUTHOR]

All existing protocols are computationally secure, each of which is based on a certain security assumption and depends on the computational power of the adversary. This paper uses the matrix masking technique to hide the coefficient matrix. Combined with the LU (lower‐ upper) decomposition algorithm, a new information ‐ theoretically secure outsourcing of linear algebraic equations method is proposed in malicious adversarial model. Compared with the previous protocol, the security of the unique solution is improved to information ‐ theoretic security (perfect secrecy) for the first time, without sacrificing the complexity of computation and communication. A formal security proof is provided, and the practicality is proved theoretically and experimentally. [ABSTRACT FROM AUTHOR]

The piracy of digital products has become an urgent problem to be solved. Robust watermarking is a promising technique for copyright protection. To go against rapidly increasing attacks, many techniques have been proposed in the transform domain for ensuring the quality of water marked image, the robustness of extracted watermark, and execution time. Among these techniques, the LU decomposition is considered an out- standing transformation in terms of computation. However, some pixel matrices cannot be decomposed as the product of a lower triangular matrix and an upper triangular matrix, and embedding them into two elements reduces the quality of the watermark image. Therefore, this paper proposes a novel image watermarking scheme based on a LU block selection strategy and an improved embedding algorithm. First, a pixel block is only applied LU decomposition and embedded a watermark bit when all determinants of sub-matrices are non-zero. Second, an effective embedding formula is proposed where the watermark is embedded into one element of suitable blocks. This solution helps to limit the modification of pixel values and enhance the quality of the watermarked image. Third, to decrease the extraction time, a novel formula is built for calculating embedded elements instead of LU decomposition. The experimental results show that our proposed method has the better visual quality of watermarked images and can effectively extract the watermark under some attacks, such as blurring, sharpening, and salt & pepper noise. [ABSTRACT FROM AUTHOR]

The AES (Advance Encryption Standard) has made the development of new block ciphers unnecessary; it is now the de facto standard for most uses of block ciphers. However, the AES is still not well-suited for very limited contexts like RFID (Radio-Frequency Identification) tags and WSN(Wireless Sensor Networks), despite recent implementation advancements. In this study, we present SHC (Simple Hybrid Cipher), a new block cipher that uses a 64-bit block length and a 128-bit key length. It offers a hardware implementation that efficiently uses limited resources, making it ideal for use as a sensor in a WSN or an RFID tag. The core function of SHC depends on S-Box-based, composite field arithmetic technology, as it consumes relatively low cost on hardware implementation while still providing sufficient security as a solid encryption algorithm. The hardware implementation of SHC-64 requires 949 LUTs; it generates a maximum operating frequency of 515.995 MHz on the Xilinx-powered Artix-7 Field Programmable Gate Array (FPGA) development board. At the same time, the National Institute of Standards and Technology (NIST) recommended standard algorithm AES consumes 3645 LUTs and generates a maximum operating frequency of 277.369 MHz. The SHC-64 cipher also shows resistance against known cryptanalytics attacks.

Broad Learning System (BLS) is proposed as an alternative to deep learning. It has a fast adaptive model selection and online incremental learning capability, which has been successfully applied in many fields. In this paper, the BLS model is introduced into intrusion detection, and considering the weakness of the BLS model in mining the internal structural information of samples, this paper proposes a Manifold Regularized Broad Learning System based on LU decomposition (LU-MRBLS) intrusion detection. Based on the manifold hypothesis, the LU-MRBLS model firstly constructs the graph Laplacian operator in the data input space to mine the potential information of the data. Then, under the manifold regularized framework, the feature nodes, enhancement nodes, and Laplacian matrix are combined to construct the objective function to regularize and optimize the BLS model to avoid the model falling into local optimization. Finally, the LU decomposition method is used to solve the output weight matrix of the MRBLS model, shorten the training time of the MRBLS model, avoid singular value problems of the solution process, and improve the intrusion detection performance of the model. In this paper, we use the KDD Cup99 dataset for parameter selection and apply it to other network models. Through rigorous experiments, the LU-MRBLS model is applied to KDD Cup99, NSL-KDD, UNSW-NB15, and CIDDS-001 datasets with better detection results than the classical machine learning models and the latest intrusion detection models. [ABSTRACT FROM AUTHOR]

In the co-localization algorithm of augmented information filtering robot, the Cholesky method is usually used to decompose the information matrix of the joint distribution. The augmented information filtering based on Cholesky decomposition has high requirements for the positive definite symmetry of the information matrix of the joint distribution, and in the case that the information matrix of the joint distribution does not meet the positive definite symmetry, the inversion produces anomalies, affects the information recovery of the joint distribution, and the robustness of the system decreases. An augmented information filtering algorithm based on LU decomposition was proposed, which ensures the accuracy of the robot co-localization algorithm, effectively solves the problem of information matrix decomposition anomaly of joint distribution, and finally the observability of the robot system was analyzed. The MATLAB software platform was used to simulate and verify the algorithm. The results show that the algorithm ensures the cooperative positioning accuracy of the robot and improves the robustness of the robot system. [ABSTRACT FROM AUTHOR]

In this paper, based on the structure of pentadiagonal CUPL-Toeplitz matrix and Sherman–Morrison–Woodbury formula, we develop a new algorithm for solving nonsingular pentadiagonal CUPL-Toeplitz linear system. Some numerical examples are given in order to illustrate the effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

Decomposing a matrix $\mathbf {A}$ A into a lower matrix $\mathbf {L}$ L and an upper matrix $\mathbf {U}$ U , which is also known as LU decomposition, is an essential operation in numerical linear algebra. For a sparse matrix, LU decomposition often introduces more nonzero entries in the $\mathbf {L}$ L and $\mathbf {U}$ U factors than in the original matrix. A symbolic factorization step is needed to identify the nonzero structures of $\mathbf {L}$ L and $\mathbf {U}$ U matrices. Attracted by the enormous potentials of the Graphics Processing Units (GPUs), an array of efforts have surged to deploy various LU factorization steps except for the symbolic factorization, to the best of our knowledge, on GPUs. This article introduces gSoFa, the first GPU-based symbolic factorization design with the following three optimizations to enable scalable LU symbolic factorization for nonsymmetric pattern sparse matrices on GPUs. First, we introduce a novel fine-grained parallel symbolic factorization algorithm that is well suited for the Single Instruction Multiple Thread (SIMT) architecture of GPUs. Second, we tailor supernode detection into a SIMT friendly process and strive to balance the workload, minimize the communication and saturate the GPU computing resources during supernode detection. Third, we introduce a three-pronged optimization to reduce the excessive space consumption problem faced by multi-source concurrent symbolic factorization. Taken together, gSoFa achieves up to 31× speedup from 1 to 44 Summit nodes (6 to 264 GPUs) and outperforms the state-of-the-art CPU project, on average, by 5×. Notably, gSoFa also achieves up to 47 percent of the peak memory throughput of a V100 GPU in the Summit Supercomputer. [ABSTRACT FROM AUTHOR]

In this paper, we argue for the utility of deterministic inference in the classical problem of numerical linear algebra, that of solving a linear system. We show how the Gaussian belief propagation solver, known to work for symmetric matrices, can be modified to handle nonsymmetric matrices. Furthermore, we introduce a new algorithm for matrix inversion that corresponds to the generalized belief propagation derived from the cluster variation method. We relate these algorithms to LU and block LU decompositions and provide certain guarantees based on theorems from the theory of belief propagation. The proposed solvers can be accelerated within a framework of geometric multigrid. We benchmark the resulting multigrid method with a set of classical multigrid solvers based on incomplete LU, Chebyshev, and pattern Gauss-Seidel smoothers. The results indicate that Gaussian belief propagation can be used to construct competitive solvers, smoothers, and preconditioners for selected elliptic test problems. [ABSTRACT FROM AUTHOR]

Abstract This paper considers time-domain spatial multiplexing in MIMO wideband system, using an LU-based polynomial matrix decomposition. Because the corresponding pre- and post-filters are not paraunitary, the noise output power is amplified and the performance of the system is degraded, compared to QR-based spatial multiplexing approach. Degradations are important as the post-filter polynomial matrix is ill-conditioned. In this paper, we introduce simple transformations on the decomposition that solve the ill-conditioning problem. We show that this results in a MIMO spatial multiplexing scheme that is robust to noise and channel estimation errors. In the latter context, the proposed LU-based beamforming compares favorably to the QR-based counterpart in terms of complexity and bit error rate.

We consider LU and QR matrix decompositions using exact computations. We show that fraction-free Gauß–Bareiss reduction leads to triangular matrices having a non-trivial number of common row factors. We identify two types of common factors: systematic and statistical. Systematic factors depend on the reduction process, independent of the data, while statistical factors depend on the specific data. We relate the existence of row factors in the LU decomposition to factors appearing in the Smith–Jacobson normal form of the matrix. For statistical factors, we identify some of the mechanisms that create them and give estimates of the frequency of their occurrence. Similar observations apply to the common factors in a fraction-free QR decomposition. Our conclusions are tested experimentally. [ABSTRACT FROM AUTHOR]

Logarithm of a square matrix is a factor that clearly affects the quality or result of something. Logarithms are used to maintain exactitude in many numerical computations than the determinant. At present cloud computing is an emerging topic to perform this kind of operations. Cloud computing qualifies resource-awkward clients to solve their immense computational workloads through a cloud server that provides heavy computational power and it is also less expensive. This rising computing paradigm undoubtedly introduces new challenges and security concerns such as security, correctness, competency and result verifiability. Matrix log-determinant is a quite simple scientific, engineering and machine learning application. It inspired us to plan a protocol that satisfies all the concerns for the outsourcing of LDC to a cloud which is malicious in this paper. The primary concept to save privacy is to put in some changes on the original matrix for encryption and send the encrypted matrix to the cloud. The cloud then uses any LU decomposition algorithm on the encrypted matrix and divide the matrix for calculating the log-determinant. After returning from the cloud some transformations are applied to the returned result to get the accurate log-determinant of the original matrix. [ABSTRACT FROM AUTHOR]

A generalization of the well-known Helmert matrix was recently introduced in [Akkus, I., and G. Kizilaslan. 2019. Generalization of a statistical matrix and its factorization. Communications in Statistics - Theory and Methods. doi:10.1080/03610926.2019.1645854.]. We consider a modified form of this generalization and we obtain the LU decomposition (in closed form) of this modified generalized matrix. We also prove some special cases, that are all orthogonal, only using the Helmert matrix. [ABSTRACT FROM AUTHOR]

Many statistical problems involve the estimation of a d × d orthogonal matrix Q . Such an estimation is often challenging due to the orthonormality constraints on Q . To cope with this problem, we use the well-known PLU decomposition, which factorizes any invertible d × d matrix as the product of a d × d permutation matrix P , a d × d unit lower triangular matrix L , and a d × d upper triangular matrix U . Thanks to the QR decomposition, we find the formulation of U when the PLU decomposition is applied to Q . We call the result as PLR decomposition; it produces a one-to-one correspondence between Q and the d d - 1 / 2 entries below the diagonal of L , which are advantageously unconstrained real values. Thus, once the decomposition is applied, regardless of the objective function under consideration, we can use any classical unconstrained optimization method to find the minimum (or maximum) of the objective function with respect to L . For illustrative purposes, we apply the PLR decomposition in common principle components analysis (CPCA) for the maximum likelihood estimation of the common orthogonal matrix when a multivariate leptokurtic-normal distribution is assumed in each group. Compared to the commonly used normal distribution, the leptokurtic-normal has an additional parameter governing the excess kurtosis; this makes the estimation of Q in CPCA more robust against mild outliers. The usefulness of the PLR decomposition in leptokurtic-normal CPCA is illustrated by two biometric data analyses. [ABSTRACT FROM AUTHOR]

We propose a novel spatial bagging workflow for predictive ensemble machine learning that improves on standard bagging models. Our proposed method integrates spatial bootstrap for bagging with the number of effective sample size, n e f f , for integration of the spatial context of the dataset. We benchmark the improved performance over standard machine learning bagging models with a large number of two-dimensional synthetic datasets with varying degrees of Gaussian noise. For noise free datasets, both methods demonstrate equivalent accuracy; however, spatial bagging achieves this with a significantly smaller sample size, showcasing its improved efficiency. As data noise increases, spatial bagging consistently outperforms standard bagging, displaying an improved Mean Squared Error (MSE) and robustness against overfitting. Our proposed spatial bagging method computes the optimal effective sample size for spatial data, reducing model overfitting. Furthermore, our proposed method requires only the additional step of variogram calculation and modeling, and can be implemented with any predictive machine learning bagging model with minimal code modification. i.e., specification of the number of bootstrap samples as the number of effective data. We recommend using spatial bagging for improved predictions for any spatial data setting across diverse scientific fields, e.g., atmospheric, agricultural, subsurface resources etc. • Ensemble learning methods employ standard bagging, which is prone to overfitting. • We introduce a novel spatial bagging technique utilizing effective sample size. • Spatial bagging mitigates overfitting, enhancing robustness against noise. • Spatial bagging provides a computational advantage by using smaller data. [ABSTRACT FROM AUTHOR]

The copyright protection of three-dimensional (3D) content is a matter of interest in artistic and creative works due to the rights of the holder for the distribution of the material. However, although stereo images are widely used for the generation of 3D content, there is a little amount of research focused on copyright protection for this type of image. In this study, a novel invisible and blind color image watermarking framework for protecting the copyright of the stereo images based on Binocular Just Noticeable Difference (BJND) and lower-and-upper (LU) decomposition is proposed. In this framework, a color watermark is encoded to reduce the information. Then, the BJND model is calculated in the R channel of a stereo image, and finally, LU decomposition is applied into the G and B channels to embed and extract a color watermark. The BJND model has demonstrated a remarkable sensitivity comparable with the Human Visual System (HVS) in luminance changes such as the ones presented in stereo images, guaranteeing a good protection performance and a high imperceptibility. The proposed framework was compared with other state-of-the-art techniques, and it has demonstrated better performance and high resistance to attacks of JPEG compression or impulsive and Gaussian noise. Additionally, this novel framework does not affect the generation of the disparity map or the 3D content when the watermarked stereo images are used for their creation. [ABSTRACT FROM AUTHOR]

Many systems like the control systems and in communication systems, there is usually a demand for matrix inversion solution. This solution requires many operations, which makes it not possible or very hard to meet the needs for real-time constraints. Methods were exists to solve this kind of problems, one of these methods by using the LU decomposition of matrix which is a good alterative to matrix inversion. The LU matrices are two matrices, the L matrix, which is a lower triangular matrix, and the U matrix, which is an upper triangular matrix. In this paper, a design of dual-core processor is used as the hardware of the work and contain software was written to enable the two cores of the dual-core processor to work simultaneously in computing the value of the L matrix and U matrix. The result of this work are compared with other works that using single-core processor, and the results found that the time required in the cores of the dual-core is more less than using single-core. The designed dual-core processor is invoked using the VHDL language. [ABSTRACT FROM AUTHOR]

In this paper, we develop a new algorithm for solving diagonally dominant symmetric quasi-pentadiagonal Toeplitz linear systems. Numerical experiments are given in order to illustrate the validity and efficiency of our algorithm. [ABSTRACT FROM AUTHOR]

The field-programmable gate array (FPGA)-based real-time simulator takes advantage of many merits of FPGA, such as small time-step, high simulation precision, rich I/O interface resources, and low cost. The sparse linear equations formed by the node conductance matrix need to be solved repeatedly within each time-step, which introduces great challenges to the performance of the real-time simulator. In this paper, a fine-grained solver of the FPGA-based real-time simulator for active distribution networks is designed to meet the computational demand. The framework of the solver, offline process design on PC and online process design on FPGA are proposed in detail. The modified IEEE 33-node system with photovoltaics is simulated on a 4-FPGA-based real-time simulator. Simulation results are compared with PSCAD/EMTDC under the same conditions to validate the solver design.

In this paper, we present some single-pass randomized algorithms to compute LU decomposition. These algorithms need only one pass over the original matrix and hence are very suitable for extremely large and high-dimensional matrix stored outside of core memory or generated in a streaming fashion. Rigorous error bounds and complexity of these algorithms are provided. Numerical experiments show that these single-pass algorithms have the similar accuracy and runtime (excluding the cost of matrix transfer) compared with the state-of-the-art randomized algorithms for LU decomposition. [ABSTRACT FROM AUTHOR]

In this paper, the elementary transformation is discussed for split quaternion matrices and the upper triangulation process is given. Then two new determinants are defined by means of real and complex representation matrices and the sufficient condition for the existence of LU decomposition is obtained. Finally, the inverse of the split quaternion matrix is studied and its existence condition and computation method are given. [ABSTRACT FROM AUTHOR]

The problem of solving overdetermined, underdetermined, singular, or ill conditioned SLAEs using matrix canonization is considered. A modification of an existing canonization algorithm based on matrix decomposition is proposed. Formulas using LU decomposition, QR decomposition, LQ decomposition, or singular value decomposition, depending on the properties of the given matrix, are obtained. A method for evaluating the condition number of the canonization problem is proposed. It is based on computing the norm of the matrices obtained as a result of canonization; this method does not require the original matrix to be inverted. A general step-by-step matrix canonization algorithm is described and implemented in MATLAB. The implementation is tested on a set of 100 000 randomly generated matrices. The testing results confirmed the validity and efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, the higher-order singular value decomposition and the LU decomposition are applied to solve the general multi-linear systems, rather than special structure like M − tensor. Here, we transform the general tensor system into a form with a special structure. In addition, an iterative refinement method is constructed by the information of the coefficient tensor itself, the higher-order singular value, instead of the spectra of iterative tensor. Finally, the numerical experimental results are given to demonstrate the efficiency of the iterative refinement method. [ABSTRACT FROM AUTHOR]

In recent years, several algorithms which approximate matrix decomposition have been developed. These algorithms are based on metric conservation features for linear spaces of random projection types. We present a new algorithm, which achieves with high probability a rank- |$r$| singular value decomposition (SVD) approximation of an |$n \times n$| matrix and derive an error bound that does not depend on the first |$r$| singular values. Although the algorithm has an asymptotic complexity similar to state-of-the-art algorithms and the proven error bound is not as tight as the state-of-the-art bound, experiments show that the proposed algorithm is faster in practice while providing the same error rates as those of the state-of-the-art algorithms. We also show that an i.i.d. sub-Gaussian matrix with large probability of having null entries is metric conserving. This result is used in the SVD approximation algorithm, as well as to improve the performance of a previously proposed approximated LU decomposition algorithm. [ABSTRACT FROM AUTHOR]

Digital watermarking technology is countered as one of the best solutions for solving the problem of copyright protection, content identification, duplication, data authentication and distribution of the digital media due to the great developments in Internet technology. Lately, protection of digital audio signals has fascinated the attention of researchers in watermarking area. In this paper, we propose a new high-capacity, transparent and robust audio watermarking scheme based on synergy DCT transform and LU decomposition using genetic algorithm (GA) for embedding watermark data into an audio signal. In the proposed scheme, the original audio signal is first split into set of segments, and then each segment is taken DCT transform. After then, DCT coefficients are framed and finally decomposed by LU factorization to obtain upper and lower-triangular matrices. GA is applied to optimally select frequency sub-band and frame length. The watermark data is embedded into upper-triangular matrices based on Fibonacci sequence. The watermark can be blindly drawn out without knowledge of the original audio signal. Experimental results show that the transparency and robustness of the proposed scheme is satisfied, and that high watermarking payload capacity (2002 bit per second) is obtained through the proposed watermarking scheme. [ABSTRACT FROM AUTHOR]

We derive the asymptotic distribution for the LU decomposition, that is, the Cholesky decomposition, of realized covariance matrix. Distributional properties are combined with an existing generalized heterogeneous autoregressive (GHAR) method for forecasting realized covariance matrix, which will be referred to as a generalized HARQ (GHARQ) method. An out-of-sample forecast comparison of a real data set shows that the proposed GHARQ method outperforms other existing methods in terms of optimizing the variances of portfolios. [ABSTRACT FROM AUTHOR]

A square matrix over a semiring is called strongly invertible if all of its leading principal submatrices are invertible. In this paper, the strongly invertible matrices over a semiring are discussed and an equivalent condition for a square matrix over a semiring to be strongly invertible is given. Also, some equivalent descriptions are obtained for a semiring over which the product of any two strongly invertible matrices with the same size is strongly invertible. Some of the results obtained in this paper generalize and develop previous results for matrices over the field of complex numbers and matrices over commutative rings. [ABSTRACT FROM AUTHOR]

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[Display omitted] • Efficient scheme for two-electron integrals (TEIs) in two-component method. • Decompositions of the matrix representation of TEIs. • Cholesky decomposition were applied to the symmetric matrices. • Lower-upper decomposition were applied to the asymmetric matrices. • Number of elements to compute TEIs were reduced without a loss of accuracy. The Cholesky decomposition (CD) and lower–upper decomposition (LUD) of two-electron integral (TEI) matrices were applied to a spin-free infinite-order two-component (IOTC) relativistic Hamiltonian. In the Hamiltonian, the evaluation of three types of TEIs of primitive basis functions and the unitary transformation of TEIs are required. CD was used to calculate the symmetric TEI matrices: Coulomb-like and specific spin-free interaction terms. LUD was used for the asymmetric TEI matrix: Darwin-like term. Unitary transformation was performed on the decomposed matrices. Numerical assessments suggest that the proposed method can reduce the computational cost without loss of accuracy. [ABSTRACT FROM AUTHOR]

Matrix inversion is a fundamental operation for solving linear equations for many computational applications, especially for various emerging big data applications. However, it is a challenging task to invert large-scale matrices of extremely high order (several thousands or millions), which are common in most Web-scale systems, such as social networks and recommendation systems. In this paper, we present an lower upper decomposition-based block-recursive algorithm for large-scale matrix inversion. We present its well-designed implementation with optimized data structure, reduction of space complexity, and effective matrix multiplication on the Spark parallel computing platform. The experimental evaluation results show that the proposed algorithm is efficient to invert large-scale matrices on a cluster composed of commodity servers and is scalable for inverting even larger matrices. The proposed algorithm and implementation will become a solid foundation for building a high-performance linear algebra library on Spark for big data processing and applications.