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[EN] We first attempt to determine conditions on a linear relation T such that µT becomes a demicompact linear relation for each µ ∈ [0, 1) (see Theorems 2.4 and 2.5). Second, we display some results on Fredholm and upper semi-Fredholm linear relations involving a demicompact one (see Theorems 3.1 and 3.2). Finally, we provide some results in which a block matrix of linear relations becomes a demicompact block matrix of linear relations (see Theorems 4.2 and 4.3).

Dynamic Time Warping (DTW) is a well-known similarity measure for time series. The standard dynamic programming approach to compute the DTW distance of two length-n time series, however, requires $$O(n^2)$$ O ( n 2 ) time, which is often too slow for real-world applications. Therefore, many heuristics have been proposed to speed up the DTW computation. These are often based on lower bounding techniques, approximating the DTW distance, or considering special input data such as binary or piecewise constant time series. In this paper, we present a first exact algorithm to compute the DTW distance of two run-length encoded time series whose running time only depends on the encoding lengths of the inputs. The worst-case running time is cubic in the encoding length. In experiments we show that our algorithm is indeed fast for time series with short encoding lengths.

The maximum weighted submatrix coverage problem is a recently introduced problem with applications in data mining. It is concerned with selecting K submatrices of a given numerical matrix such that the sum of the matrix-entries, which occur in at least one of the selected submatrices, is maximized. In the paper introducing the problem, a problem-specific constraint programming approach was developed and embedded in a large neighborhood-search to obtain a heuristic. A compact integer linear programming formulation was also presented, but deemed inefficient due to its size. In this paper, we introduce new integer linear programming formulations for the problem, one of them is based on Benders decomposition. The obtained Benders decomposition-based formulation has a nice combinatorial structure, i.e., there is no need to solve linear programs to separate Benders cuts. We present preprocessing procedures and valid inequalities for all formulations. We also develop a greedy randomized adaptive search procedure for the problem, which is enhanced with a local search. A computational study using the instances from literature is done to evaluate the effectiveness of our new approaches. Our algorithms manage to find improved primal solutions for ten out of 17 real-world instances, and optimality is proven for two real-world instances. Moreover, for over 700 of 1617 large-scale synthetic instances, our algorithms find improved primal solutions compared to the heuristics from the literature.

Let A,B be subsets of a finite abelian group G. Suppose that A+B does not contain a unique sum, i.e., there is no g∈G with a unique representation g=a+b, a∈A, b∈B. From such sets A,B, sparse linear systems over the rational numbers arise. We obtain a new determinant bound on invertible submatrices of the coefficient matrices of these linear systems. Under the condition that |A|+|B| is small compared to the order of G, these bounds provide essential information on the Smith Normal Form of these coefficient matrices. We use this information to prove that A and B admit coset partitions whose parts have properties resembling those of A and B. As a consequence, we improve previously known sufficient conditions for the existence of unique sums in A+B and show how our structural results can be used to classify sets A and B for which A+B does not contain a unique sum when |A|+|B| is relatively small. Our method also can be applied to subsets of abelian groups which have no unique differences. Ministry of Education (MOE) This research is supported by the Ministry of Education, Singapore, under its Academic Research Fund Tier 1 (RG27/18).

We provide evidence for partial deconfinement -- the deconfinement of a SU($M$) subgroup of the SU($N$) gauge group -- by using lattice Monte Carlo simulations. We take matrix models as concrete examples. By appropriately fixing the gauge, we observe that the $M\times M$ submatrices deconfine. This gives direct evidence for partial deconfinement at strong coupling. We discuss the applications to QCD and holography., Comment: 49 pages, 24 figures, 3 tables. Version accepted for publication on JHEP

Multi-view clustering has become an active topic in artificial intelligence. Yet, similar investigation for graph-structured data clustering has been absent so far. To fill this gap, we present a Multi-View Graph embedding Clustering network (MVGC). Specifically, unlike traditional multi-view construction methods, which are only suitable to describe Euclidean structure data, we leverage Euler transform to augment the node attribute, as a new view descriptor, for non-Euclidean structure data. Meanwhile, we impose block diagonal representation constraint, which is measured by the l 1 , 2 -norm, on self-expression coefficient matrix to well explore the cluster structure. By doing so, the learned view-consensus coefficient matrix well encodes the discriminative information. Moreover, we make use of the learned clustering labels to guide the learnings of node representation and coefficient matrix, where the latter is used in turn to conduct the subsequent clustering. In this way, clustering and representation learning are seamlessly connected, with the aim to achieve better clustering performance. Extensive experimental results indicate that MVGC is superior to 11 state-of-the-art methods on four benchmark datasets. In particular, MVGC achieves an Accuracy of 96.17% (53.31%) on the ACM (IMDB) dataset, which is an up to 2.85% (1.97%) clustering performance improvement compared with the strongest baseline.

Multi-view subspace clustering is an effective method to partition data into their corresponding categories. Nevertheless, existing multi-view subspace clustering approaches generally operate in a purely unsupervised manner, while ignoring the valuable weakly supervised information that can be readily obtained in many practical applications. In this paper, we consider the weakly supervised form of sample pair constraints, and devote to promoting the performance of multi-view subspace clustering with the aid of such prior knowledge. To achieve this goal, inspired by the intrinsic block diagonal structure of ideal low-rank representation (LRR), we propose a novel regularization to integrate must-link, cannot-link and normalization constraints into a unified formulation. The proposed regularization can be regarded as a general description for sample pairwise constraints, and thus provides a flexible framework for multi-view semi-supervised subspace clustering task. Furthermore, we devise a contrained tensor representation learning (CTRL) model that takes advantage of our proposed regularization to facilitate the learning of the desired representation tensor. An efficient optimization algorithm based on alternating direction minimization strategy is carefully designed to solve the proposed CTRL model. Extensive experiments on eight challenging real-world datasets are conducted, and the results validate the effectiveness of our designed pairwise constraints regularization, as well as the superiority of the proposed CTRL model.

We give a unified proof of the interlacing properties of eigenvalues of principle submatrices of totally positive matrices.

Last decades, there has been an increased interest in the use of the self-potential (SP) method in hydrogeophysics. In response to this strong interest, we develop a novel approach to the inversion of SP data. Mathematically, the SP inverse problem is the source identification problem for the Poisson equation. Our approach substantially differs from the standard regularization approach, which explicitly includes the forward-problem operator into the cost functional. We formulated the inverse problem is as an optimal control problem and then translate it into a variational system. The system is approximated in suitable finite-element spaces giving rise to an algebraic problem with the saddle-point structure. In contrast to the standard approach, which leads to a dense linear system, our method results in a system with a sparse block matrix. It can be efficiently solved by either direct sparse solvers or preconditioned iterative solvers. In this paper, we present the formulation of the problem and its finite-element approximation. We discuss the iterative solution and preconditioning strategies. Our software implementation is based on an industrial finite-element package. We also present a numerical experiment with node-based linear finite elements on tetrahedral grids. Our results suggest that the proposed approach may serve as a rapid and reliable tool for large-scale SP inverse problems. Moreover, the same technique can easily be extended to a wide range of geophysical linear inverse problems, such as inversions of magnetic and gravity data.

This letter proposes a novel sparse recovery method of multiple-input and multiple-output (MIMO) radar compressed sensing (CS) imaging algorithms. This method leverages the prior structure of the measurement matrix to judge targets' locations and to estimate the sparsity level in the grid roughly and finally inhabits the emergence of false targets in the imaging figure. Explicitly, the algorithm we propose is inspired by the orthogonal matching pursuit (OMP) algorithm. First, the measurement matrix can be divided into some submatrices by column. Then, we estimate which submatrices do not contain signal components by the algorithms we propose in this literature to achieve the reconstruction of the measurement matrix. Finally, we use the sparse Bayesian learning algorithm and the sparsity adaptive matching pursuit algorithm to recover the target location and scattering intensity. Experiments validate that the reconstruction error of the algorithm we propose is much lower than other sparse recovery algorithms, and targets in the imaging are more obvious than other algorithms.

International audience; This paper is concerned with the in-domain stabilization of a class of block diagonal infinite-dimensional systems in the presence of an uncertain and time-varying delay in the distributed control input. Two actuation schemes are considered. The first one assumes a control input that is fully distributed over the domain. The second one assumes that the control input is finite-dimensional, acting over the domain via a bounded operator. In both cases, the control design strategy consists in a predictor feedback law synthesized on a finite-dimensional truncated LTI model capturing the unstable dynamics of the original infinitedimensional system. The predictor feedback law is designed based on the knowledge of the nominal value of the uncertain and time-varying input delay. In the second actuation scheme, the case of distinct input delays in the different scalar control input channels is considered. Index Terms-Delayed distributed actuation, partial differential equations, predictor feedback law. H. Lhachemi and R. Shorten acknowledge the support of Science Foundation Ireland (Grant numbers 16/RC/3872 and 16/IA/4610) and I-Form industry partners. C. Prieur acknowledges the support from MIAI@Grenoble Alpes, (ANR-19-P3IA-0003).

There is a dearth of compressed-sensing localization in the domain of underwater acoustic sensor networks (UASNs). To address this limitation, this paper proposes SAP2: a Sparse-sensing Acoustic Positioning scheme for Stratified and Perturbed UASNs. Three conditions of underwater perturbation are defined, namely, Spin perturbation, Rectilinear dispersion condition, and Radially perturbed condition to model the sudden disturbances in the underwater network topology. Then, “Footprint Variation” is proposed to express the behavior of UASN node topology. Theorem 2.2.b is proposed for the exact retrieval of sparsely deployed UASN location under perturbation conditions. The derivation of a unique solution to the insertion vector in Lemma 3.1.a holds the estimated location of target, necessitating the incoherence of the sensing matrix with the basis vector. Block matrix representation of the coefficients is derived in Lemma 3.1.b. Invertibility of the block matrix is proven in Lemma 3.1.c to establish sparsity of underwater sensor networks. An example is illustrated with a realistic scenario to observe parameters such as correlation of sparse coefficients, distinctness of discretized samples and footprint variation based on perturbation. Footprint variation is found to be below 100m2 for uncorrelated discrete measurements using SAP2 technique. Through rigorous mathematical justification and realistic underwater channel scenario, it is established that compressed sensing based underwater localization is feasible for a sparse sensor network.

Two-level fractional factorial designs are often used in screening scenarios to identify active factors. This article investigates the block diagonal structure of the information matrix of nonregul...

The subspace clustering methods for motion segmentation are widely used in the field of computer vision. However, the existing methods ignore the low-rank property of motion trajectory with nonlinear structure and are sensitive to non-Gaussian noise. To this end, we seek to improve the performance of motion segmentation by effectively modeling some important characteristics of the motion trajectories, such as nonlinear structure and contained non-Gaussian noise. Specifically, we propose to use kernel function to model motion trajectory, design a variant of the correntropy-induced metric to measure noise, and integrate the block diagonal regularizer into the kernel subspace clustering to strengthen the block diagonal structure of the learned affinity matrix. More importantly, we propose a unified rank-constrained block diagonal subspace clustering method for motion segmentation, which can handle not only rigid body motion segmentation, but also non-rigid motion segmentation. And we further extend this method to deal with various noises in motion data, such as missing trajectories, corrupted trajectory and outlying trajectory. An effective algorithm HQ& AM, which is integrated by Half-quadratic theory and alternating minimization, is designed to optimize these models. Experimental results on several commonly used motion datasets indicate the effectualness and robustness of our methods.

The main goal of this paper is to elaborate some results on the spectral properties of $$2 \times 2$$ block matrix linear relations with unbounded entries and with a domain consisting of vectors which satisfy certain relations between their components. We present some conditions to prove some Frobenius–Schur decompositions for linear relations and characterize the stability of the essential spectra of these linear relations. Our results generalize many known ones in the literature, in particular those obtained by Alvarez et al. (in Math Methods Appl Sci 37:620–644, 2014) and Batkai et al. (in Math Nachr 278:1408–1429, 2005).

LU factorization is extensively used in engineering and scientific computations for solution of large set of linear equations. Particularly, circuit simulators rely heavily on sparse version of LU factorization for solution involving circuit matrices. One of the recent advances in this field is exploiting the emerging computing platform of graphics processing units (GPUs) for parallel and sparse LU factorization. In this article, following contributions are made to advance the state of the art in hybrid right-looking algorithm (RLA): 1) a novel GPU kernel based on parallel column and block size optimization (PCBSO) is developed for adaptively allocating the block size while optimizing the number of columns for parallel execution based on the size of their associated submatrices at every level. The proposed approach helps to minimize the resource contention and to improve the computational performance and 2) an algorithm is developed to enable the execution of the new adaptive mode with dynamic parallelism. Also, a comprehensive performance comparison using a set of benchmark circuit examples is presented. The results indicate that, the proposed advancements can improve the results of state-of-the-art right looking sparse LU factorization in GPU by $1.54\times $ (Arithmetic Mean).

This article considers two inverse eigenvalue problems for Lefkovitch and doubly Lefkovitch matrices. The problems consist of constructing these matrices from the maximal eigenvalues of all its leading principal submatrices, eigenvectors associated with some of these eigenvalues, and some diagonal entries. We give sufficient conditions for the existence of such matrices. In particular, when the matrices have constant diagonal entries. In addition, we provide numerical examples to show the results obtained.

Resistive random access memory (RRAM) has been demonstrated to implement multiply-and-accumulate (MAC) operations using a highly parallel analog fashion, which dramatically accelerates the convolutional neural networks (CNNs). Since CNNs require considerable converters between analog crossbars and digital peripheral circuits, recent studies map the binary neural networks (BNNs) onto RRAM and binarize the weights to {+1, −1}. However, two mainstream representations for BNN weights introduce patterns of redundant 0s and 1s when dealing with negative weights. In this work, we reduce the area of redundant 0s and 1s by proposing a BNN weight representation framework based on the novel pattern representation and a corresponding architecture. First, we spilt the weight matrix into several small matrices by clustering adjacent columns together. Second, we extract 1s’ patterns, i.e., the submatrices only containing 1s, from the small weight matrix, such that each final output can be represented by the sum of several patterns. Third, we map these patterns onto RRAM crossbars, including pattern computation crossbars (PCCs) and pattern accumulation crossbars (PACs). Finally, we compare the pattern representation with two mainstream representations and adopt the more area efficient one. The evaluation results demonstrate that our framework can save over 20% of crossbar area effectively, compared with two mainstream representations.

The aim of this paper is to establish formulae for the Drazin inverse of anti-triangular block matrices under new assumptions in literature. Precisely, we consider the Drazin inverse of three kinds of anti-triangular block matrices. Applying these results, we present new expressions for the Drazin inverse of an arbitrary block matrix. In this way, we generalize a list of earlier results and illustrate it with three examples.

This paper is concerned with the identification of the bilinear systems in the state-space form. The parameters to be identified of the considered systems are coupled with the unknown states, which makes the identification problem difficult. To deal with such a difficulty, the iterative estimation theory is considered to derive the joint parameter and state estimation algorithm. Specifically, a moving data window least squares-based iterative (MDW-LSI) algorithm is derived to estimate the parameters of the systems by using the window data, and the unknown states are estimated by a bilinear state estimator. Furthermore, in order to improve the computational efficiency, a matrix decomposition-based MDW-LSI algorithm and a hierarchical MDW-LSI algorithm are developed according to the block matrix inversion lemma and the hierarchical identification principle. Finally, the computational efficiency is discussed and the numerical examples are employed to test the proposed approaches.

This paper concerns test matrices for numerical linear algebra using an error-free transformation of floating-point arithmetic. For specified eigenvalues given by a user, we propose methods of generating a matrix whose eigenvalues are exactly known based on, for example, Schur or Jordan normal form and a block diagonal form. It is also possible to produce a real matrix with specified complex eigenvalues. Such test matrices with exactly known eigenvalues are useful for numerical algorithms in checking the accuracy of computed results. In particular, exact errors of eigenvalues can be monitored. To generate test matrices, we first propose an error-free transformation for the product of three matrices YSX. We approximate S by ${S^{\prime }}$ S ′ to compute ${YS^{\prime }X}$ Y S ′ X without a rounding error. Next, the error-free transformation is applied to the generation of test matrices with exactly known eigenvalues. Note that the exactly known eigenvalues of the constructed matrix may differ from the anticipated given eigenvalues. Finally, numerical examples are introduced in checking the accuracy of numerical computations for symmetric and unsymmetric eigenvalue problems.

It is an open question to learn an over-complete dictionary from a limited number of face samples, and the inherent attributes of the samples are underutilized. Besides, the recognition performance may be adversely affected by the noise (and outliers), and the strict binary label based linear classifier is not appropriate for face recognition. To solve above problems, we propose a virtual samples based robust block-diagonal dictionary learning for face recognition. In the proposed model, the original samples and virtual samples are combined to solve the small sample size problem, and both the structure constraint and the low rank constraint are exploited to preserve the intrinsic attributes of the samples. In addition, the fidelity term can effectively reduce negative effects of noise (and outliers), and the ε-dragging is utilized to promote the performance of the linear classifier. Finally, extensive experiments are conducted in comparison with many state-of-the-art methods on benchmark face datasets, and experimental results demonstrate the efficacy of the proposed method.

In this paper, an L-stable third derivative multistep method has been proposed for the solution of stiff systems of ordinary differential equations. The continuous hybrid method is derived using interpolation and collocation techniques of power series as the basis function for the approximate solution. The method consists of the main method and an additional method which are combined to form a block matrix and implemented simultaneously. The stability and convergence properties of the block were investigated and discussed. Numerical examples to show the efficiency and accuracy of the new method were presented.

Let A ∈ C n × n be normal with eigenvalues λ 1 , … , λ n . For a given z in the numerical range of A, consider the set B A , k ( z ) of k × k matrices W for which [ z ⁎ 0 W ] is a compression of A. Assuming that no three eigenvalues lie on the same line, we prove that for any compact subset K of the numerical range of A that avoids the eigenvalues, the mappings z ↦ B A , k ( z ) and z ↦ σ ( B A , k ( z ) ) are uniformly continuous on K.

We consider the structure of the point-line incidence matrix of the projective space $$\mathrm {PG}(3,q)$$ connected with orbits of points and lines under the stabilizer group of the twisted cubic. Structures of submatrices with incidences between a union of line orbits and an orbit of points are investigated. For the unions consisting of two or three line orbits, the original submatrices are split into new ones, in which the incidences are also considered. For each submatrix (apart from the ones corresponding to a special type of lines), the numbers of lines through every point and of points lying on every line are obtained. This corresponds to the numbers of ones in columns and rows of the submatrices.

A motor imagery EEG (MI-EEG) signal is often selected as the driving signal in an active brain computer interface (BCI) system, and it has been a popular field to recognize MI-EEG images via convolutional neural network (CNN), which poses a potential problem for maintaining the integrity of the time-frequency-space information in MI-EEG images and exploring the feature fusion mechanism in the CNN. However, information is excessively compressed in the present MI-EEG image, and the sequential CNN is unfavorable for the comprehensive utilization of local features. In this paper, a multidimensional MI-EEG imaging method is proposed, which is based on time-frequency analysis and the Clough-Tocher (CT) interpolation algorithm. The time-frequency matrix of each electrode is generated via continuous wavelet transform (WT), and the relevant section of frequency is extracted and divided into nine submatrices, the longitudinal sums and lengths of which are calculated along the directions of frequency and time successively to produce a 3 × 3 feature matrix for each electrode. Then, feature matrix of each electrode is interpolated to coincide with their corresponding coordinates, thereby yielding a WT-based multidimensional image, called WTMI. Meanwhile, a multilevel and multiscale feature fusion convolutional neural network (MLMSFFCNN) is designed for WTMI, which has dense information, low signal-to-noise ratio, and strong spatial distribution. Extensive experiments are conducted on the BCI Competition IV 2a and 2b datasets, and accuracies of 92.95% and 97.03% are yielded based on 10-fold cross-validation, respectively, which exceed those of the state-of-the-art imaging methods. The kappa values and p values demonstrate that our method has lower class skew and error costs. The experimental results demonstrate that WTMI can fully represent the time-frequency-space features of MI-EEG and that MLMSFFCNN is beneficial for improving the collection of multiscale features and the fusion recognition of general and abstract features for WTMI.

We characterize the involutiveness of the linear combinations of the form $a{\bf{A}} + b{\bf{B}}$ when $a,b$ are nonzero complex numbers, ${\bf{A}}$ is a quadratic $n \times n$ nonzero matrix and ${\bf{B}}$ is an arbitrary $n \times n$ nonzero matrix, under certain properties imposed on $\bf{A}$ and $\bf{B}$. Moreover, we give some examples illustrating our main results.

Multi-view subspace clustering (MVSC) finds a shared structure in latent low-dimensional subspaces of multi-view data to enhance clustering performance. Nonetheless, we observe that most existing MVSC methods neglect the diversity in multi-view data by considering only the common knowledge to find a shared structure either directly or by merging different similarity matrices learned for each view. In the presence of noise, this predefined shared structure becomes a biased representation of the different views. Thus, in this article, we propose a MVSC method based on coupled low-rank representation to address the above limitation. Our method first obtains a low-rank representation for each view, constrained to be a linear combination of the view-specific representation and the shared representation by simultaneously encouraging the sparsity of view-specific one. Then, it uses the k -block diagonal regularizer to learn a manifold recovery matrix for each view through respective low-rank matrices to recover more manifold structures from them. In this way, the proposed method can find an ideal similarity matrix by approximating clustering projection matrices obtained from the recovery structures. Hence, this similarity matrix denotes our clustering structure with exactly k connected components by applying a rank constraint on the similarity matrix’s relaxed Laplacian matrix to avoid spectral post-processing of the low-dimensional embedding matrix. The core of our idea is such that we introduce dynamic approximation into the low-rank representation to allow the clustering structure and the shared representation to guide each other to learn cleaner low-rank matrices that would lead to a better clustering structure. Therefore, our approach is notably different from existing methods in which the local manifold structure of data is captured in advance. Extensive experiments on six benchmark datasets show that our method outperforms 10 similar state-of-the-art compared methods in six evaluation metrics.

An efficient improved spectrum integral method (ISIM) is proposed to accelerate the scattering simulation of cuboid object by method of moment (MoM) in multiplanar layered medium (LM). The layered medium Green’s functions (LMGFs) are studied and decomposed into transmission and reflection terms, which hold the translational invariance and reverse translational invariance, respectively. By virtue of the (reverse) translational invariance of LMGF from the uniform square grids on the surface of the cuboid object, most of the coupling between the expanding basis functions and the testing basis functions can be represented by two kinds of submatrices, i.e., the Toeplitz matrix when transmission term is involved, and the conjugate Toeplitz matrix when reflection term is involved. Moreover, the submatrix coupling the expanding/testing basis functions defined on the edges with those defined on the parallel surfaces/edges is also extended to the (conjugate) Toeplitz matrixes. This will significantly alleviate the heavy computation and memory burden from the conventional SIM. With the improved operations on the basis functions associated with the edges of the cuboid object, the proposed ISIM in LM can reach the computational complexity as low as ${O}$ ( ${N} ^{1.5}$ log( N)) in the matrix vector production, while the computational complexity of matrix filling and the memory requirement are both ${O}$ ( ${N}^{1.5}$ ). Moreover, compared with conventional SIM, both theoretical analysis and numerical results demonstrate that the memory requirement and matrix filling time are dramatically decreased.

Structured representation is of remarkable significance in subspace clustering. However, most of the existing subspace clustering algorithms resort to single-structured representation, which may fail to fully capture the essential characteristics of data. To address this issue, a novel multi-structured representation subspace clustering algorithm called block diagonal sparse representation (BDSR) is proposed in this paper. It takes both sparse and block diagonal structured representations into account to obtain the desired affinity matrix. The unified framework is established by integrating the block diagonal prior into the original sparse subspace clustering framework and the resulting optimization problem is iteratively solved by the inexact augmented Lagrange multipliers (IALM). Extensive experiments on both synthetic and real-world datasets well demonstrate the effectiveness and efficiency of the proposed algorithm against the state-of-the-art algorithms.

In control engineering, numerical stability and real time are the two most important requirements for the matrix inversion. This article presents an efficient and robust method for the field-programmable gate array (FPGA) calculation of the matrix inversion. We initially consider the scenario that the matrix to be processed is a nonsingular Hermitian matrix. The proposed computation procedures are composed of the matrix decomposition, triangular matrix inversion, and matrices multiplication. The first procedure is completed by LDL factorization based on the outer form of Cholesky's method, whereas the recursive algorithm for block submatrices is adopted to achieve the triangular matrix inversion. The new method has the high level in the parallel pipelining mechanism and steals the characteristics of both the upper triangular matrix and its inversion to reduce the computation load and improve the numerical stability. Furthermore, the non-Hermitian matrix inversion can be easily solved if another procedure is added in the new method. Finally, we compare our method with the exiting FPGA-based techniques on one Xilinx Virtex-7 XC7VX690T FPGA. Meanwhile, it has solved one array antenna control problem of the adaptive digital beam forming for one phased array radar successfully.

In this paper, we study an inverse eigenvalue problem, referred to as the minimax inverse eigenvalue problem, of constructing matrices whose graph is a special type of tree called a generalized star of depth 2. A scheme of labelling the vertices of such a tree is introduced in order to express the corresponding matrices in a number of special forms. We then solve the minimax inverse eigenvalue problem of constructing such matrices from given eigen data consisting of the minimal and maximal eigenvalues of each of the leading principal submatrices of the required matrices. We pose an open problem of solving the minimax inverse eigenvalue problem for matrices described by arbitrary trees.

Let a norm on the set M n of real or complex n-by-n matrices be given. We investigate the question of finding the largest constants α n and β n such that for each A ∈ M n the average of the norms of its (n−1)-by-(n−1) principal submatrices is at least α n times the norm of A, and such that the maximum of the norms of those principal submatrices is at least β n times the norm of A. For a variety of classical norms including induced l p -norms, weakly unitarily invariant norms, and entrywise norms we give lower and upper bounds for α n and β n . In several cases α n and β n are explicitly determined.

Several trace inequalities for positive semi-definite 2×2 block matrices are revisited. We present shorter proofs of these inequalities along with their refinements. We obtain bounds for the smalle...

The kernel subspace clustering algorithm aims to tackle the nonlinear subspace model. The block diagonal representation subspace clustering has a more promising capability in pursuing the k-block diagonal matrix. Therefore, the low-rankness and the adaptivity of the kernel subspace clustering can boost the clustering performance, so an adaptive low-rank kernel block diagonal representation (ALKBDR) subspace clustering algorithm is put forward in this work. On the one hand, for the nonlinear nature of the practical visual data, a kernel block diagonal representation (KBDR) subspace clustering algorithm is put forward. The proposed KBDR algorithm first maps the original input space into the kernel Hilbert space which is linearly separable, and next applies the spectral clustering on the feature space. On the other hand, the ALKBDR algorithm uses the adaptive kernel matrix and makes the feature space low-rank to further promote the clustering performance. The experimental results on the Extended Yale B database and the ORL dataset have proved the excellent quality of the proposed KBDR and ALKBDR algorithm in comparison with other advanced subspace clustering algorithms that also are tested in this paper.

We investigate the spectrum of the non-backtracking matrix of a graph. In particular, we show how to obtain eigenvectors of the non-backtracking matrix in terms of eigenvectors of a smaller matrix. In doing so, we create a block diagonal decomposition of the non-backtracking matrix, more clearly expressing its eigenvalues. Furthermore, we find an expression for the eigenvalues of the non-backtracking matrix in terms of eigenvalues of the adjacency matrix, and use this to upper-bound the spectral radius of the non-backtracking matrix, and to give a lower bound on the spectrum. Specifically, an upper-bound on the spectral radius is found in terms of the number of nodes and edges of the graph. We also investigate properties of a graph that can be determined by the spectrum. Specifically, we prove that the number of components, the number of degree 1 vertices, and whether or not the graph is bipartite are all determined by the spectrum of the non-backtracking matrix.

We present a new necessary and sufficient condition of exponential stability of neutral-type linear time invariant (LTI) systems with one delay. The new stability criterion is given in terms of a block matrix, which exclusively depends on the delay Lyapunov matrix and is of finite dimension.

We develop a nonoverlapping Additive Schwarz Preconditioner for the p-version finite element with $$H^2$$ -conforming elements on triangles based on Argyris elements. With a particular choice of basis functions corresponding to the $$C^2$$ degrees of freedom (dofs), we give a preconditioner consisting of eliminating the interior dofs, global solves of the $$C^0$$ and $$C^1$$ vertex dofs, diagonal solves of the $$C^2$$ vertex dofs, and block diagonal solves of the edge dofs. We show that the condition number of the preconditioner system grows at most like $${\mathscr {O}}(1+\log ^3 p)$$ independent of the mesh size h. The analysis permits the use of inexact interior solves which lends itself to a more efficient implementation while maintaining the same $${\mathscr {O}}(1 + \log ^3 p)$$ asymptotic growth of the preconditioned system.

In the mechanical design industry, it is very time-consuming to convert actual engineering design problems into finite element models, and it is repeatedly modified during the product design process. Each time the structure is modified, computer simulation analysis is repeated, and the product development cycle becomes longer. Therefore, in this paper, a re-analysis theoretical method based on the inversion of the block matrix based on the change threshold is proposed to quickly obtain responses with different accuracy after the structure is modified according to the initial analysis. By modifying the size of the change threshold, the accuracy and calculation time of the reanalysis results are controlled. This reanalysis method shortens a lot of calculation time under the premise of controllable response accuracy. In this paper, numerical calculations are given to show the accuracy of the response under different thresholds, and the efficiency of this method is proved by the finite element model under the same accuracy.

RNA molecules play crucial roles in various biological processes. Their three-dimensional configurations determine the functions and, in turn, influences the interaction with other molecules. RNAs and their interaction structures, the so-called RNA–RNA interactions, can be abstracted in terms of secondary structures, i.e., a list of the nucleotide bases paired by hydrogen bonding within its nucleotide sequence. Each secondary structure, in turn, can be abstracted into cores and shadows. Both are determined by collapsing nucleotides and arcs properly. We formalize all of these abstractions as arc diagrams, whose arcs determine loops. A secondary structure, represented by an arc diagram, is pseudoknot-free if its arc diagram does not present any crossing among arcs otherwise, it is said pseudoknotted. In this study, we face the problem of identifying a given structural pattern into secondary structures or the associated cores or shadow of both RNAs and RNA–RNA interactions, characterized by arbitrary pseudoknots. These abstractions are mapped into a matrix, whose elements represent the relations among loops. Therefore, we face the problem of taking advantage of matrices and submatrices. The algorithms, implemented in Python, work in polynomial time. We test our approach on a set of 16S ribosomal RNAs with inhibitors of Thermus thermophilus, and we quantify the structural effect of the inhibitors.

There has been a rise in the popularity of algebraic methods for graph algorithms given the development of the GraphBLAS library and other sparse matrix methods. An exemplar for these approaches is Breadth-First Search (BFS). The algebraic BFS algorithm is simply a recurrence of matrix-vector multiplications with the $n \times n$ adjacency matrix, but the many redundant operations over nonzeros ultimately lead to suboptimal performance. Therefore an optimal algebraic BFS should be of keen interest especially if it is easily integrated with existing matrix methods. Current methods, notably in the GraphBLAS, use a Sparse Matrix masked-Sparse Vector (SpMmSpV) multiplication in which the input vector is kept in a sparse representation in each step of the BFS, and nonzeros in the vector are masked in subsequent steps. This has been an area of recent research in GraphBLAS and other libraries. While in theory these masking methods are asymptotically optimal on sparse graphs, many add work that leads to suboptimal runtime. We give a new optimal, algebraic BFS for sparse graphs, thus closing a gap in the literature. Our method multiplies progressively smaller submatrices of the adjacency matrix at each step. Let $n$ and $m$ refer to the number of vertices and edges, respectively. On a sparse graph, our method takes $O(n)$ algebraic operations as opposed to $O(m)$ operations needed by theoretically optimal sparse matrix approaches. Thus for sparse graphs it matches the bounds of the best-known sequential algorithm and on a Parallel Random Access Machine (PRAM) it is work-optimal. Our result holds for both directed and undirected graphs. Compared to a leading GraphBLAS library our method achieves up to 24x faster sequential time and for parallel computation it can be 17x faster on large graphs and 12x faster on large-diameter graphs., Will appear in ACM Transactions on Knowledge Discovery from Data, Vol. 15, No. 5, 2021

Subspace clustering is a technique utilized to find clusters within multiple subspaces. However, most existing methods cannot obtain an accurate block diagonal clustering structure to improve clustering performance. This drawback exists because these methods learn the similarity matrix in advance by utilizing a low dimensional matrix obtained directly from the data, where two unrelated data samples can stay related easily due to the influence of noise. This paper proposes a novel method based on coupled low-rank representation to tackle the above problem. First, our method constructs a manifold recovery structure to correct inadequacy in the low-rank representation of data. Then it obtains a clustering projection matrix that obeys the k-block diagonal property to learn an ideal similarity matrix. This similarity matrix denotes our clustering structure with a rank constraint on its normalized Laplacian matrix. Therefore, we avoid k-means spectral post-processing of the low dimensional embedding matrix, unlike most existing methods. Furthermore, we couple our method to allow the clustering structure to adaptively approximate the low-rank representation so as to find more optimal solutions. Several experiments on benchmark datasets demonstrate that our method outperforms similar state-of-the-art methods in Accuracy, Normalized Mutual Information, F-score, Recall, Precision, and Adjusted Rand Index evaluation metrics.

A simplified decoding method for polar codes with large kernels is presented. The proposed approach consists in decoder-side substitution of some submatrices of the polarizing transformation with matrices, which admit simpler evaluation of log-likelihood ratios. This approach enables complexity reduction for the successive cancellation, list and sequential decoding algorithms.

For a class of optimal control problems constrained with certain time- and space-fractional diffusive equations, by making use of mixed discretizations of temporal finite-difference and spatial finite-element schemes along with Lagrange multiplier approach, we obtain specially structured block two-by-two linear systems. We demonstrate positive definiteness of the coefficient matrices of these discrete linear systems, construct rotated block-diagonal preconditioning matrices, and analyze spectral properties of the corresponding preconditioned matrices. Both theoretical analysis and numerical experiments show that the preconditioned Krylov subspace iteration methods, when incorporated with these rotated block-diagonal preconditioners, can exhibit optimal convergence property in the sense that their convergence rates are independent of both discretization stepsizes and problem parameters, and their computational workloads are linearly proportional with the number of discrete unknowns.

This article considers the problem of solving linear algebraic equations of the form $Ax=b$ among multiagents, which seek a solution by using local information in presence of random communication topologies. The equation is solved by $m$ agents where each agent only knows a subset of rows of the partitioned matrix $[A,b]$ . The problem is formulated such that this formulation does not need the distribution of random interconnection graphs. Therefore, this framework includes asynchronous updates and/or unreliable communication protocols. The random Krasnoselskii–Mann iterative algorithm is applied that converges almost surely and in mean square to a solution of the problem for any matrices $A$ and $b$ and any initial conditions of agents’ states. The algorithm is a totally asynchronous algorithm without requiring a priori B-connectivity and distribution dependency assumptions. The algorithm is able to solve the problem even if the weighted matrix of the graph is periodic and irreducible for synchronous protocol. It is demonstrated that the limit point to which the agents’ states converge is determined by the unique solution of a convex optimization problem regardless of the distribution of random communication graphs. Finally, some numerical examples are given to show the results.

This paper proposes a sphere-triangle contact model for three-dimensional sphere discontinuous deformation analysis to simulate complex boundary face problems often encountered in engineering. First, the detection of sphere-triangle contact and the calculation of corresponding sphere-triangle contact submatrices for building a sphere-triangle contact model are described. Then, the correctness of the proposed model is preliminarily verified by simulating the sphere-plane/edge/vertex contact, sphere-triangle friction problem and a complex boundary face problem. Furthermore, a rigorous open-close iteration is proposed to guarantee the correctness of setting penalty springs for sphere-triangle contacts, to ensure computational stability of three-dimensional sphere discontinuous deformation analysis. Meanwhile, two imperceptible fatal errors (i.e., direction deflection error and oscillation error) hidden in the sphere-triangle contact model are estimated, which reveals that a larger sphere or a higher collision velocity at a sphere-triangle contact leads to larger errors. In addition, an algorithm for excluding invalid contacts is proposed to avoid these fatal errors, enhancing the robustness of the proposed contact model. Finally, a landslide simulation with 80,974 spheres, 76,382 triangles and 150,000 calculation steps is conducted, further validating the proposed model in large-scale simulations. This work indicates the feasibility of the proposed model for simulating complex boundary face problems.

Mutual coupling in finite arrays of antennas with strongly overlapping minimum spheres is quickly calculated by computing the general translation matrix between spherical modes. This matrix is obtained by using the transformation properties of spherical and plane vector waves. Although this approach is less efficient than the classical one, which is based on addition theorems, it allows to overcome the well-known limitation of addition theorems that requires nonintersecting minimum spheres. Symmetry relations are provided for the translation coefficients that greatly increase the speed of computation of the general translation matrix. By computing the reflection and the transmission submatrices of the generalized scattering matrix of a finite antenna array, accurate results are obtained for the S-parameters and the radiation patterns of arrays, in comparison with commercial software or a purely numerical in-house full-wave method. For this purpose, different types of antennas with strongly overlapping hemispheres in an array environment on a ground plane are used, such as apertures, monopoles, cavity-backed patch antennas, or dielectric resonator antennas.

Summary This paper introduces a new data-driven methodology for estimating sparse covariance matrices of the random coefficients in logit mixture models. Researchers typically specify covariance matrices in logit mixture models under one of two extreme assumptions: either an unrestricted full covariance matrix (allowing correlations between all random coefficients), or a restricted diagonal matrix (allowing no correlations at all). Our objective is to find optimal subsets of correlated coefficients for which we estimate covariances. We propose a new estimator, called MISC (mixed integer sparse covariance), that uses a mixed-integer optimization (MIO) program to find an optimal block diagonal structure specification for the covariance matrix, corresponding to subsets of correlated coefficients, for any desired sparsity level using Markov Chain Monte Carlo (MCMC) posterior draws from the unrestricted full covariance matrix. The optimal sparsity level of the covariance matrix is determined using out-of-sample validation. We demonstrate the ability of MISC to correctly recover the true covariance structure from synthetic data. In an empirical illustration using a stated preference survey on modes of transportation, we use MISC to obtain a sparse covariance matrix indicating how preferences for attributes are related to one another.

A conference matrix of order n is an $$n\times n$$ n × n matrix C with diagonal entries 0 and off-diagonal entries $$\pm 1$$ ± 1 satisfying $$CC^\top =(n-1)I$$ C C ⊤ = ( n - 1 ) I . If C is symmetric, then C has a symmetric spectrum $$\Sigma $$ Σ (that is, $$\Sigma =-\Sigma $$ Σ = - Σ ) and eigenvalues $$\pm \sqrt{n-1}$$ ± n - 1 . We show that many principal submatrices of C also have symmetric spectrum, which leads to examples of Seidel matrices of graphs (or, equivalently, adjacency matrices of complete signed graphs) with a symmetric spectrum. In addition, we show that some Seidel matrices with symmetric spectrum can be characterized by this construction.

We consider discrete Dirac systems as an alternative (to the famous Szeg\H{o} recurrencies and matrix orthogonal polynomials) approach to the study of the corresponding block Toeplitz matrices. We prove an analog of the Christoffel--Darboux formula and derive the asymptotic relations for the analog of reproducing kernel (using Weyl--Titchmarsh functions of discrete Dirac systems). We study also the case of rational Weyl--Titchmarsh functions (and GBDT version of the B\"acklund-Darboux transformation of the trivial discrete Dirac system). We show that block diagonal plus block semi-separable Toeplitz matrices appear in this case., Comment: This work is an important development of the idea presented in our paper arXiv:1711.03064