Block-diagonal Research Articles

Parity-Check Matrix Partitioning for Efficient Layered Decoding of QC-LDPC Codes

In this paper, we consider how to partition the parity-check matrices (PCMs) to reduce the hardware complexity and increase decoding throughput for the row layered decoding of quasi-cyclic low-density parity-check (QC-LDPC) codes. First, we formulate the PCM partitioning as an optimization problem, which targets to minimize the maximum column weight of each layer while maintaining a block cyclic shift property among different layers. As a result, we derive all the feasible solutions for the problem and propose a tight lower bound ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>LB</i></sub> on the minimum possible maximum column weight to evaluate a solution. Second, we define a metric called layer distance to measure the data dependency between consecutive layers and further illustrate how to identify the solutions with desired layer distance from those achieving the minimum value of ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>LB</i></sub> = 1, which is preferred to reduce computation delay. Next, we demonstrate that up-to-now, finding an optimal solution for the optimization problem with polynomial time complexity is unachievable. Therefore, both enumerative and greedy partition algorithms are proposed instead. After that, we modify the quasi-cyclic progressive edge-growth (QC-PEG) algorithm to directly construct PCMs that have a straightforward partition scheme to achieve ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>LB</i></sub> or the desired layer distance. Simulation results showed that the constructed codes have better error correction performance and achieve less average number of iterations than the underlying 5G LDPC codes.

Energy Storage Optimization Method for Flexible Interconnected Low-voltage Distribution Network Based on Positive Semi-definite Programming

With the fast development of the electricity market, the number of small and medium-sized new energy generation in the urban low-voltage distribution networks is increasing. These “retail investors” hope to sell the extra electricity for financial gain. However, these renewable energy generation units have small capacity and obvious intermittent output. Thus, they can not be added to the operation of the power system alone. In this paper, the model is understood from the perspective of optimization theory, and the solution method is given. The optimization problem of the model is given, and the required gradient and Hessian functions are defined. By applying mathematical induction and the properties of the special matrix column, a decomposition formula of the matrix column is derived. Through using this inequality and the standard decomposition of a positive semi-definite matrix, a feasible scheme is formed. The eigenvalue relationship between the sum of elements of a positive semi-definite block matrix and its quasi-diagonal block is given under certain conditions, while the number of such blocks is not required. The standard form of the dual problem is drawn by the method of Lagrange multipliers, and the relationship between the dual solution and the solution of the original problem is studied. The stability of the system is further improved, which provides reference significance for improving the utilization rate of photovoltaic new energy, optimizing the charging and discharging of the energy storage system, and ensuring the safe and reliable operation of the distribution network.

Reversible Data Hiding in Encrypted Image Based on Bit-Plane Redundancy of Prediction Error

In this paper, we propose a reversible data hiding scheme in an encrypted image based on bit-plane redundancy of prediction error. The scheme greatly improves the embedding capacity while maintaining lossless image recovery and error-free secret data extraction. Firstly, the original image is preprocessed to obtain the prediction error image. After the error matrix is divided into blocks, the corresponding block type is obtained. Secondly, the predicted error image is encrypted with stream cipher and the encryption matrix blocks are scrambled to ensure the security of the scheme. Finally, after embedding the block type value into the encrypted image, the spare room corresponding to each block was obtained, which was used to embed the secret data. The scheme makes full use of the spatial correlation of the pixels in the block, so it improves the embedding rate. By selecting 100 images in each dataset of BOSSbase and BOWS-2, when the block size is 3×3, the average embedding rate of our scheme can reach 3.56 bpp and 3.81 bpp, respectively. The performance of the proposed method is better than the other schemes with similar properties.

Positivity preservers forbidden to operate on diagonal blocks

The question of which functions acting entrywise preserve positive semidefiniteness has a long history, beginning with the Schur product theorem [Crelle 1911], which implies that absolutely monotonic functions (i.e., power series with nonnegative coefficients) preserve positivity on matrices of all dimensions. A famous result of Schoenberg and of Rudin [Duke Math. J. 1942, 1959] shows the converse: there are no other such functions. Motivated by modern applications, Guillot and Rajaratnam [Trans. Amer. Math. Soc. 2015] classified the entrywise positivity preservers in all dimensions, which act only on the off-diagonal entries. These two results are at “opposite ends”, and in both cases the preservers have to be absolutely monotonic. We complete the classification of positivity preservers that act entrywise except on specified “diagonal/principal blocks”, in every case other than the two above. (In fact we achieve this in a more general framework.) This yields the first examples of dimension-free entrywise positivity preservers - with certain forbidden principal blocks - that are not absolutely monotonic.

An Energy-Efficient Sensing Matrix for Wireless Multimedia Sensor Networks

A measurement matrix is essential to compressed sensing frameworks. The measurement matrix can establish the fidelity of a compressed signal, reduce the sampling rate demand, and enhance the stability and performance of the recovery algorithm. Choosing a suitable measurement matrix for Wireless Multimedia Sensor Networks (WMSNs) is demanding because there is a sensitive weighing of energy efficiency against image quality that must be performed. Many measurement matrices have been proposed to deliver low computational complexity or high image quality, but only some have achieved both, and even fewer have been proven beyond doubt. A Deterministic Partial Canonical Identity (DPCI) matrix is proposed that has the lowest sensing complexity of the leading energy-efficient sensing matrices while offering better image quality than the Gaussian measurement matrix. The simplest sensing matrix is the basis of the proposed matrix, where random numbers were replaced with a chaotic sequence, and the random permutation was replaced with random sample positions. The novel construction significantly reduces the computational complexity as well time complexity of the sensing matrix. The DPCI has lower recovery accuracy than other deterministic measurement matrices such as the Binary Permuted Block Diagonal (BPBD) and Deterministic Binary Block Diagonal (DBBD) but offers a lower construction cost than the BPBD and lower sensing cost than the DBBD. This matrix offers the best balance between energy efficiency and image quality for energy-sensitive applications.

Computerized Block Games for Automated Cognitive Assessment: Development and Evaluation Study

Cognitive assessment using tangible objects can measure fine motor and hand-eye coordination skills along with other cognitive domains. Administering such tests is often expensive, labor-intensive, and error prone owing to manual recording and potential subjectivity. Automating the administration and scoring processes can address these difficulties while reducing time and cost. e-Cube is a new vision-based, computerized cognitive assessment tool that integrates computational measures of play complexity and item generators to enable automated and adaptive testing. The e-Cube games use a set of cubes, and the system tracks the movements and locations of these cubes as manipulated by the player. The primary objectives of the study were to validate the play complexity measures that form the basis of developing the adaptive assessment system and evaluate the preliminary utility and usability of the e-Cube system as an automated cognitive assessment tool. This study used 6 e-Cube games, namely, Assembly, Shape-Matching, Sequence-Memory, Spatial-Memory, Path-Tracking, and Maze, each targeting different cognitive domains. In total, 2 versions of the games, the fixed version with predetermined sets of items and the adaptive version using the autonomous item generators, were prepared for comparative evaluation. Enrolled participants (N=80; aged 18-60 years) were divided into 2 groups: 48% (38/80) of the participants in the fixed group and 52% (42/80) in the adaptive group. Each was administered the 6 e-Cube games; 3 subtests of the Wechsler Adult Intelligence Scale, Fourth Edition (WAIS-IV; Block Design, Digit Span, and Matrix Reasoning); and the System Usability Scale (SUS). Statistical analyses at the 95% significance level were applied. The play complexity values were correlated with the performance indicators (ie, correctness and completion time). The adaptive e-Cube games were correlated with the WAIS-IV subtests (r=0.49, 95% CI 0.21-0.70; P<.001 for Assembly and Block Design; r=0.34, 95% CI 0.03-0.59; P=.03 for Shape-Matching and Matrix Reasoning; r=0.51, 95% CI 0.24-0.72; P<.001 for Spatial-Memory and Digit Span; r=0.45, 95% CI 0.16-0.67; P=.003 for Path-Tracking and Block Design; and r=0.45, 95% CI 0.16-0.67; P=.003 for Path-Tracking and Matrix Reasoning). The fixed version showed weaker correlations with the WAIS-IV subtests. The e-Cube system showed a low false detection rate (6/5990, 0.1%) and was determined to be usable, with an average SUS score of 86.01 (SD 8.75). The correlations between the play complexity values and performance indicators supported the validity of the play complexity measures. Correlations between the adaptive e-Cube games and the WAIS-IV subtests demonstrated the potential utility of the e-Cube games for cognitive assessment, but a further validation study is needed to confirm this. The low false detection rate and high SUS scores indicated that e-Cube is technically reliable and usable.

Fast verified computation for real powers of large matrices with Kronecker structure

Let ⊗ be the Kronecker product, In be the n×n identity matrix, α∈R, A∈Cm×m, B∈Cn×n, C∈Cm×n, and vec(C) be a column vector by stacking the columns of C. We propose two fast numerical algorithms for computing interval vectors containing (In⊗A+B⊗Im)αvec(C), where C has rank one. Particular emphasis is put on the computational costs of these algorithms, which are only O(m3+n3) if |α|≪min(m,n). The first algorithm is based on numerical spectral decomposition of A and B. Radii given by the first algorithm are smaller than those by the second algorithm when numerically computed eigenvector matrices for A and B are well-conditioned. The second algorithm is based on numerical block diagonalization, and applicable even when the computed eigenvector matrices are singular or ill-conditioned. Numerical results show efficiency of the algorithms.

A Graduated Non-Convexity Technique for Dealing Large Point Spread Functions

This paper focuses on reducing the computational cost of a GNC Algorithm for deblurring images when dealing with full symmetric Toeplitz block matrices composed of Toeplitz blocks. Such a case is widespread in real cases when the PSF has a vast range. The analysis in this paper centers around the class of gamma matrices, which can perform vector multiplications quickly. The paper presents a theoretical and experimental analysis of how γ-matrices can accurately approximate symmetric Toeplitz matrices. The proposed approach involves adding a minimization step for a new approximation of the energy function to the GNC technique. Specifically, we replace the Toeplitz matrices found in the blocks of the blur operator with γ-matrices in this approximation. The experimental results demonstrate that the new GNC algorithm proposed in this paper reduces computation time by over 20% compared with its previous version. The image reconstruction quality, however, remains unchanged.

Properties and characterizations of dual sharp orders

In this paper, we extend the sharp order to dual ring. To be specific, we introduce D-sharp, C-sharp, G- and G-sharp binary relations of dual matrices, and discuss relevant properties of these dual binary relations by using dual group generalized inverse(DGGI for short), group dual generalized inverse(GDGI for short) and matrix decomposition. We conclude that D-sharp and G-sharp binary relations are partial orders. The dual binary relations are connected with the sharp order by using the upper triangular block matrix. Furthermore, we discuss relationships among dual binary relations mentioned above. When dual parts of dual matrix pairs are all zero matrices, dual binary relations(D-sharp, C-sharp, G- and G-sharp binary relations) are equivalent to the sharp order.

Squared distance matrices of trees with matrix weights

Let T be a tree on n vertices whose edge weights are positive definite matrices of order s. The squared distance matrix of T, denoted by Δ, is the ns × ns block matrix with , where d(i, j) is the sum of the weights of the edges in the unique (i, j)-path. In this article, we obtain a formula for the determinant of Δ and find under some conditions.

Deep Multi-Modal Discriminative and Interpretability Network for Alzheimer's Disease Diagnosis.

Multi-modal fusion has become an important data analysis technology in Alzheimer's disease (AD) diagnosis, which is committed to effectively extract and utilize complementary information among different modalities. However, most of the existing fusion methods focus on pursuing common feature representation by transformation, and ignore discriminative structural information among samples. In addition, most fusion methods use high-order feature extraction, such as deep neural network, by which it is difficult to identify biomarkers. In this paper, we propose a novel method named deep multi-modal discriminative and interpretability network (DMDIN), which aligns samples in a discriminative common space and provides a new approach to identify significant brain regions (ROIs) in AD diagnosis. Specifically, we reconstruct each modality with a hierarchical representation through multilayer perceptron (MLP), and take advantage of the shared self-expression coefficients constrained by diagonal blocks to embed the structural information of inter-class and the intra-class. Further, the generalized canonical correlation analysis (GCCA) is adopted as a correlation constraint to generate a discriminative common space, in which samples of the same category gather while samples of different categories stay away. Finally, in order to enhance the interpretability of the deep learning model, we utilize knowledge distillation to reproduce coordinated representations and capture influence of brain regions in AD classification. Experiments show that the proposed method performs better than several state-of-the-art methods in AD diagnosis.

Efficiency of local Vanka smoother geometric multigrid preconditioning for space‐time finite element methods to the Navier–Stokes equations

AbstractNumerical simulation of incompressible viscous flow continues to remain a challenging task, in particular if three space dimensions are involved. Space‐time finite element methods feature the natural construction of higher order discretization schemes. They offer the potential to achieve accurate results on computationally feasible grids. Using a temporal test basis supported on the subintervals and linearizing the resulting algebraic problems by Newton's method yield linear systems of equations with block matrices built of (k + 1) × (k + 1) saddle point systems, where k denotes the polynomial order of the variational time discretization. We demonstrate numerically the efficiency of preconditioning GMRES iterations for solving these linear systems by a V‐cycle geometric multigrid approach based on a local Vanka smoother. The studies are done for the two‐ and three‐dimensional benchmark problem of flow around a cylinder. The robustness of the solver with respect to the piecewise polynomial orderkin time is analyzed and confirmed numerically.

In‐depth investigation of inter‐turn short‐circuit faults of surface‐mounted permanent magnet machines with series‐parallel coil connections

AbstractA general analytical model is developed in a concise block matrix form for surface‐mounted permanent magnet (SPM) machines with series‐parallel coil connections under inter‐turn short circuit (ITSC) fault. In the model, branch currents are used as state variables and inductances of different series‐parallel coil connections are calculated using an analytical method, namely winding function approach (WFA) together with slot permeance method. Based on the characteristics of the calculated inductances and the developed fault model, the multiphase Clarke transformation has been proposed to simplify the fault model. In the process of model simplification, the healthy machine model using branch currents as state variables have been proven to be equivalent to that using 3‐phase currents as state variables. The proposed fault models of a 3kW 96‐slot 32‐pole SPM machine with different series‐parallel coil connections have been built in Matlab/Simulink and validated by time‐stepping 2D FE simulations. Simulation results show that different series‐parallel coil connections have little influence on the amplitude of the ITSC current. Finally, a small scale 24‐slot 8‐pole SPM machine prototype has been built to further validate the accuracy of the proposed fault model.

Optical solitary wave solutions in generalized determinant form for Kundu–Eckhaus equation

The Kundu–Eckhaus (KE) equation describes the propagation of ultra-short femtosecond pulses in optical fibers. In this paper, on the basis of Hirota bilinear form of KE equation, a complex matrix is introduced into the differential relation satisfied by determinant elements. By constructing relations among the matrix elements, the solution in generalized double Wronskian determinant form for the KE equation is derived. When the complex matrix introduced in the differential relation of the determinant elements take diagonal matrix and Jordan block matrix respectively, the soliton solutions and Jordan block solutions of the KE equation are obtained and propagation situations are discussed via different parameters.

Load balanced locality-aware parallel SGD on multicore architectures for latent factor based collaborative filtering

We investigate the parallelization of Stochastic Gradient Descent (SGD) for matrix completion on multicore architectures. We provide an experimental analysis of current SGD algorithms to find out their bottlenecks and limitations. Grid-based methods suffer from load imbalance among 2D blocks of the rating matrix, especially when datasets are skewed and sparse. Asynchronous methods, on the other hand, can face cache issues due to their memory access pattern. We propose bin-packing-based block balancing methods that are alternative to the recently proposed BaPa method. We then introduce Locality Aware SGD (LASGD), a grid-based asynchronous parallel SGD algorithm that efficiently utilizes cache by changing nonzero update sequence without affecting factor update order and carefully arranging latent factor matrices in the memory. Combined with our proposed load balancing methods, our experiments show that LASGD performs significantly better than alternative approaches in parallel shared-memory systems.

Industrial Machinery Components Classification: A Case of D-S Pooling

Industries are increasingly shifting towards unmanned and intelligent systems that require efficient processing and monitoring of structures across various applications, ranging from machine manufacturing to waste disposal. In order to achieve the goal of intelligent processing, it is crucial to accurately classify and differentiate various components and parts. However, existing studies have not focused on simultaneously classifying electro-mechanical machinery components. This poses a challenge as these components, including capacitors, transistors, ICs, inductors, springs, locating pins, washers, nuts, and bolts, exhibit high intra- and inter-class similarity, making their accurate classification a tedious task. Furthermore, many of these components have symmetrical shapes but are asymmetrical among different classes. To address these challenges, this article introduces a new double-single (D-S) pooling method that focuses on the higher resemblance of seventeen electro-mechanical component classifications with minimum trainable parameters and achieves maximum accuracy. The industrial machine component classification model (IMCCM) consists of two convolutional neural network (CNN) blocks designed with a D-S pooling method that facilitates the model to effectively highlight the differences for the higher similar classes, and one block of grey-level co-occurrence matrix (GLCM) to strengthen the classification outcome. The extracted fused features from these three blocks are then forwarded to the random forest classifier to distinguish components. The accuracy achieved by this proposed model is 98.15%—outperforming the existing state of the arts (SOTAs) models, and has 141,346 trainable parameters– hence, highly effective for industrial implementation.

A new formulation of sparse multiple kernel k$$ k $$‐means clustering and its applications

AbstractMultiple kernel ‐means (MKKM) clustering has been an important research topic in statistical machine learning and data mining over the last few decades. MKKM combines a group of prespecified base kernels to improve the clustering performance. Although many efforts have been made to improve the performance of MKKM further, the present works do not sufficiently consider the potential structure of the partition matrix. In this paper, we propose a novel sparse multiple kernel ‐means (SMKKM) clustering by introducing a ‐norm to induce the sparsity of the partition matrix. We then design an efficient alternating algorithm with curve search technology. More importantly, the convergence and complexity analysis of the designed algorithm are established based on the optimality conditions of the SMKKM. Finally, extensive numerical experiments on synthetic and benchmark datasets demonstrate that the proposed method outperforms the state‐of‐the‐art methods in terms of clustering performance and robustness.

The differential graded Verlinde formula and the Deligne Conjecture

AbstractA modular category gives rise to a differential graded modular functor, that is, a system of projective mapping class group representations on chain complexes. This differential graded modular functor assigns to the torus the Hochschild chain complex and, in the dual description, the Hochschild cochain complex of . On both complexes, the monoidal product of induces the structure of an ‐algebra, to which we refer as the differential graded Verlinde algebra. At the same time, the modified trace induces on the tensor ideal of projective objects in a Calabi–Yau structure so that the cyclic Deligne Conjecture endows the Hochschild cochain and chain complex of with a second ‐structure. Our main result is that the action of a specific element in the mapping class group of the torus transforms the differential graded Verlinde algebra into this second ‐structure afforded by the Deligne Conjecture. This result is established for both the Hochschild chain and the Hochschild cochain complex of . In general, these two versions of the result are inequivalent. In the case of Hochschild chains, we obtain a block diagonalization of the Verlinde algebra through the action of the mapping class group element . In the semisimple case, both results reduce to the Verlinde formula. In the non‐semisimple case, we recover after restriction to zeroth (co)homology earlier proposals for non‐semisimple generalizations of the Verlinde formula.

Inequalities for functions of $$2\times 2$$ block matrices

Let $$T=\left[ \begin{array}{cc} T_{11} &{} T_{12} \\ T_{21} &{} T_{22} \end{array} \right] $$ be accretive-dissipative, where $$T_{11},T_{12},T_{21},$$ and $$T_{22} $$ are $$n\times n$$ complex matrices. Let f be a non-negative function on $$ [0,\infty )$$ such that $$f(0)=0$$ , and let $$\alpha ,\beta \in (0,1)$$ such that $$\alpha +\beta =1$$ . For every unitarily invariant norm $$\left| \left| \left| \cdot \right| \right| \right| $$ , it is shown that $$\begin{aligned} \sum _{j=1}^{2}\left| \left| \left| f\left( \frac{\left| T_{jj}+(2\alpha -1)T_{jj}^{*}\right| }{2\sqrt{2}}\right) +f\left( \sqrt{\frac{\alpha \beta }{2}}\left| T_{jj}^{*}\right| \right) \right| \right| \right| \\ \le 2\max (\alpha ,\beta )\left| \left| \left| f(\left| T\right| )\right| \right| \right| \end{aligned}$$ whenever $$t\rightarrow f\left( \sqrt{t}\right) $$ is convex and $$\begin{aligned} \sum _{j=1}^{2}\left| \left| \left| \alpha f\left( \frac{ \left| T_{jj}+(2\alpha -1)T_{jj}^{*}\right| }{\sqrt{2\alpha }} \right) +\beta f\left( \sqrt{2\alpha }\left| T_{jj}^{*}\right| \right) \right| \right| \right| \\ \le 4\left| \left| \left| f\left( \sqrt{ \max (\alpha ,\beta )}\left| T\right| \right) \right| \right| \right| \end{aligned}$$ whenever f is concave.

Local spatial similarity based joint-sparse regression for hyperspectral image unmixing

Hyperspectral unmixing is indispensable for hyperspectral remote sensing technology. Exploration of spatial and spectral information helps to obtain accurate abundances. In recent years, utilizing a joint-sparse prior of the abundance matrix has earned a lot of interest. The joint-sparse-blocks structure considers the joint sparsity among column blocks (neighboring pixels) of an abundance matrix. Considering different ways of tensor expansion, the bilateral joint-sparse structure is extended to consider the joint sparsity along both horizontal and vertical directions. But it is still hard to fully utilize spectral and spatial information in hyperspectral data. In this article, we propose a local spatial similarity based joint-sparse structure by imposing the joint sparse on similar pixels in a local region, instead of only horizontally and vertically adjacent pixels. Specifically, we assume that similar pixels share identical endmembers. In this vein, we enhance the sparsity of local related pixels and have a deeper mining of the spatial information in hyperspectral data. Moreover, combined with a low-rank property, we propose a new unmixing model and derive an algorithm under the alternating direction method of multipliers framework. The validity of the proposed algorithm is verified by experiments on both synthetic and real data, compared with several state-of-the-art unmixing algorithms.