Decomposition Of Matrices Research Articles

Graph total variation and low-rank regularization for heterogeneous change detection

Heterogeneous change detection (HCD) is challenging because different imaging mechanisms for various sensors make images difficult to compare directly. To address this problem, a graph-based regression algorithm is proposed for HCD, by leveraging the Graph Total Variation regularization and Low-Rank matrices decomposition (GTVLR). Utilizing graph signal processing (GSP) theory, a directed graph (digraph) model is employed to effectively represent the orientation and correlation information of images, thereby enabling direct comparison of heterogeneous data within the same domain after graph filtering. The GTVLR framework facilitates the decomposition of post-event images into regression and changed images. This decomposition ensures that the regression image mirrors the structure similarity of the pre-event image, while the changed image highlights areas of alteration, aiding in change detection. The model characterizes the piecewise smoothness and Low-Rank properties of data through GTV regularization and Low-Rank penalty, respectively. Moreover, by integrating the higher-order neighboring information within the digraph to refine the model. Experiments conducted on three real-world datasets and comparison with several state-of-the-art methods demonstrate the effectiveness of the proposed algorithm.

Decomposition of matrices into products of commutators of quadratic matrices

Let F be a field, p ( x ) be a quadratic polynomial in F [ x ] with a non-zero constant term, and n ≥ 2 be a positive integer. We say that T ∈ M n ( F ) is p ( x ) -quadratic if p ( T ) = 0 . In this paper, given A ∈ SL n ( F ) , we show that A can be decomposed into a product of at most two commutators of p ( x ) -quadratic matrices if either p ( x ) has two distinct roots in F and A is non-scalar, or if F is algebraically closed and has characteristic different from 2. A similar result for matrices over real quaternion division rings is also provided.

Factoring perfect reconstruction filter banks into causal lifting matrices: A Diophantine approach

The elementary theory of bivariate linear Diophantine equations over polynomial rings is used to construct causal lifting factorizations (elementary matrix decompositions) for causal two-channel FIR perfect reconstruction transfer matrices and wavelet transforms. The Diophantine approach generates causal factorizations satisfying certain polynomial degree-reducing inequalities, enabling a new factorization strategy called the Causal Complementation Algorithm. This provides a causal (i.e., polynomial, hence realizable) alternative to the noncausal lifting scheme developed by Daubechies and Sweldens using the Extended Euclidean Algorithm for Laurent polynomials. The new approach replaces the Euclidean Algorithm with Gaussian elimination employing a slight generalization of polynomial division that ensures existence and uniqueness of quotients whose remainders satisfy user-specified divisibility constraints. The Causal Complementation Algorithm is shown to be more general than the causal version of the Euclidean Algorithm approach by generating additional causal lifting factorizations beyond those obtainable using the polynomial Euclidean Algorithm.

Spectral packing decomposition of CMV matrices through power-law subordinacy

This note gives a classification for the spectral measures of CMV matrices, which are special unitary five-diagonal matrices related to orthogonal polynomials on the unit circle, with respect to packing measures.

Variations in fluorescence properties of humic acids from calcareous soils amended with different swine manures in a long-term soil experiment.

Molecular properties of soil humic acid (HA) can play an important role in the mechanisms regulating plant nutrient availability. This study explores how the structure of HA is altered by long-term treatment with different forms of swine manure and how these changes may influence nutrient availability. Liquid swine manure (LSM), solid swine manure (SSM), and swine manure compost (SMC) were applied to a calcareous soil over 17 years in a long-term soil fertility study. HA was extracted from site soil samples and analyzed using fluorescence spectroscopic techniques, including a Cu2+ quenching experiment, in order to assess differences in the structure and functionality of the soil organic matter (SOM) resulting from these different treatments. Emission spectra of the SSM-HA and SMC-HA are similar, while the LSM-HA is distinct. Procedures such as parallel factor analysis (PARAFAC) decomposition of emission-excitation matrices showed that structures in the SSM-HA and SMC-HA samples have lower complexity, whereas the structures of LSM-HA are of higher complexity. Interactions with Cu2+ at different pH levels indicate that the LSM-HA shows more dynamic conformational changes as well as stronger interactions and higher quenching efficiency compared to the other treatments. Conversely, SMC-HA demonstrates relatively stable binding constant (Ka) values across different pH levels. The binding constants and quenching efficiency of SSM-HA are significantly affected by changes in pH. This study shows distinct structural characteristics of HA formed under different manure management systems and provides valuable insights into how these variations may impact nutrient dynamics in soils.

Tensorized Pauli decomposition algorithm

This paper introduces a novel general-purpose algorithm for Pauli decomposition that employs matrix slicing and addition rather than expensive matrix multiplication, significantly accelerating the decomposition of multi-qubit matrices. In a detailed complexity analysis, we show that the algorithm admits the best known worst-case scaling and more favorable runtimes for many practical examples. Numerical experiments are provided to validate the asymptotic speed-up already for small instance sizes, underscoring the algorithm’s potential significance in the realm of quantum computing and quantum chemistry simulations.

A genuine extension of the Moore–Penrose inverse to dual matrices

The Moore–Penrose (MP) inverse is a genuine extension of the matrix inverse. In the last one and a half decades, in the study of approximate synthesis in kinematic, two generalizations of the MP inverse appeared for dual real matrices. One generalization, dual MP generalized inverse (DMPGI), satisfies the four MP conditions, but does not exist for uncountably many dual real matrices. Another generalization, MP dual generalized inverse (MPDGI), always exists, but does not satisfy the four MP conditions even if DMPGI exists. Recently, a new dual MP inverse (NDMPI) appeared. In this paper, based on the singular value decomposition of dual matrices given in this article, we show that the NDMPI can be regarded as a combination of the DMPGI and MPDGI. It coincides with the DMPGI when the DMPGI exists. When the DMPGI does not exist, it is an MPDGI extension of the DMPGI. It always satisfies the last three MP conditions, and minimizes the error of the first MP condition when that MP condition is impossible to be satisfied. Given a dual matrix, its NDMPI is the unique dual matrix that satisfies the extended version of the first MP condition, and the other three original MP conditions. If the original matrix is complex, its NDMPI is exactly the MP inverse of that matrix. Furthermore, NDMPI may be used to generate the minimum-norm solution to the dual least squares problem. Numerical results on kinematic analysis and dual linear regression are reported.

Effects of polymer matrix and temperature on pyrolysis of tetrabromobisphenol A: Product profiles and transformation pathways

Plastics are crucial constituents in electronic waste (e-waste) and part of the issue in e-waste recycling and environmental protection. However, previous studies have mostly focused on plastic recovery or thermal behavior of flame retardants, but not both simultaneously. The present study simulated the process of e-waste thermal treatment to explore tetrabromobisphenol A (TBBPA) pyrolysis at various temperatures using polystyrene (PS), polyvinyl chloride (PVC), and e-waste plastics as polymer matrices. Pyrolysis of TBBPA produced bromophenol, bromoacetophenone, bromobenzaldehyde, and bromobisphenol A. Co-pyrolysis with the polymer matrices increased emission factors by 1 − 2 orders of magnitude. The pyrolytic products of TBBPA, TBBPA+PS, and TBBPA+PVC were mainly low-brominated bisphenol A, while that of TBBPA in e-waste plastics was consistently bromophenol. Increasing temperature drove up the proportions of gaseous and particulate products, but lowered the relative abundances of inner wall adsorbed and residual products in pyrolysis of pure TBBPA. In co-pyrolysis of TBBPA with polymer matrix, the proportions of products in different phases were no longer governed solely by temperature, but also by polymer matrix. Co-pyrolysis of TBBPA with PS generated various bromophenols, while that with PVC produced chlorophenols and chlorobrominated bisphenol A. Transformation pathways, deduced by ab initio calculations, include hydrogenation-debromination, isopropylphenyl bond cleavage, oxidation, and chlorination.

Tensor Golub–Kahan method based on Einstein product

The Singular Value Decomposition (SVD) of matrices is a widely used tool in scientific computing. In many applications of machine learning, data analysis, signal and image processing, the large datasets are structured into tensors, for which generalizations of SVD have already been introduced, for various types of tensor–tensor products. In this paper, we present innovative methods for approximating this generalization of SVD to tensors in the framework of the Einstein tensor product. These singular elements are called singular values and singular tensors respectively. The proposed method uses the tensor Lanczos bidiagonalization applied to the Einstein product. In most applications, as in the matrix case, the extremal singular values are of special interest. To enhance the approximation of the largest or the smallest singular triplets (singular values and left and right singular tensors), a restarted method based on Ritz augmentation is proposed. Numerical results are proposed to illustrate the effectiveness of the presented method. In addition, applications to video compression and facial recognition are presented.

Robust [formula omitted] time-varying formation control of singular multiagent systems under jointly connected topologies

This study investigates time-varying formation design and analysis problems for the singular multiagent systems (SMASs) with external disturbances and jointly connected topologies. Firstly, the novel nonlinear distributed formation protocol is formulated and the finite-time reachability of sliding mode surfaces is proved. Another improvement emphasizes that a common matrix can be extracted from the decomposition of all switching Laplacian matrices, which simplifies the formation control problem to admissible analysis. Secondly, by using the feedback impulse elimination and Cauchy convergence criterion methods, the solvability of the time-varying formation control is obtained. Thirdly, the external disturbances are well suppressed by the robust H∞ stability analysis. Finally, to verify the validity of the proposed theory, the simulation examples of hexagonal time-varying formation are provided.

On the total positivity of q-Bernstein mass matrices and their accurate computations

In this paper the total positivity of the Gramian (mass) matrices of q-Bernstein bases is analyzed. Furthermore, we provide an efficient method to obtain a bidiagonal decomposition of these mass matrices allowing us to calculate their singular values and inverses to high relative accuracy. Numerical examples are provided to illustrate the high accuracy of the performed computations using the proposed decompositions.

Efficient subspace clustering and feature extraction via [formula omitted]-norm and [formula omitted]-norm minimization

The nuclear norm-based Latent Low-Rank Representation (LatLRR) has gained much attention due to its success in subspace clustering and feature extraction. However, it suffers from high computational costs due to the calculation of singular value decomposition for large matrices. To this end, we develop an efficient subspace clustering and feature extraction method (ESCFE) which substitutes the nuclear norm with the ℓ2,1-norm and ℓ1,2-norm respectively. Theoretically proof shows both the ℓ2,1-norm and ℓ1,2-norm can serve as the convex surrogates of the nuclear norm while can derive closed-form solutions. Furthermore, the ℓ2,1-norm (or ℓ1,2-norm) regularization promotes column (or row) structure sparsity due to the discriminative nature inherited from the ℓ1-norm. Thus the proposed ESCFE is robust to outliers in data and can extract features with joint sparsity. Extensive experiments on multiple benchmark datasets demonstrate the superiority of our method in both efficiency and effectiveness.

QR decomposition of dual matrices and its application

Dual number matrix decompositions have played an important role in fields such as kinematics and computer graphics in recent years. In this paper, we present a QR decomposition algorithm for dual number matrices. When dealing with large-scale problems, we present the thin QR decomposition of dual number matrices, along with its algorithm with column pivoting. In numerical experiments, we discuss the suitability of different QR algorithms when confronted with various large-scale dual matrices, providing their respective domains of applicability. Finally, we employ the QR decomposition of dual matrices to compute the DMPGI, attaining results of higher precision.

The Chevalley--Jordan decomposition and spectral projections of complex matrices

In this paper, a novel and simple method for obtaining the Chevalley-Jordan decomposition and the spectral projections of matrices is presented. Our method is direct and elementary, it gives tractable and manageable formulas with minimum mathematical prerequisites. Moreover, knowing only some associated matrices of the matrix, we can simply provide the minimal polynomial of this matrix.

Discovering quantum circuit components with program synthesis

Despite rapid progress in the field, it is still challenging to discover new ways to leverage quantum computation: all quantum algorithms must be designed by hand, and quantum mechanics is notoriously counterintuitive. In this paper, we study how artificial intelligence, in the form of program synthesis, may help overcome some of these difficulties, by showing how a computer can incrementally learn concepts relevant to quantum circuit synthesis with experience, and reuse them in unseen tasks. In particular, we focus on the decomposition of unitary matrices into quantum circuits, and show how, starting from a set of elementary gates, we can automatically discover a library of useful new composite gates and use them to decompose increasingly complicated unitaries.

Reciprocal polarization imaging of optical activity in reflection.

We present reciprocal polarization imaging for the optical activity of chiral media in reflection geometry. The method is based on the reciprocal polar decomposition of backscattering Mueller matrices accounting for the reciprocity of light waves in forward and backward scattering paths. Anisotropic depolarization is introduced to gain sensitivity to optical activity in backscattering. Experiments with glucose solutions show that while the Lu-Chipman decomposition of the backscattering Mueller matrices produces erroneous results, reciprocal polarization imaging correctly retrieves the optical activity of chiral media. The recovered optical rotation agrees with that obtained in the forward geometry and increases linearly with the concentration and thickness of the chiral media. The potential for in vivo glucose monitoring based on optical activity sensing using reciprocal polarization imaging is then discussed.

A Block Householder–Based Algorithm for the QR Decomposition of Hierarchical Matrices

A Block Householder–Based Algorithm for the QR Decomposition of Hierarchical Matrices

A novel structure-preserving algorithm for the singular value decomposition of biquaternion matrices

In this paper, we study the singular value decomposition of biquaternion matrices. We prove that the singular value decomposition of biquaternion matrices can be equivalently converted to the singular value decomposition of quaternion adjoint matrices. By means of the Cheng product, we propose a novel representation called Q -representation and design a Q structure-preserving algorithm to deal with the singular value decomposition of quaternion matrices, which makes use of high-level operations. The algorithm is more efficient than that in quaternion toolbox for matlab using quaternion arithmetics and real structure-preserving algorithm. The Q structure-preserving algorithm for singular value decomposition of biquaternion matrices is presented through the application of the Q structure-preserving algorithm for singular value decomposition of quaternion matrices. Numerical experiments are provided to demonstrate the efficiency of the Q structure-preserving algorithm. We apply the Q structure-preserving algorithm of quaternion singular value decomposition to improve the image denoising process.

VDR decomposition of Chebyshev-Vandermonde matrices with the Arnoldi Process

This paper introduces the VDR decomposition of Chebyshev-Vandermonde matrices, where V represents an ordinary Vandermonde matrix, D is diagonal, and R is upper triangular. Our motivation for this work stems from the study by Brubeck et al. [Vandermonde with Arnoldi. SIAM Rev. 2021;63(2):405–415]. We explore the VDR decomposition and combine it with a QR decomposition of V for Chebyshev-Vandermonde matrices. Furthermore, we demonstrate that the upper triangular factor in the QR decomposition can be recursively obtained from the upper Hessenberg matrix in the Arnoldi process. To compare our approach with the results obtained using the default QR code in Matlab, we present experimental results. Finally, we provide an algorithm for solving linear systems with Chebyshev-Vandermonde matrices as coefficient matrices, and experimental results for an upper bound on the relative error by applying an a posteriori error analysis for the computed solutions of the systems.

On Rotfel'd type inequalities for sectorial matrices

Using a matrix decomposition for sectorial matrices and an AM-GM inequality for singular values, we present a new Rotfel'd type inequality for sectorial matrices, which generalizes and partially improves some results by Zhang. We also give alternative simpler proofs of a main result of a recent paper by Yang, Lu and Chen as well as that by Fu and Liu.