Generalized Singular Value Decomposition Research Articles

Comparative Analysis of NOMA and OMA Schemes: GSVD-based NOMA Systems and the Role of Mobile Edge Computing

This paper presents a comprehensive study that examines the fundamental concept of the non-orthogonal multiple access (NOMA) scheme and provides its detailed comparison with the orthogonal multiple access (OMA) technique. Furthermore, the paper explores the application of the generalized singular value decomposition (GSVD) method in conjunction with NOMA, accompanied by a detailed review of GSVD-based NOMA systems. This study also introduces the concept of mobile edge computing (MEC) and extensively discusses its key parameters. Furthermore, a comprehensive analysis of NOMA MEC is presented, shedding light on its potential advantages and challenges. The aims of this study are to provide a comprehensive understanding of the aforementioned topics and contribute to the advancement of MIMO-NOMA systems.

Thick-restarted joint Lanczos bidiagonalization for the GSVD

The computation of the partial generalized singular value decomposition (GSVD) of large-scale matrix pairs can be approached by means of iterative methods based on expanding subspaces, particularly Krylov subspaces. We consider the joint Lanczos bidiagonalization method, and analyze the feasibility of adapting the thick restart technique that is being used successfully in the context of other linear algebra problems. Numerical experiments illustrate the effectiveness of the proposed method. We also compare the new method with an alternative solution via equivalent eigenvalue problems, considering accuracy as well as computational performance. The analysis is done using a parallel implementation in the SLEPc library.

Optical double-image cryptosystem based on generalized singular value decomposition and five-dimensional hyperchaotic maps.

In this paper, we propose an asymmetric optical double-image cryptosystem based on generalized singular value decomposition (GSVD) and five-dimensional (5D) hyperchaotic maps. In the proposed cryptosystem, the two plain images are first decomposed into five components by the GSVD operation. The two unitary matrices obtained by GSVD are encoded as a complex function, which is then modulated by the chaotic random phase masks (CRPMs). The private key and the final encryption result are generated by phase-truncation and amplitude-truncation operations. The GSVD operation can decompose two images at the same time and is used to generate the private key that enables the encryption process to be asymmetric. Compared with the existing phase-truncated-based cryptosystems, our cryptosystem can improve security against a special attack. In addition, the CRPMs are generated by 5D hyperchaotic maps, which have a larger parameter space and better randomness. Numerical simulation results are shown to verify the feasibility and robustness of our cryptosystem. Furthermore, the proposed cryptosystem can be extended to encrypt multiple images conveniently.

Text encryption by image encryption key with based on generalized singular value decomposition (GSVD)

This paper we will address the encryption process text is generally through a number of processes of dispersion of the later values in a matrix is A through remittances linear in addition to the use of GSVD enters, including one key which is gray image or to be possible colored but to be converted to a grayscale image, where work will begin the application process distracting initial. In a compound to generate new matrices B1 and B2 of the original text matrix and which enters the mean of the matrix of images key.The second phase of the dispersion is using GSVD on the last matrices with the matrix of the key and switch C matrix among the two spectral for the new matrices B1 with key and B2 with key to generation new matrices B11,B22, which represents putting as correlative F = [B11;B22] who represents the matrix of the initial of the encoded text. For purpose of ascertaining the accuracy of the work we calculated the matrix of absolute value of the difference between the matrix of numbers for original text characters lettering before encryption and matrix of numbers for text characters after decryption to show us zero matrix, as well as the calculation of the average values of this matrix was zero.

Depth-wise Decomposition for Accelerating Separable Convolutions in Efficient Convolutional Neural Networks

Very deep convolutional neural networks (CNNs) have been firmly established as the primary methods for many computer vision tasks. However, most state-of-the-art CNNs are large, which results in high inference latency. Depth-wise separable convolution has been proposed for image recognition tasks on platforms with limited computation power, such as robots and self-driving cars. Any regular deep CNN has a depth-wise separable counterpart, which is faster, but less accurate, when equally trained. In this paper, we propose a novel decomposition approach based on SVD, namely depth-wise decomposition, for converting regular convolutions into depth-wise separable convolutions post-training, while maintaining high accuracy. We show that our approach generalizes to the multi-channel and multi-layer cases, by applying Generalized Singular Value Decomposition (GSVD). We conduct thorough experiments with the ShuffleNet V2 model on a large-scale image recognition dataset: ImageNet. Our approach outperforms the baseline, channel decomposition. Moreover, our approach improves the Top-1 accuracy of ShuffleNet V2 by ∼2%.

A Cross-Product Free Jacobi–Davidson Type Method for Computing a Partial Generalized Singular Value Decomposition of a Large Matrix Pair

The FEAST eigensolver is extended to the computation of the singular triplets of a large matrix $A$ with the singular values in a given interval. The resulting FEAST SVDsolver is subspace iteration applied to an approximate spectral projector of $A^TA$ corresponding to the desired singular values in a given interval, and constructs approximate left and right singular subspaces corresponding to the desired singular values, onto which $A$ is projected to obtain Ritz approximations. Differently from a commonly used contour integral-based FEAST solver, we propose a robust alternative that constructs approximate spectral projectors by using the Chebyshev--Jackson polynomial series, which are symmetric positive semi-definite with the eigenvalues in $[0,1]$. We prove the pointwise convergence of this series and give compact estimates for pointwise errors of it and the step function that corresponds to the exact spectral projector. We present error bounds for the approximate spectral projector and reliable estimates for the number of desired singular triplets, establish numerous convergence results on the resulting FEAST SVDsolver, and propose practical selection strategies for determining the series degree and for reliably determining the subspace dimension. The solver and results on it are directly applicable or adaptable to the real symmetric and complex Hermitian eigenvalue problem. Numerical experiments illustrate that our FEAST SVDsolver is at least competitive with and is much more efficient than the contour integral-based FEAST SVDsolver when the desired singular values are extreme and interior ones, respectively, and it is also more robust than the latter.

Improved key performance indicator-partial least squares method for nonlinear process fault detection based on just-in-time learning

Key performance indicator-partial least squares (KPI-PLS) achieves linear indicator fault detection through the appropriate decomposition of variables into related and unrelated parts. This approach uses the projection matrix obtained by general singular value decomposition to decompose the residual space of PLS and thus achieves a more reasonable division of data space. However, KPI-PLS has limitations in handling nonlinear issues. Such problems can be effectively addressed by a local model, just-in-time learning (JITL). Therefore, in this study, an improved KPI-PLS method combined with JITL (JITL-KPI-PLS) is proposed to achieve nonlinear fault detection. The local model is built using hierarchical clustering to improve computational efficiency of JITL for searching relevant samples. By updating the model and control limit in real time, the current status of online samples can be better tracked. Finally, better fault detection performance of the proposed method is confirmed on the Tennessee Eastman benchmark and the Zhoushan thermal power plant.

Machine learning conservation laws from differential equations.

We present a machine learning algorithm that discovers conservation laws from differential equations, both numerically (parametrized as neural networks) and symbolically, ensuring their functional independence (a nonlinear generalization of linear independence). Our independence module can be viewed as a nonlinear generalization of singular value decomposition. Our method can readily handle inductive biases for conservation laws. We validate it with examples including the three-body problem, the KdV equation, and nonlinear Schrödinger equation.

Hybrid NOMA based MIMO offloading for mobile edge computing in 6G networks

Non-orthogonal multiple access (NOMA), multiple-input multiple-output (MIMO) and mobile edge computing (MEC) are prominent technologies to meet high data rate demand in the sixth generation (6G) communication networks. In this paper, we aim to minimize the transmission delay in the MIMO-MEC in order to improve the spectral efficiency, energy efficiency, and data rate of MEC offloading. Dinkelbach transform and generalized singular value decomposition (GSVD) method are used to solve the delay minimization problem. Analytical results are provided to evaluate the performance of the proposed Hybrid-NOMA-MIMO-MEC system. Simulation results reveal that the H-NOMA-MIMO-MEC system can achieve better delay performance and lower energy consumption compared to OMA.

Two Harmonic Jacobi–Davidson Methods for Computing a Partial Generalized Singular Value Decomposition of a Large Matrix Pair

Two harmonic extraction based Jacobi--Davidson (JD) type algorithms are proposed to compute a partial generalized singular value decomposition (GSVD) of a large regular matrix pair. They are called cross product-free (CPF) and inverse-free (IF) harmonic JDGSVD algorithms, abbreviated as CPF-HJDGSVD and IF-HJDGSVD, respectively. Compared with the standard extraction based JDGSVD algorithm, the harmonic extraction based algorithms converge more regularly and suit better for computing GSVD components corresponding to interior generalized singular values. Thick-restart CPF-HJDGSVD and IF-HJDGSVD algorithms with some deflation and purgation techniques are developed to compute more than one GSVD components. Numerical experiments confirm the superiority of CPF-HJDGSVD and IF-HJDGSVD to the standard extraction based JDGSVD algorithm.

Multilinear Generalized Singular Value Decomposition (ML-GSVD) and Its Application to Multiuser MIMO Systems

In this paper, we introduce a Multilinear Generalized Singular Value Decomposition (ML-GSVD) for two or more matrices with one common dimension. The ML-GSVD extends the Generalized Singular Value decomposition (GSVD) of two matrices to higher orders. The proposed decomposition allows us to jointly factorize a set of matrices with one common dimension. In comparison with other approaches that extend the GSVD, the ML-GSVD preserves the essential properties of the original (matrix-based) GSVD, such as orthogonality of the second-mode factor matrices as well as the subspace structure of the third-mode factor matrices. We introduce an ALS-based algorithm to compute the ML-GSVD, which has been inspired by PARAFAC2 decomposition algorithms. In addition, we present an application of the ML-GSVD for transceiver optimization in multicast and unicast MIMO-OFDM systems. Our numerical results show that the proposed ML-GSVD multicast and unicast beamforming outperforms existing state-of-the-art schemes in terms of the sum rate.

Improved Generalized Cross-Validation and Unbiased Predictive Risk Estimator Methods Using the RGSVD: Application to Inversion of Potential Field Data

The inversion of potential field data has widely utilized the generalized cross-validation (GCV) and the unbiased predictive risk estimator (UPRE) methods to determine the regularization parameter. However, these two methods are time-consuming and it is difficult for them to determine the optimal linear search range including the optimal regularization. To solve these problems, this article improves the GCV and UPRE methods using the RGSVD (randomized generalized singular value decomposition) algorithm. The improved methods first use the randomized algorithm to compute an approximate generalized singular value decomposition (GSVD) with less computational time. Then, the optimal linear search range is determined based on the generalized singular values. Finally, the GCV and the UPRE functions are efficiently computed on the basis of the results from the RGSVD algorithm. In this way, the GCV and UPRE methods using the RGSVD algorithm are able to determine the optimal regularization parameter fast and effectively. One comparative test shows the effectiveness and efficiency of the GCV and the UPRE methods using the RGSVD algorithm.

GSVD-Based MIMO-NOMA Security Transmission

Recently, the physical layer security over multiple-input multiple-output (MIMO) non-orthogonal multiple-access (NOMA) systems has been taken more and more seriously. In this letter, a generalized singular value decomposition (GSVD) based MIMO-NOMA security transmission scheme is proposed. The distribution characteristics of the GSVD are applied as tools for the outage performance analysis for the scheme. To minimize the outage probability, the optimal power allocation coefficients are obtained. The theoretical contrast with the Orthogonal Multiple Access (OMA) scheme is also presented, which proves the more superior performance, especially at high signal-noise ratio (SNR) condition. Simulation results are developed to verify the analytical results.

Randomized algorithms for generalized singular value decomposition with application to sensitivity analysis

AbstractThe generalized singular value decomposition (GSVD) is a valuable tool that has many applications in computational science. However, computing the GSVD for large‐scale problems is challenging. Motivated by applications in hyper‐differential sensitivity analysis (HDSA), we propose new randomized algorithms for computing the GSVD which use randomized subspace iteration and weighted QR factorization. Detailed error analysis is given which provides insight into the accuracy of the algorithms and the choice of the algorithmic parameters. We demonstrate the performance of our algorithms on test matrices and a large‐scale model problem where HDSA is used to study subsurface flow.

An element-wise generalized singular value decomposition (GSVD) based image encryption algorithm

The image encryption algorithm aids a secure data transmission. In our proposed method, we use the generalized singular value decomposition (GSVD) as the encryption tool. Here, the secret image will be decomposed with respect to the key and generates the ciphered text. Further, to improve its security, the Arnold transforms and the XOR operation is used respectively. The region-wise applied GSVD reduce the time consumption of the conventional image encryption methods. Here, the GSVD decomposition is element-wise, so, the algorithm is more robust. The generated cipher text is transmitted and the key is generated with the combination of the key used to decompose the secret image pixel in GSVD module, the Arnold transform key and the XOR operator key. In receiver end, the ciphered image is synthesized with the key received in a separate channel. This proposed model is lossless encryption method.

The joint bidiagonalization process with partial reorthogonalization

The joint bidiagonalization(JBD) process is a useful algorithm for the computation of the generalized singular value decomposition(GSVD) of a matrix pair. However, it always suffers from rounding errors, which causes the Lanczos vectors to loss their mutual orthogonality. In order to maintain some level of orthongonality, we present a semiorthogonalization strategy. Our rounding error analysis shows that the JBD process with the semiorthogonalization strategy can ensure that the convergence of the computed quantities is not affected by rounding errors and the final accuracy is high enough. Based on the semiorthogonalization strategy, we develop the joint bidiagonalization process with partial reorthogonalization(JBDPRO). In the JBDPRO algorithm, reorthogonalizations occur only when necessary, which saves a big amount of reorthogonalization work compared with the full reorthogonalization strategy. Numerical experiments illustrate our theory and algorithm.

Polynomial GSVD Beamforming for Two-User Frequency-Selective MIMO Channels

In this paper, we propose a generalized singular value decomposition (GSVD) for polynomial matrices, or polynomial GSVD (PGSVD). We then consider PGSVD-based beamforming for two-user, frequency-selective, multiple-input multiple-output (MIMO) multicasting. The PGSVD can jointly factorize two frequency-selective MIMO channels, producing a set of virtual channels (VCs), split into: private channels (PCs) and common channels (CCs). An important advantage of the proposed PGSVD-based beamformer, over the application of GSVD independently to each frequency bin of the orthogonal frequency division multiplexing (OFDM) scheme, is that it can facilitate different modulation and/or access schemes to various users. Using computer simulations, we characterize the bit error rate performance of our two-user MIMO multicasting system for different PCs/CCs configurations. Here, we also propose an OFDM-GSVD benchmark system, and show that our PGSVD-based beamformer compares favorably to this benchmark under erroneous and uncertain MIMO channel conditions, in addition to its advantage of facilitating heterogeneous modulation and access for various users.

Implicit Hari–Zimmermann algorithm for the generalized SVD on the GPUs

A parallel, blocked, one-sided Hari–Zimmermann algorithm for the generalized singular value decomposition (GSVD) of a real or a complex matrix pair [Formula: see text] is here proposed, where F and G have the same number of columns, and are both of the full column rank. The algorithm targets either a single graphics processing unit (GPU), or a cluster of those, performs all non-trivial computation exclusively on the GPUs, requires the minimal amount of memory to be reasonably expected, scales acceptably with the increase of the number of GPUs available, and guarantees the reproducible, bitwise identical output of the runs repeated over the same input and with the same number of GPUs.

Key-performance-indicator-related state monitoring based on kernel canonical correlation analysis

As a multivariate statistical analysis method, canonical correlation analysis (CCA) performs well for state monitoring of linear processes, but most industrial processes are nonlinear. To solve this problem, kernel canonical correlation analysis (KCCA) has been adopted; however, KCCA still has key performance indicators (KPI)-related issue. In this paper, two improved KCCA methods are proposed to deal with KPI-related issue. One is performing singular value decomposition (SVD) on the correlation coefficient matrix, then the kernel matrix can be divided into KPI-related and KPI-unrelated parts. Another one is performing general singular value decomposition (GSVD) on two coefficient matrices. In addition, this paper also performs fault detectability analysis and computational complexity analysis on these two methods. Finally, the Tennessee Eastman (TE) process is used in this study to verify the efficacy of these two proposed methods.

Design and optimization for UAV-enabled two-way relaying system with SWIPT

In this paper, we investigate the resource allocation scheme for an unmanned-aerial-vehicle-enable (UAV-enabled) two-way relaying system with simultaneous wireless information and power transfer (SWIPT), where two user equipment exchange information with the help of UAV relay and harvest energy through power splitting (PS) scheme. Under the transmission power constraints at UEs and UAV relay, a non-convex intractable optimization problem is formulated which maximizes the sum retained energy of two UEs while satisfying the minimum signal-to-noise ratio requirement. We decouple the complicated beamforming and PS factor optimization problem into three solvable subproblems and propose an efficient alternating optimization scheme. Subsequently, in order to reduce the complexity, a robust scheme based on generalized singular value decomposition (GSVD) is designed. Finally, numerical results verify the robustness and effectiveness of the two proposed schemes.