Sum Of Matrices Research Articles

Concave function inequalities for sum of matrices

Concave function inequalities for sum of matrices

Enhanced Results on Sampled-Data Synchronization for Chaotic Neural Networks With Actuator Saturation Using Parameterized Control.

This article investigates a novel sampled-data synchronization controller design method for chaotic neural networks (CNNs) with actuator saturation. The proposed method is based on a parameterization approach which reformulates the activation function as the weighted sum of matrices with the weighting functions. Also, controller gain matrices are combined by affinely transformed weighting functions. The enhanced stabilization criterion is formulated in terms of linear matrix inequalities (LMIs) based on the Lyapunov stability theory and weighting function's information. As shown in the comparison results of the bench marking example, the presented method much outperforms previous methods, and thus the enhancement of the proposed parameterized control is verified.

Spectral properties of the Laplacian of temporal networks following a constant block Jacobi model.

We study the behavior of the eigenvectors associated with the smallest eigenvalues of the Laplacian matrix of temporal networks. We consider the multilayer representation of temporal networks, i.e., a set of networks linked through ordinal interconnected layers. We analyze the Laplacian matrix, known as supra-Laplacian, constructed through the supraadjacency matrix associated with the multilayer formulation of temporal networks, using a constant block Jacobi model which has closed-form solution. To do this, we assume that the interlayer weights are perturbations of the Kronecker sum of the separate adjacency matrices forming the temporal network. Thus we investigate the properties of the eigenvectors associated with the smallest eigenvalues (close to zero) of the supra-Laplacian matrix. Using arguments of perturbation theory, we show that these eigenvectors can be approximated by linear combinations of the zero eigenvectors of the individual time layers. This finding is crucial in reconsidering and generalizing the role of the Fielder vector in supra-Laplacian matrices.

Voltage differences as functions of a matrix eigensystem

We firstly consider an electrical network that consists of multiple subnetworks. We refer to some basic definitions which relate net current flows and voltages, the edges and nodes respectively of each subnetwork, and define matrix G, as the direct sum of the admittance matrices that correspond to each subnetwork. Then, we prove that any voltage difference at two arbitrary nodes can be explicitly written as a function only of the eigensystem of G, namely its eigenvalues and their corresponding eigenvectors. Next, we consider a DC/AC circuit network and by using duality properties of a matrix pencil related to the network, we obtain expressions for voltage differences as functions of the pencil’s eigenvalues and their corresponding eigenvectors. To validate our findings, we present illustrative numerical examples.

Scattering matrices of particle ensembles analytically decomposed into pure Mueller matrices

Scattering matrices of particle ensembles are analytically decomposed into sums of pure Mueller matrices. The ensembles are assumed to have equal numbers of particles and their mirror particles, both in random orientation. In the general case, there are four pure Mueller matrices in the decomposition. In the present spectral decomposition, of these four matrices, there is a single matrix that qualifies as a scattering matrix, whereas the remaining three matrices represent other classes of pure matrices. For ensembles of spherical particles, there are two pure Mueller matrices in the decomposition. Again, there is a single matrix qualifying as a scattering matrix. The analytical decomposition unveils the explicit dependencies of the pure Mueller matrices on the ensemble-averaged scattering matrix. Applications are identified in scattering by single particles and by random media of particles.

Spectral co-clustering in multi-layer directed networks

Modern network analysis often involves multi-layer network data in which the nodes are aligned, and the edges on each layer represent one of the multiple relations among the nodes. Current literature on multi-layer network data is mostly limited to undirected relations. However, direct relations are more common and may introduce extra information. This study focuses on community detection (or clustering) in multi-layer directed networks. To take into account the asymmetry, a novel spectral-co-clustering-based algorithm is developed to detect co-clusters, which capture the sending patterns and receiving patterns of nodes, respectively. Specifically, the eigendecomposition of the debiased sum of Gram matrices over the layer-wise adjacency matrices is computed, followed by the k-means, where the sum of Gram matrices is used to avoid possible cancellation of clusters caused by direct summation. Theoretical analysis of the algorithm under the multi-layer stochastic co-block model is provided, where the common assumption that the cluster number is coupled with the rank of the model is relaxed. After a systematic analysis of the eigenvectors of the population version algorithm, the misclassification rates are derived, which show that multi-layers would bring benefits to the clustering performance. The experimental results of simulated data corroborate the theoretical predictions, and the analysis of a real-world trade network dataset provides interpretable results.

A GenEO Domain Decomposition method for Saddle Point problems

We introduce an adaptive element-based domain decomposition (DD) method for solving saddle point problems defined as a block two by two matrix. The algorithm does not require any knowledge of the constrained space. We assume that all sub matrices are sparse and that the diagonal blocks are spectrally equivalent to a sum of positive semi definite matrices. The latter assumption enables the design of adaptive coarse space for DD methods that extends the GenEO theory (Spillane et al., 2014) to saddle point problems. Numerical results on three dimensional elasticity problems for steel-rubber structures discretized by a finite element with continuous pressure are shown for up to one billion degrees of freedom.

Distributed Consensus for Global Matrix Formation in the Principal Component Pursuit Scenario

The aim behind principal component pursuit is to recover a low-rank matrix and a sparse matrix from a noisy signal which is the sum of both matrices. This optimization problem is a priori and non-convex and is useful in signal processing, data compression, image processing, machine learning, fluid dynamics, and more. Here, a distributed scheme described by a static undirected graph, where each agent only observes part of the noisy or corrupted matrix, is applied to achieve a consensus; then, a robust approach that can also handle missing values is applied using alternating directions to solve the convex relaxation problem, which actually solves the non-convex problem under some weak assumptions. Some examples of image recovery are shown, where the network of agents achieves consensus exponentially fast.

Stability Region of Grid-Forming Wind Turbine with Variable Parameters Using Bialternate Sum Matrix Approach

Although the stability regions of wind turbines in the islanding mode have been widely researched, small-signal modeling of grid-forming wind turbines (GFWTs) in the islanding mode has yet to be explored. In addition, the state-space matrix of the wind turbine system has yet to be fully represented. Therefore, this paper proposes a small-signal modeling of GFWT and a method for identifying the stabilization region of a system with variable parameters. First, small-signal modeling of a GFWT based on virtual synchronous generator control is developed. Second, the bialternate sum matrix approach is used to determine the system stabilization region. The system matrix with multiple variable parameters is first decomposed into the sum of several matrices in this paper. Furthermore, the rotor-side generator control is simplified. It can reduce the dimensionality of the system matrix model. Finally, the simulation shows that the proposed method for determining the stabilization region of the variable system is accurate.

3-D LiDAR Localization Based on Novel Nonlinear Optimization Method for Autonomous Ground Robot

3D Light Detection and Ranging (LiDAR)-based localization for autonomous ground robot in unknown environment is a critical ability, which has been extensively studied. In this paper, we propose a novel nonlinear optimization method to promote the 3D LiDAR localization performance. Firstly, due to that the objective function is the summation of point correspondence matching error items, a novel item quality evaluation criterion is proposed. The quality of each item is directly proportional to the degree of its matching error distance descent during the optimization process. Then, an attention mechanism based on the criterion is proposed and the objective function is iteratively refined by putting different attention weights on the matching error items. Thirdly, in the Gauss-Newton optimization framework for LiDAR localization, we point out that the Hessian matrix is essentially the sum of the Hessian matrices of high-quality and low-quality point correspondence subsets. To increase the dominance of the Hessian matrix derived from the high-quality point correspondences, the Hessian matrix of the estimated high-quality point correspondence subset in the last updating step is added to the current Hessian matrix. The proposed LiDAR localization is extensively evaluated on public and custom datasets. The experimental results demonstrate that the proposed mechanisms can effectively promote the LiDAR localization performance. We will make our source code open to serve as a new baseline for the robotic community.

Dimension-free bounds for sums of independent matrices and simple tensors via the variational principle

Dimension-free bounds for sums of independent matrices and simple tensors via the variational principle

Dimension-free bounds for sums of dependent matrices and operators with heavy-tailed distributions

Dimension-free bounds for sums of dependent matrices and operators with heavy-tailed distributions

Sums of GUE matrices and concentration of hives from correlation decay of eigengaps

Sums of GUE matrices and concentration of hives from correlation decay of eigengaps

Adaptive multi-channel contrastive graph convolutional network with graph and feature fusion

The domain of multi-view semi-supervised classification is an appealing topic in real-world applications. Due to the powerful capability of gathering information from neighbors, Graph Convolutional Network (GCN) has become a hotspot in the classification task. However, most of multi-view classification works based on GCN only assign weights for feature fusion, and directly consider the weighted sum of the adjacency matrices, ignoring the interaction and correlation among features. These may be problematic since aggregating the matrices from less relevant views may destroy the original topology space, leading to undesired performance. To tackle the aforementioned challenges, this paper presents an Adaptive Multi-Channel Graph Convolutional Network (AMC-GCN). To extract the interactive information, AMC-GCN designs a deep interactive feature integration network to incorporate consensus and complementary information. To fuse the graph structures, AMC-GCN exploits the relevance between views and imposes an adjacency matrix fusion network on constructing multiple GCN channels, thereby delivering discriminative information on graphs. To enhance the homogeneity of the framework, AMC-GCN applies a contrastive loss to joint learning during the optimization for classification. With these considerations, AMC-GCN exploits relevant and interactive information between views to promote graph and feature fusion. Substantial experimental results on real-world datasets verify the superiority of AMC-GCN.

Computational Complexity of Decomposing a Symmetric Matrix as a Sum of Positive Semidefinite and Diagonal Matrices

Computational Complexity of Decomposing a Symmetric Matrix as a Sum of Positive Semidefinite and Diagonal Matrices

Singular vectors of sums of rectangular random matrices and optimal estimation of high-rank signals: The extensive spike model.

Across many disciplines spanning from neuroscience and genomics to machine learning, atmospheric science, and finance, the problems of denoising large data matrices to recover hidden signals obscured by noise, and of estimating the structure of these signals, is of fundamental importance. A key to solving these problems lies in understanding how the singular value structure of a signal is deformed by noise. This question has been thoroughly studied in the well-known spiked matrix model, in which data matrices originate from low-rank signal matrices perturbed by additive noise matrices, in an asymptotic limit where matrix size tends to infinity but the signal rank remains finite. We first show, strikingly, that the singular value structure of large finite matrices (of size ∼1000) with even moderate-rank signals, as low as 10, is not accurately predicted by the finite-rank theory, thereby limiting the application of this theory to real data. To address these deficiencies, we analytically compute how the singular values and vectors of an arbitrary high-rank signal matrix are deformed by additive noise. We focus on an asymptotic limit corresponding to an extensive spike model, in which both the signal rank and the size of the data matrix tend to infinity at a constant ratio. We map out the phase diagram of the singular value structure of the extensive spike model as a joint function of signal strength and rank. We further exploit these analytics to derive optimal rotationally invariant denoisers to recover the hidden high-rank signal from the data, as well as optimal invariant estimators of the signal covariance structure. Our extensive-rank results yield several conceptual differences compared to the finite-rank case: (1) as signal strength increases, the singular value spectrum does not directly transition from a unimodal bulk phase to a disconnected phase, but instead there is a bimodal connected regime separating them; (2) the signal singular vectors can be partially estimated even in the unimodal bulk regime, and thus the transitions in the data singular value spectrum do not coincide with a detectability threshold for the signal singular vectors, unlike in the finite-rank theory; (3) signal singular values interact nontrivially to generate data singular values in the extensive-rank model, whereas they are noninteracting in the finite-rank theory; and (4) as a result, the more sophisticated data denoisers and signal covariance estimators we derive, which take into account these nontrivial extensive-rank interactions, significantly outperform their simpler, noninteracting, finite-rank counterparts, even on data matrices of only moderate rank. Overall, our results provide fundamental theory governing how high-dimensional signals are deformed by additive noise, together with practical formulas for optimal denoising and covariance estimation.

Singular value and norm inequalities for products and sums of matrices

Singular value and norm inequalities for products and sums of matrices

The sum of all width-one matrices

A nonnegative integer matrix is said to be width-one if its nonzero entries lie along a path consisting of steps to the south and to the east. These matrices are important in optimal transport theory: the northwest corner algorithm, for example, takes supply and demand vectors and outputs a width-one matrix. The problem in this paper is to write down an explicit formula for the sum of all width-one matrices (with given dimensions n×n and given sum d of the entries). We prove two strikingly different formulas. The first, a 4F3 hypergeometric series with unit argument, is obtained by applying the Robinson–Schensted–Knuth correspondence to the width-one matrices; the second is obtained via Stanley–Reisner theory. Computationally, our two formulas are complementary to each other: the first formula outperforms the second if d is fixed and n increases, while the second outperforms the first if n is fixed and d increases. We also show how our result yields a new non-recursive formula for the mean value of the discrete earth mover’s distance (i.e., the solution to the transportation problem), whenever the cost matrix has the Monge property.

Proper improvements of well-known perturbation estimations for the parallel sum

Let A:B be the parallel sum of the positive semidefinite matrices A and B. In this paper, we first establish an operator inequality involving A:B and then apply it to obtain some norm upper bounds for the perturbations of A:B. Our results lead to considerable improvements of the well-known inequalities due to Anderson and Duffin (1969) [2]. Examples are also presented to illustrate the sharpness of the obtained norm upper bounds.

Проектный анализ подкрепленной композитной оболочки

The paper formulates the design calculation problem of a reinforced composite shell as a related problem of two shells — the layered and the lattice (ribbed). The layered shell is formed by a single double-helical monolayer; the lattice shell is made up of longitudinal and transverse ribs. The reinforced shell continuum model is based on a stiffness matrix equal to the sum of stiffness matrices of the quasi-homogeneous layered and ribbed shells, which are rigidly connected to each other and are deformed without slipping. External axial load between the two shells is distributed under the static conditions. The shells’ thickness is determined from the strength condition. The layered skin winding angle is found through iterations in achieving the goal function, for example, the minimum mass of a structure. The ribs transverse dimensions are found from the strength condition with preliminary setting of one of the ribs’ parameters. The obtained relationships make it possible to determine the stress-strain state of reinforced composite cylindrical shells.