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Real-time object tracking is an important step of many modern image processing applications. The efficient hardware design of real-time object tracker must achieve the desired accuracy while satisfying the frame rate requirements for a variety of image sizes. The existing methods of visual tracking employ sophisticated algorithms and challenge the capabilities of most embedded architectures. Discriminative scale space tracking is one algorithm that is capable of demonstrating good performance with affordable complexity. It has a high degree of parallelism which can be exploited for efficient implementation of reconfigurable hardware architectures. This paper proposes a real-time implementation of the discriminative scale-space tracker on FPGA for the major blocks. A careful design exploration of core mathematical operations of the tracking algorithm is performed to improve their hardware utilization and timing performance. Among the core functional units optimized in this work, the discrete Fourier transform achieves a computational time improvement of 92% relative to existing works, QR factorization achieves a 2.3$$\times$$ × reduction in resource utilization, and singular value decomposition yields a 3.8$$\times$$ × improvement in processing time. The proposed data path architecture is designed using Vivado HLS tool set and implemented for Zync Zed Board (xc7z020clg484-1). For an input image size of 320 $$\times$$ × 240, the proposed architecture achieves a mean 25.38 fps.

In this study, we investigate a subspace-system identification method that estimates the modal characteristics and damping ratio of structures using a small number of sensor measurement responses. With this method, a Kalman state vector is first constructed from the input and measurement data, and then the state-space system matrix is optimized from it. For this purpose, a block Hankel matrix is constructed using the input data of the load applied to the structure and the measurement data of the response, and then the matrix is decomposed through QR factorization. Then, singular value decomposition is used to obtain the extended observability matrix and state-space system order $$n$$ , and the state-space system matrix constituting the Kalman state vector is obtained via least-squares optimization. The natural frequency and damping ratio of the structural system are calculated from the eigenvalue of the obtained system matrix. To verify the effectiveness of this subspace state-space system identification method, the natural frequency and mode shape of a simply supported beam and cantilever beam were calculated using a limited number of measurement data, and the results were compared with an exact solution to evaluate the accuracy of the estimation results of the dynamic characteristics. Furthermore, the modal characteristics and damping ratio of a real bridge were estimated from acceleration data. The stability chart calculated from the example shows that the subspace state-space system identification method discussed in this study can accurately identify the modal characteristics of structures using only a small number of measurement data.

For a matrix X ∈ Rn×p, we provide an analytic formula that keeps track of an orthonormal basis for the range of X under rank-one modifications. More precisely, we consider rank-one adaptations Xnew = X + abT of a given X with known matrix factorization X = UW, where U ∈ Rn×p is column-orthogonal and W ∈ Rp×p is invertible. Arguably, the most important methods that produce such factorizations are the singular value decomposition (SVD), where X = UW = U(ΣV T), and the QR-decomposition, where X = UW = QR. We give a geometric description of rank-one adaptations and derive a closed-form expression for the geodesic line that travels from the subspace S = ran(X) to the subspace Snew = ran(Xnew) = ran(UnewWnew). This leads to update formulas for orthogonal matrix decompositions, where both Unew and Wnew are obtained via elementary rank-one matrix updates in O(np) time for n » p. Moreover, this allows us to determine the subspace distance and the Riemannian midpoint between the subspaces S and Snew without additional computational effort.

We consider cost-constrained sparse sensor selection for full-state reconstruction, applying a well-known greedy algorithm to dynamical systems for which the usual singular value decomposition (SVD) basis may not be available or preferred. We apply the cost-modified, column-pivoted QR decomposition to a physically relevant basis -- the pivots correspond to sensor locations, and these locations are penalized with a heterogeneous cost function. In considering different bases, we are able to account for the dynamics of the particular system, yielding sensor arrays that are nearly Pareto optimal in sensor cost and performance in the chosen basis. This flexibility extends our framework to include actuation and dynamic estimation, and to select sensors without training data. We provide three examples from the physical and engineering sciences and evaluate sensor selection in three dynamically relevant bases: truncated balanced modes for control systems, dynamic mode decomposition (DMD) modes, and a basis of analytic modes. We find that these bases all yield effective sensor arrays and reconstructions for their respective systems. When possible, we compare to results using an SVD basis and evaluate tradeoffs between methods., 12 pages, 12 figures

It is well known that the singular value decomposition (SVD), as the best rank-revealing factorization, furnishes the best rank- $k$ approximation to a dense matrix with expensive computational cost of $\mathcal O(M^{2}N+N^{2}M+\min (M,N)^{3})$ . Moreover, it is hard to implement in parallel, which challenges the memory storage in wireless communications data-driven system. In this letter, a fast rank-revealing technique, namely, bilateral random projections (BRP) with $\mathcal O(MN)$ operations, is exploited for two-dimensional (2-D) frequency estimation of a complex sinusoid in noisy environment. Based on the resulting data matrix, whereafter, two-stage QR factorization frequency estimation method with the weighted least squares (WLS) as solver, is proposed to reduce the computational complexity of those SVD-based frequency estimators. Simulation results demonstrate the efficiency of the proposed algorithm in comparison with several frequency estimation approaches and the Cramer-Rao lower bound (CRLB) as benchmark.

This chapter provides a mathematical background to define a mathematical model of cellular systems, and design and optimize wireless communications. It describes the approximation problems and establishes a solution in vector spaces. The chapter reviews some important matrix factorization. Most matrix computation from original data is not easy to calculate in an explicit way. Matrix factorization (or decomposition) allows us to rephrase some matrix forms in such a way that they can be solved more easily. Various matrix decompositions covered include lower upper decomposition, Cholesky decomposition, QR decomposition, and singular value decomposition. The chapter explains the least square algorithm in the context of a regression problem. Regression analysis is a useful tool for forecasting, error reduction or data analysis.

This paper is concerned with normwise errors in the LS estimation for linear regression. It provides probabilistic tail bounds for the normwise error between the computed least squares estimator and the parameter vector, when the least squares problem is solved in floating point arithmetic using either the normal equations method or a backward stable method (for example, using the Householder QR factorization or the singular value decomposition). These bounds are used to provide a condition under which the computationally more efficient normal equations method can safely be used instead of a backward stable method, without any loss of accuracy in the computed estimator.

Rank-revealing matrix decompositions provide an essential tool in spectral analysis of matrices, including the Singular Value Decomposition (SVD) and related low-rank approximation techniques. QR with Column Pivoting (QRCP) is usually suitable for these purposes, but it can be much slower than the unpivoted QR algorithm. For large matrices, the difference in performance is due to increased communication between the processor and slow memory, which QRCP needs in order to choose pivots during decomposition. Our main algorithm, Randomized QR with Column Pivoting (RQRCP), uses randomized projection to make pivot decisions from a much smaller sample matrix, which we can construct to reside in a faster level of memory than the original matrix. This technique may be understood as trading vastly reduced communication for a controlled increase in uncertainty during the decision process. For rank-revealing purposes, the selection mechanism in RQRCP produces results that are the same quality as the standard algorithm, but with performance near that of unpivoted QR (often an order of magnitude faster for large matrices). We also propose two formulas that facilitate further performance improvements. The first efficiently updates sample matrices to avoid computing new randomized projections. The second avoids large trailing updates during the decomposition in truncated low-rank approximations. Our truncated version of RQRCP also provides a key initial step in our truncated SVD approximation, TUXV. These advances open up a new performance domain for large matrix factorizations that will support efficient problem-solving techniques for challenging applications in science, engineering, and data analysis., Comment: Revised from Randomized QR with Column Pivoting for publication in SIGEST

The periodicity of an object is one of its most important visual characteristics. Recently, several low-rank/sparse matrix decomposition techniques have indicated that a relationship exists between the frequency components of the motion matrix and its decomposition components. This relationship was mostly identified based on empirical evidence without proper analysis, which led to an unclear understanding and poor utilization. This paper attempts to establish the relationship between the periodic components in the motion matrix and its singular value decomposition (SVD) components. The transformation of the periodic components of the motion matrix through QR factorization and Golub–Kahan bidiagonalization, which are the two essential steps of SVD, is thoroughly discussed and analyzed. Furthermore, the two cases of fully and partially periodic motion matrices are considered where the relationships between their frequency components and left singular vectors are established. The performance of the proposed SVD-based scheme is validated using real videos with ground truth. The results show that the proposed scheme correctly estimates the singular vectors that contain the frequency information.

This paper presents the transform domain image steganography schemes using three popular matrix factorization techniques and contourlet transform. It is known that security of image steganography is mainly evaluated using undetectability of stego image when steganalyzer examines it in order to detect the presence of hidden secret information. Good imperceptibility only suggests eavesdropper’s inability to suspect about the hidden information; however stego image may be analyzed by applying certain statistical checks when it is being transmit- ted through the channel. This work focusses on improving undetectability by employing ma- trix decomposition techniques along with transform domain image steganography. Singular value decomposition (SVD), QR factorization, Nonnegative matrix factorization (NMF) are employed to decompose contourlet coefficients of cover image and secret is embedded into its matrix factorized coefficients. The variety of investigations include the effect of matrix decomposition techniques on major attributes of image steganography like imperceptibility, robustness to a variety of image processing operations, and universal steganalysis perfor- mance. Better imperceptibility, large capacity, and poor detection accuracy compared to existing work validate the efficacy of the proposed image steganography algorithm. Compa- rative analysis amongst three matrix factorization methods is also presented and analyzed.

International audience; Element-wise techniques based on SVD or QR Decomposition completely get rid of the usual ill-conditioning inherent to the plane-wave discretizations of the Ultra-Weak Variational Formulation (UWVF) for the Helmholtz equation. Associated preconditioning strategies lead to very low condition numbers of the corresponding linear system matrices, without any limitation on the number of plane waves per element. In addition, some of these procedures have the advantage of considerably reducing the size of the final system to be solved without altering the accuracy of the numerical solution.

Network embedding approaches have recently attracted considerable interest as they learn low-dimensional vector representations of nodes. Embeddings based on the matrix factorization are effective but they are usually computationally expensive due to the eigen-decomposition step. In this paper, we propose a Random RangE FInder based Network Embedding (REFINE) algorithm, which can perform embedding on one million of nodes (YouTube) within 30 seconds in a single thread. REFINE is 10x faster than ProNE, which is 10-400x faster than other methods such as LINE, DeepWalk, Node2Vec, GraRep, and Hope. Firstly, we formulate our network embedding approach as a skip-gram model, but with an orthogonal constraint, and we reformulate it into the matrix factorization problem. Instead of using randomized tSVD (truncated SVD) as other methods, we employ the Randomized Blocked QR decomposition to obtain the node representation fast. Moreover, we design a simple but efficient spectral filter for network enhancement to obtain higher-order information for node representation. Experimental results prove that REFINE is very efficient on datasets of different sizes (from thousand to million of nodes and edges) for node classification, while enjoying a good performance.

Tucker decomposition is a low-rank tensor approximation that generalizes a truncated matrix singular value decomposition (SVD). Existing parallel software has shown that Tucker decomposition is particularly effective at compressing terabyte-sized multidimensional scientific simulation datasets, computing reduced representations that satisfy a specified approximation error. The general approach is to get a low-rank approximation of the input data by performing a sequence of matrix SVDs of tensor unfoldings, which tend to be short-fat matrices. In the existing approach, the SVD is performed by computing the eigendecomposition of the Gram matrix of the unfolding. This method sacrifices some numerical stability in exchange for lower computation costs and easier parallelization. We propose using a more numerically stable though more computationally expensive way to compute the SVD by preprocessing with a QR decomposition step and computing an SVD of only the small triangular factor. The more numerically stable approach allows us to achieve the same accuracy with half the working precision (for example, single rather than double precision). We demonstrate that our method scales as well as the existing approach, and the use of lower precision leads to an overall reduction in running time of up to a factor of 2 when using 10s to 1000s of processors. Using the same working precision, we are also able to compute Tucker decompositions with much smaller approximation error.

Massive Multiple In and Multiple Out (MIMO) is being used in the fifth generation of wireless communication systems. As the number of antennas increases, the computational complexity grows dramatically, and this involves matrix calculations with complex numbers. We have designed and implemented a general matrix arithmetic processor to accelerate these calculations, including matrix multiplication, Singular Value Decomposition (SVD), and QR decomposition, on an FPGA. The design can be implemented in fixed-point or single-precision floating-point depending on the requirements of the application. The system behavior can be controlled by instructions such as elementary multiplication, rotation and vector projection, which allows the system to work as a coprocessor in a baseband System on a Chip (SoC). Latency for changing from one matrix computation to another is just a few clock cycles, providing the low latency required for edge processing. The design is implemented and verified on the Xilinx RFSoC development board using fixed point and single precision floating point numbers and matrix sizes of 8 × 8 and 16 × 16.

Datacenter disaggregation provides numerous benefits to both the datacenter operator and the application designer. However switching from the server-centric model to a disaggregated model requires developing new programming abstractions that can achieve high performance while benefiting from the greater elasticity. To explore the limits of datacenter disaggregation, we study an application area that near-maximally benefits from current server-centric datacenters: dense linear algebra. We build NumPyWren, a system for linear algebra built on a disaggregated serverless programming model, and LAmbdaPACK, a companion domain-specific language designed for serverless execution of highly parallel linear algebra algorithms. We show that, for a number of linear algebra algorithms such as matrix multiply, singular value decomposition, Cholesky decomposition, and QR decomposition, NumPyWren's performance (completion time) is within a factor of 2 of optimized server-centric MPI implementations, and has up to 15% greater compute efficiency (total CPU-hours), while providing fault tolerance.

This paper aims to the solution of large-scale discrete-time algebraic Riccati equation D(X) ≡ −X + AT X(I + GX)−11A + H = 0 with numerically low-ranked solutions. Based on the original structure-preserving doubling algorithm (SDA), we develop the large-scale doubling algorithm by applying Sherman-Morrison-Woodbury formula, using QR decomposition, SVD to compress and truncate the approximate solutions properly. The resulting large-scale doubling algorithm has an O(n) computational complexity and memory requirement per iteration and converges essentially quadratically. A detailed error analysis is given.

Low-rank matrix completion aims to recover matrices with missing entries and has attracted considerable attention from machine learning researchers. Most of the existing methods, such as weighted nuclear-norm-minimization-based methods and Qatar Riyal (QR)-decomposition-based methods, cannot provide both convergence accuracy and convergence speed. To investigate a fast and accurate completion method, an iterative QR-decomposition-based method is proposed for computing an approximate singular value decomposition. This method can compute the largest $r (r>0)$ singular values of a matrix by iterative QR decomposition. Then, under the framework of matrix trifactorization, a method for computing an approximate SVD based on QR decomposition (CSVD-QR)-based $L_{2,1}$ -norm minimization method (LNM-QR) is proposed for fast matrix completion. Theoretical analysis shows that this QR-decomposition-based method can obtain the same optimal solution as a nuclear norm minimization method, i.e., the $L_{2,1}$ -norm of a submatrix can converge to its nuclear norm. Consequently, an LNM-QR-based iteratively reweighted $L_{2,1}$ -norm minimization method (IRLNM-QR) is proposed to improve the accuracy of LNM-QR. Theoretical analysis shows that IRLNM-QR is as accurate as an iteratively reweighted nuclear norm minimization method, which is much more accurate than the traditional QR-decomposition-based matrix completion methods. Experimental results obtained on both synthetic and real-world visual data sets show that our methods are much faster and more accurate than the state-of-the-art methods.

Fueled by the surge of ever expanding successful applications of deep neural networks and the great computational power demanded, modern computer processors and accelerators are beginning to offer half precision floating point arithmetic support, and special units (neural engines) such as NVIDIA TensorCore on GPU and Google Tensor Processing Unit (TPU) to accelerate the training and prediction of deep neural networks. It remains unclear how neural engines can be profitably used in applications other than neural networks. In this paper we present an endeavor of accelerating and stabilizing a fundamental matrix factorization on neural engines---the QR factorization---which may open doors to much wider relevance to scientific, engineering, and data science. We show that traditional Householder QR algorithms and implementations do not have the necessary data locality, parallelism, accuracy, and robustness on neural engines which are characterized by extreme speed and low precision/range. We demonstrate that neural engines can be effectively used to accelerate matrix computations (QR 3.0x-14.6x speedup compared to cuSOLVER, reaching up to 36.6TFLOPS); however different algorithms (recursive Gram-Schmidt) are needed to expose more locality and parallelism, even at the cost of increased computations. Moreover, scaling, iterative refinement, and other safeguarding procedures are also needed to regain accuracy and avoid overflowing. Our experience seems to suggest that presently with neural engines the matrix factorizations (QR, LU, Cholesky) are best to be co-designed with its applications (linear solver, least square, orthogonalization, SVD, etc) to achieve high performance and adequate accuracy and reliability.

In this paper we present some techniques for construction of two dimensional mapping arrays, Appropriate for signal processing, called systolic arrays. These type of array algorithms are implemented on VLSI arrays. Taking into the consideration that matrix operations and applications are crucial for signal processing, we make an overview of the systolic processing for the case of matrix-matrix multiplication, QR decomposition and Singular value decomposition (SVD) of the matrix. Systolic arrays are very appropriate for parallel manipulations with matrices. Application domain of these arrays cover different areas especially the signal and image processing. Generally, all signal processing algorithms can be classified in two groups: Algorithms which perform the matrix operations and other special algorithms for signal processing. Most of the algorithms fall into the algorithms of matrix computations and other matrix transformations. Therefore, we consider that the techniques of construction of such arrays are very important in developing the new ideas and transformations which will enable to optimize and make them more efficient.

In target tracking, the tracking process needs to constantly update the data information. However, during data acquisition and transmission of sensors, outliers may occur frequently, and the model is disturbed by non-Gaussian noise, that affects the performance of system state estimation. In this paper, a new filtering algorithm is proposed based on QR decomposition and singular value decomposition (SVD) method, namely adaptive robust unscented Kalman filter (QS-ARUKF) to suppress the interference of outliers, nonGaussian noise as well as a model error to achieve high accuracy state estimation. An adaptive filtering algorithm based on strong tracking idea is used in modifying the state equation of unscented Kalman filter (UKF), so that the algorithm can effectively improve the tracking ability of the state model. By using the robust filtering method to construct a new cost function used to modify the measurement covariance formula of the Kalman filter, the error of measurement model can be effectively suppressed. The QR decomposition is introduced to the time update and measurement update to avoid the covariance non-positive definite. We propose the SVD method to address the problem of numerical sensitivity in the filtering process. The purpose of this method is to replace the calculation of the inverse of the filter gain matrix and further improve the robustness of the algorithm. The simulation results showed that the proposed algorithm has higher accuracy and better robustness than the traditional filtering method.

The typical model selection strategy applied in most Extreme Learning Machine (ELM) papers depends on a k-fold cross-validation and a grid search to select the best pair {L, C} of two adjustable hyper-parameters, namely the number L of hidden ELM nodes and the regularisation parameter, C, that minimizes the validation error. However, by testing only 30 values for L and 30 values for C via 10-fold cross-validation, the learning phase must build 9000 different ELM models, each with a different pair {L, C}. Since these models are not independent from one another, the essence of managing and drastically reducing the computational cost of the ELM model selection relies on matrix decompositions that avoid direct matrix inversion and allow producing reusable matrices during the cross-validations. Still, one can find many matrix decompositions and cross-validation versions that result in several combinations. In this paper, we identify these combinations and analyse them theoretically and experimentally to discover which is the fastest. We compare Singular Value Decomposition (SVD), Eigenvalue Decomposition (EVD), Cholesky decomposition, and QR decomposition, which produce re-usable matrices (orthogonal, Eigen, singular, and upper triangular). These decompositions can be combined with different cross-validation approaches, and we present a direct and thorough comparison of many k-fold cross-validation versions as well as leave-one-out cross-validation. By analysing the computational cost, we demonstrate theoretically and experimentally that while the type of matrix decomposition plays one important role, another equally important role is played by the version of cross-validation. Finally, a scalable and computationally-effective algorithm is presented that significantly reduces computational time.

A new model of statistical shape theory under elliptical models is proposed by using the polar decomposition. This work completes the group of SVD and QR shape densities obtained from the transpose of the square root of a non singular Wishart matrix. The associated non isotropic and non central polar shape distributions are set in the context of consistent computable series of zonal polynomials. Then the inference procedures with elliptical assumptions can be performed at the same computational cost of the published routines based on Gaussian models. As an example of the technique, a classical application in Biology is studied under three models, the usual Gaussian and two Kotz type models; then the best model is selected by a modified BIC∗ criterion, and a test for equality in polar shapes is performed. The published results for this landmark data under isotropic Gaussian models and procrustes theory are also discussed.