Eigenvectors Of Matrix Research Articles

Event-triggered distributed Nash equilibrium seeking over directed graphs and its application to power management

In this paper, the problem of distributed discrete-time Nash equilibrium computation with event-triggered communication over directed graphs is studied. Each agent can only receive strategies from its neighbors through a directed and unbalanced graph, while its cost function may rely on the information of all agents. To this end, each agent needs to estimate the strategy profile. Meanwhile, an event-triggered communication mechanism is introduced, which can effectively compensate for network resource loss. Concretely, specific moments that each agent transmits its information are decided by the pre-designed event-triggered condition. In this setting, two discrete-time Nash equilibrium seeking algorithms with event-triggered communication are designed, respectively considering that the Perron-Frobenius eigenvector of the adjacency matrix associated the directed communication graph is known and unknown in advance. It is rigorously proved that both algorithms can linearly converge to the Nash equilibrium with a small error. Finally, a power management example in smart grids is presented to verify the performance of proposed algorithms.

Airpak-based art and design talent training innovation model framework

Abstract In this paper, the 15 original variables of the Airpak-based influence factor scale for art and design talent training were first determined, and the five-level scoring method was applied to score the 15 influence factors. Then, we used factor analysis to factorize the data, standardize the sample data, construct a coordinate system with common factors, and use the projection of each variable in the coordinate system instead of the original variables to construct the correlation coefficient matrix R. Then, we solved the correlation coefficient matrix between each variable, selected the number of common factors, and calculated the eigenvectors of the factor loading matrix A to construct the influence factor analysis model. Finally, the model is used to analyze the weight of each influencing factor of Airpak-based art and design talent cultivation and to build an innovative path for talent cultivation. In terms of students’ own factors, the influence weight of learning enthusiasm on talent cultivation is 0.48, and the influence weights of learning achievement and design talent are 0.43 and 0.545, respectively; in terms of teaching level factors, the influence weights of teachers’ teaching ability, professional level and design experience on talent cultivation are 0.44, 0.5 and 0.37 respectively. The research of this paper has an important reference and reference value for the cultivation of art and design talents.

Super-resolution DOA estimation methods based on element multiplication and summation

In this article, we propose two direction-of-arrival (DOA) estimation methods to further improve the resolution and performance of subspace-based algorithms. While most subspace-based algorithms employ either noise or signal subspaces, we utilize the signal and noise subspaces jointly. First, we project the steering vectors to the estimated signal and noise subspaces respectively to obtain the projection vectors, then, by investigating the numerical characteristics behind the inner products between the steering vectors and the estimated eigenvectors of the sample covariance matrix, an initial estimator is proposed based on the multiplication of elements of the projection vectors instead of the summation. Though the initial estimator achieves higher resolution than most existing subspace-based methods, the appearance of pseudo peaks, especially in high SNR, may affect the estimation accuracy. By analyzing the main elements in the initial objective function that cause pseudo peaks, two improved algorithms incorporating element summation are proposed to balance the resolution capacity and estimation accuracy, which are especially suitable for small sample size, low SNR and closely spaced sources. The numerical simulations verify the effectiveness and superiority of the proposed algorithms.

Azuma-Hoeffding bounds for a class of urn models

We obtain Azuma-Hoeffding bounds, which are exponentially decreasing, for the probabilities of being away from the limit for a class of urn models. The method consists of relating the variables to certain linear combinations using eigenvectors of the replacement matrix, thus bringing in appropriate martingales. Some cases of repeated eigenvalues are also considered using cyclic vectors. Moreover, strong convergence of proportions is proved as an application of these bounds.

An eigenvector-assisted estimation framework for signal-plus-noise matrix models

Summary In this paper, we develop an eigenvector-assisted estimation framework for a collection of signal-plus-noise matrix models arising in high-dimensional statistics and many applications. The framework is built upon a novel asymptotically unbiased estimating equation using the leading eigenvectors of the data matrix. However, the estimator obtained by directly solving the estimating equation could be numerically unstable in practice and lacks robustness against model misspecification. We propose to use the quasi-posterior distribution by exponentiating a criterion function whose maximizer coincides with the estimating equation estimator. The proposed framework can incorporate heteroskedastic variance information, but does not require the complete specification of the sampling distribution and is also robust to the potential misspecification of the distribution of the noise matrix. Computationally, the quasi-posterior distribution can be obtained via a Markov chain Monte Carlo sampler, which exhibits superior numerical stability over some of the existing optimization-based estimators and is straightforward for uncertainty quantification. Under mild regularity conditions, we establish the large sample properties of the quasi-posterior distributions. In particular, the quasi-posterior credible sets have the correct frequentist nominal coverage probability provided that the criterion function is carefully selected. The validity and usefulness of the proposed framework are demonstrated through the analysis of synthetic datasets and the real-world ENZYMES network datasets.

A Novel Method Based on GPU for Real-Time Anomaly Detection in Airborne Push-Broom Hyperspectral Sensors

The airborne hyperspectral remote sensing systems (AHRSSs) acquire images with high spectral resolution, high spatial resolution, and high temporal dimension. While the AHRSS captures more detailed information from the terrain objects, the computational complexity of data processing is greatly increased. As an important application technology in the hyperspectral domain, anomaly detection (AD) processing must be real-time and high-precision in many cases, such as post-disaster rescue, military battlefield search, and natural disaster detection. In this paper, the real-time AD technology for the push-broom AHRSS is studied, the mathematical model is established, and a novel implementation framework is proposed. Firstly, the optimized kernel minimum noise fraction (OP-KMNF) transformation is employed to extract informative and discriminative features between the background and anomalies. Secondly, the Nyström method is introduced to reduce the computational complexity of OP-KMNF transformation by decomposing and extrapolating the sub-kernel matrix to estimate the eigenvector of the entire kernel matrix. Thirdly, the extracted features are transferred to hard disks for data storage. Then, taking the extracted features as input data, the background separation model-based CEM anomaly detector (BSM-CEMAD) is imported to detect anomalies. Finally, graphics processing unit (GPU) parallel computing is utilized in the Nyström-based OP-KMNF (NOP-KMNF) transformation and the BSM-CEMAD to improve the execution efficiency, and the real-time AD for the push-broom AHRSS could be realized. To test the feasibility of the implementation framework proposed in this paper, the experiment is carried out with the Airborne Multi-Modular Imaging Spectrometer (AMMIS) developed by the Shanghai Institute of Technical Physics as the data acquisition platform. The experimental results show that the proposed method outperforms many other state-of-the-art AD methods in anomalies detection and background suppression. Moreover, under the condition that the downlink data could retain most of the hyperspectral data information, the proposed method achieves real-time detection of pixel-level anomalies, with the initial delay not exceeding 1 s, the false alarm rate (FAR) less than 5%, and the true positive rate (TPR) close to 98%.

On testing the equality of latent roots of scatter matrices under ellipticity

In the present paper, we tackle the problem of testing H0q:λq>λq+1=⋯=λp, where λ1,…,λp are the scatter matrix eigenvalues of an elliptical distribution on Rp. This is a classical problem in multivariate analysis which is very useful in dimension reduction. We analyse the problem using the Le Cam asymptotic theory of experiments and show that contrary to the testing problems on eigenvalues and eigenvectors of a scatter matrix tackled in Hallin et al. (2010), the non-specification of nuisance parameters has an asymptotic cost for testing H0q. We moreover derive signed-rank tests for the problem that enjoy the property of being asymptotically distribution-free under ellipticity. The van der Waerden rank test uniformly dominates the classical pseudo-Gaussian procedure for the problem. Numerical illustrations show the nice finite-sample properties of our tests.

Spectral domain graph convolutional deep neural network for predicting unsteady and nonlinear flows

Mode decomposition methods, such as proper orthogonal decomposition and dynamic mode decomposition (DMD), have introduced a novel data-driven approach for flow prediction. These methods aim to identify a collection of modes that capture the essential flow features. Subsequently, the flow field data are projected onto these modes to reconstruct and predict the evolution of the flow field. However, due to their inherent linearity, mode decomposition methods are limited in effectively handling unsteady and nonlinear flow exhibiting significant nonlinearities. In this study, we propose a spectral graph convolutional deep neural network (SGC-DNN). It employs the eigenvectors of the Laplacian matrix as modes to fully utilize the adjacency information within the graph structure to solve flow on an unstructured grid better. Additionally, we employ a DNN (deep neural network) to model the temporal evolution of each mode, thereby enhancing the model's adaptability to nonlinear flow fields. To evaluate the performance of our proposed SGC-DNN, we compare its prediction results with those obtained using DMD and DNN for the flow around a cylinder on unstructured grids at various Reynolds numbers (ranging from 1000 to 500 000). We also compared the predictive results of these three models for flow with complex geometries, such as the Da Vinci pipeline flow and intracranial aneurysm blood flow. The comparative analysis demonstrates that SGC-DNN outperformed the other models, yielding lower L2 relative errors and higher R2 values. These outcomes highlight the superiority of SGC-DNN in accurately predicting unsteady and nonlinear flow characterized by graph structures.

A weighted local‐spherical grid based lightweight descriptor for 3D point cloud registration

AbstractEstablishing effective feature descriptors is a crucial step in 3D point cloud registration task. Existing manual‐based methods are noise‐susceptible and time‐consuming when running on low‐cost edge computing devices. To this end, the authors proposed a Local Reference Frame (LRF) based approach that can quickly and robustly register point clouds by using a novel lightweight local‐spherical grid weighted descriptor (LSGWD). Firstly, the LRF of the proposed algorithm is established by the covariance matrix eigenvector of KeyPoint's spherical support set and the centroid vector's projection on its orthogonal plane. Then the spherical support is grided to 32 bins, and the 4D geometric features of each subset are constructed by the centroid moment and the cosine value of the angles between the centroid vector and axes of LRF. Secondly, to restrain the insufficient discriminative information presented in the purely geometric features, the Gaussian projection and gradient mapping are proposed to calculate the smooth density and the correlation of structural characteristics, which are obtained as the distribution information of each bin to weigh the feature representation. Finally, the 32 × 4‐dimensional KeyPoint descriptor is obtained and used in the 3D point cloud registration framework. Experiments are carried out on three test datasets and real scene data. Compared to previous baselines, our descriptor achieves the state‐of‐the‐art performance in terms of efficiency and accuracy owing to its compact structure and noise robustness. The proposed method enhances the recognition and registration performance of 3D point cloud matching in low‐cost edge computing applications.

A new spectral representation of the strain energy function for linear anisotropic elasticity with a generalization for damage

A new spectral representation of the strain energy function for linear anisotropic elasticity is proposed in the form of a sum of 6 scalar eigen-stiffnesses times the squares of their associated scalar eigen-strains. Since this new representation is general, it is no less or more general than the standard representation. However, the spectral form reveals fundamental coupled eigen-modes of deformation and energy associated with general anisotropic response. Specifically, each eigen-strain is the inner product of the strain tensor with its symmetric eigen-tensor. The 6 eigen-strains are linearly independent and the 6 eigen-tensors are also linearly independent. Moreover, unlike the eigenvectors of a symmetric matrix, distinct pairs of eigen-tensors are not all orthogonal. The 15 constants that define the eigen-tensors together with the 6 eigen-stiffnesses form 21 independent material constants that characterize the fundamental physics of material anisotropy. These 21 constants are invariant to the orientation of the base vectors fixed in the material. This spectral representation also simplifies the restrictions on these material constants which ensure that the strain energy is a positive-definite function of strain. In addition, a simple damage model is proposed based on this spectral representation and a generalization for finite deformations is briefly discussed.

Principal force pattern and impulse response mode for structural equivalent force estimation and full-field response reconstruction

An impulse response mode-based approach is proposed in this study to estimate the equivalent force and achieve full-field monitoring under complex loading conditions. The structural impulse responses to quasi-static and dynamic forces are decomposed as the superposition of a set of predefined representative basis to provide the low-rank property of responses. The principal impulse response modes are calculated from the correlation matrix of the impulse responses in a time step using the principal component analysis. The principal relative force pattern is the eigenvector of the correlation matrix, and the truncation number is determined by the distribution of the eigenvalues. The equivalent force is subsequently defined and solved by the dynamic stiffness method. The full-field responses are then reconstructed by calculating the model responses to the estimated equivalent force. The systematic uncertainty is also quantified by an error propagation method for sensor placement optimization. Numerical examples and experimental tests demonstrate that the proposed methodology can estimate the equivalent force and reconstruct the structural responses accurately under various load scenarios.

Spectral Clustering Approach with K-Nearest Neighbor and Weighted Mahalanobis Distance for Data Mining

This paper proposes a spectral clustering method using k-means and weighted Mahalanobis distance (Referred to as MDLSC) to enhance the degree of correlation between data points and improve the clustering accuracy of Laplacian matrix eigenvectors. First, we used the correlation coefficient as the weight of the Mahalanobis distance to calculate the weighted Mahalanobis distance between any two data points and constructed the weighted Mahalanobis distance matrix of the data set; then, based on the weighted Mahalanobis distance matrix, we used the K-nearest neighborhood (KNN) algorithm construct similarity matrix. Secondly, the regularized Laplacian matrix was calculated according to the similarity matrix, normalized and decomposed, and the feature space for clustering was obtained. This method fully considered the degree of linear correlation between data and special spatial structure and achieved accurate clustering. Finally, various spectral clustering algorithms were used to conduct multi-angle comparative experiments on artificial and UCI data sets. The experimental results show that MDLSC has certain advantages in each clustering index and the clustering quality is better. The distribution results of the eigenvectors also show that the similarity matrix calculated by MDLSC is more reasonable, and the calculation of the eigenvectors of the Laplacian matrix maximizes the retention of the distribution characteristics of the original data, thereby improving the accuracy of the clustering algorithm.

Image segmentation by selecting eigenvectors based on extended information entropy

AbstractFor spectral clustering algorithm, the quality of eigenvectors of graph affinity matrix is very important for the clustering result. So, how to obtain high‐quality eigenvectors is crucial. In this paper, the authors aim to propose some new measurement methods to evaluate each eigenvector of affinity matrix for spectral selection. Based on extended information entropy, three criteria, i.e. Spectral Distinguishability (SD), Spectral Distinguishability Validity (SDV) and pectral Distinguishability ‐Degree (SDD), are defined respectively. The compactness of clusters for each eigenvector is measured by SD; SDV is used to remove the inefficient eigenvectors for clustering; SDD is used to evaluate the contribution of eigenvectors to clustering and is exploited to build a selective spectral ensemble scheme. To indicate the merits of the authors’ algorithm, the authors consider varied artificial data and natural images, including Berkeley image segmentation data set as benchmark data set. The authors’ simulation results confirm the superior performance of the proposed method in developing spectral clustering compared to conventional clustering methods and recent eigenvectors‐selection‐based algorithms.

Classification of Emotion using Eeg Signals: an FPGA Based Implementation

An electroencephalograph is a device that records all electrical energy in the human brain using wearable metal electrodes placed on the skull. Electrical impulses connect brain cells and are always mobile, even at rest. This activity appears as a squiggly line in EEG recordings. Activity gaze data is pre-processed to a frequency range of 0 to 75 Hz. This creates a new matrix with a sample rate of 200 Hz and a range of 0-75 Hz. A finite-impulse-response low-pass filter was used because the bandpass would distort his EEG data after processing. Each pre-processed EEG signal has an output, which completes feature extraction. Principal Component Analysis or PCA is passed in the feature reduction phase. PCA is an analytical process that uses singular value decomposition to transform a collection of corresponding features into mutually uncorrelated features or principal components. Principal component analysis: (a) mean normalization of features (b) covariance matrix (c) eigenvectors (d) reduced features or principal components. The above steps are passed to the SVM classifier for sentiment output. His VHDL code and testbench for 2*2 matrices were written, waveforms and RTL schemes were created in Xilinx 14.5. For the FPGA implementation, a Simulink model was designed, and the eigenvalues were pre-determined using a system generator.

MeT: A graph transformer for semantic segmentation of 3D meshes

Polygonal meshes have become the standard for discretely approximating 3D shapes, thanks to their efficiency and high flexibility in capturing non-uniform shapes. This non-uniformity, however, leads to irregularity in the mesh structure, making tasks like segmentation of 3D meshes particularly challenging. Semantic segmentation of 3D mesh has been typically addressed through CNN-based approaches, leading to good accuracy. Recently, transformers have gained enough momentum both in NLP and computer vision fields, achieving performance at least on par with CNN models, supporting the long-sought architecture universalism. Following this trend, we propose a transformer-based method for semantic segmentation of 3D mesh motivated by a better modeling of the graph structure of meshes, by means of global attention mechanisms. In order to address the limitations of standard transformer architectures in modeling relative positions of non-sequential data, as in the case of 3D meshes, as well as in capturing the local context, we perform positional encoding by means the Laplacian eigenvectors of the adjacency matrix, replacing the traditional sinusoidal positional encodings, and by introducing clustering-based features into the self-attention and cross-attention operators. Experimental results, carried out on three sets of the Shape COSEG Dataset (Wang et al., 2012), on the human segmentation dataset proposed in Maron et al. (2017) and on the ShapeNet benchmark (Chang et al., 2015), show how the proposed approach yields state-of-the-art performance on semantic segmentation of 3D meshes.

The two-steps eigenvector centrality in complex networks

Eigenvector centrality refers to the principal eigenvector of the adjacency matrix of a graph. The adjacency matrix can be regarded as the matrix describing the one-step walks on the graph, shortly, the one-walk matrix, and then eigenvector centrality can refer to the principal eigenvector of the one-walk matrix. In this paper, we consider the effect of combining the first and second neighbors of a given node in a network. Analogously to the one-walk matrix, we construct a three-order tensor, namely the two-steps tensor, to represent the first and second neighbor relation among nodes. We adopt the positive eigenvector of the two-steps tensor corresponding to its spectral radius as a new centrality measure. This new centrality measure is referred to as the two-steps eigenvector centrality, which extends the notion of eigenvector centrality. Experimental results show that the new centrality measure is effective, and in certain networks, it gives an additional insight regarding node importance.

Predicting attractors from spectral properties of stylized gene regulatory networks.

How the architecture of gene regulatory networks shapes gene expression patterns is an open question, which has been approached from a multitude of angles. The dominant strategy has been to identify nonrandom features in these networks and then argue for the function of these features using mechanistic modeling. Here we establish the foundation of an alternative approach by studying the correlation of network eigenvectors with synthetic gene expression data simulated with a basic and popular model of gene expression dynamics: Boolean threshold dynamics in signed directed graphs. We show that eigenvectors of the network adjacency matrix can predict collective states (attractors). However, the overall predictive power depends on details of the network architecture, namely the fraction of positive 3-cycles, in a predictable fashion. Our results are a set of statistical observations, providing a systematic step towards a further theoretical understanding of the role of network eigenvectors in dynamics on graphs.

ADSVAE: An Adaptive Density-aware Spectral Clustering Method for Multi-omics Data Based on Variational Autoencoder

Introduction: The discovery of tumor subtypes helps to explore tumor pathogenesis, determine the operability of clinical treatment, and improve patient survival. Clustering analysis is increasingly applied to multi-genomic data. However, due to the diversity and complexity of multi-omics data, developing a complete clustering algorithm for tumor molecular typing is still challenging. Methods: In this study, we present an adaptive density-aware spectral clustering method based on a variational autoencoder (ADSVAE). ADSVAE first learns the underlying spatial information of each omics data using a variational autoencoder (VAE) based on the Wasserstein distance metric. Secondly, a similarity matrix is built for each gene set using an adaptive density-aware kernel. Thirdly, tensor product graphs (TPGs) are used to merge different data sources and reduce noise. Finally, ADSVAE employs a spectral clustering algorithm and utilizes the Gaussian mixture model (GMM) to cluster the final eigenvector matrix to identify cancer subtypes. Results: We tested ADSVAE on 5 TCGA datasets, all with good performance in comparison with several advanced multi-omics clustering algorithms. Compared with the existing multi-group clustering algorithms, the variational autoencoder based on the Wasserstein distance measure in the ADSVAE algorithm can learn the underlying spatial information on each omics data, which has a better effect on learning complex data distribution. The self-tuning density-aware kernel used by the ADSVAE algorithm enhances the similarity between shared near neighbor points, and the process of tensor product plot data integration and diffusion can better reduce the noise and reveal the underlying structure, improving the performance. Conclusion: Due to the inherent pitfalls of computational biology in the study of cancer subtype identification, although some research conclusions have been made in this paper on the related issues, as the research in related fields continues to deepen, the clustering study of cancer subtype identification based on genomic data needs further improvement and refinement.

Sequenced Steering Vector Estimation for Eigen-Subspace Projection-Based Robust Adaptive Beamformer

Robust adaptive beamforming (RAB) is essential in many applications to ensure signal-receiving quality when model errors exist. Eigen-subspace projection (ESP), one of the most popular RAB methods, can be used when there are arbitrary model errors. However, a major challenge of ESP is projection subspace selection. Traditional ESP (TESP) treats the signal subspace as the projection subspace; thus, source enumeration is required to obtain prior information. Another inherent defect is its poor performance at low signal-to-noise ratios (SNRs). To overcome these drawbacks, two improved ESP-based RAB methods are proposed in this study. Considering that a reliable signal-of-interest steering vector needs to be obtained via the subspace projection, the main idea underlying the proposed methods is to use sequenced steering vector estimation to invert the subspace dimension estimate for an arranged eigenvector matrix. As the proposed methods do not require source enumeration, they are simple to implement. Numerical examples demonstrate the effectiveness and robustness of the proposed methods in terms of output signal-to-interference-plus-noise ratio performance. Specifically, compared with TESP, the proposed methods present at least a 2.6 dB improvement at low SNRs regardless of the error models.

Pressure analysis in nonlinear waves by revisiting the breaking wave onset

Breaking waves are numerically simulated by solving the two dimensional nonlinear free surface boundary conditions in potential theory. Onset of breaking wave is determined for two configurations: A focused wave in intermediate depth and a dambreaking flow leading to a soliton. Two criteria are identified: first, a kinematic criterion follows from the ratio of the horizontal velocity at the crest to the phase velocity; second, a dynamic criterion is formulated in terms of the vertical pressure gradient. Thresholds of these criteria are proposed in the literature. For the two configurations studied here, it is shown that these thresholds are not universal but they are reasonable estimates of the onset of breaking wave. To facilitate this study, the Hessian matrix of the pressure is analyzed. It is shown that there exists a continuous curve that links the crest of the wave to the sea bottom. This line follows from the colinearity of the pressure gradient and one of the two eigenvectors of the Hessian matrix. The analysis of the pressure, its gradient and its second derivatives along this line gives more insights into the definition of the thresholds associated to the onset of breaking wave.