Maximum Eigenvalue Research Articles

Large deviations and phase transitions in spectral linear statistics of Gaussian random matrices

We evaluate, in the large-N limit, the complete probability distribution P(A,m) of the values A of the sum ∑i=1N|λi|m , where λ i ( i=1,2,…,N ) are the eigenvalues of a Gaussian random matrix, and m is a positive real number. Combining the Coulomb gas method with numerical simulations using a matrix variant of the Wang–Landau algorithm, we found that, in the limit of N→∞ , the rate function of P(A,m) exhibits phase transitions of different characters. The phase diagram of the system on the (A, m) plane is surprisingly rich, as it includes three regions: (i) a region with a single-interval support of the optimal spectrum of eigenvalues, (ii) a region emerging for m < 2 where the optimal spectrum splits into two separate intervals, and (iii) a region emerging for m > 2 where the maximum or minimum eigenvalue ‘evaporates’ from the rest of eigenvalues and dominates the statistics of A. The phase transition between regions (i) and (iii) is of second order. Analytical arguments and numerical simulations strongly suggest that the phase transition between regions (i) and (ii) is of (in general) fractional order p=1+1/|m−1| , where 0<m<2 . The transition becomes of infinite order in the special case of m = 1, where we provide a more complete analytical and numerical description. Remarkably, the transition between regions (i) and (ii) for m⩽1 and the transition between regions (i) and (iii) for m > 2 occur at the ground state of the Coulomb gas which corresponds to the Wigner’s semicircular distribution.

Bounds for the maximum eigenvalues of the Fibonacci-Frank and Lucas-Frank matrices

Frank matrix is one of the popular test matrices for eigenvalue routines because it has well-conditioned and poorly conditioned eigenvalues. In this paper, we investigate the bounds for the maximum eigenvalues of the special cases of the generalized Frank matrices which are called Fibonacci-Frank and Lucas-Frank matrices. Then, we obtain the Euclidean norms and the upper bounds for the spectral norms of these matrices.

On the spectra and spectral radii of token graphs

Let G be a graph on n vertices. The k-token graph (or symmetric k-th power) of G, denoted by Fk(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_k(G)$$\\end{document}, has as vertices the nk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${n\\atopwithdelims ()k}$$\\end{document}k-subsets of vertices from G, and two vertices are adjacent when their symmetric difference is a pair of adjacent vertices in G. In particular, Fk(Kn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_k(K_n)$$\\end{document} is the Johnson graph J(n, k), which is a distance-regular graph used in coding theory. In this paper, we present some results concerning the (adjacency and Laplacian) spectrum of Fk(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_k(G)$$\\end{document} in terms of the spectrum of G. For instance, when G is walk-regular, an exact value for the spectral radius ρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho $$\\end{document} (or maximum eigenvalue) of Fk(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_k(G)$$\\end{document} is obtained. When G is distance-regular, other eigenvalues of its 2-token graph are derived using the theory of equitable partitions. A generalization of Aldous’ spectral gap conjecture (which is now a theorem) is proposed.

Complete signed graphs with largest maximum or smallest minimum eigenvalue

In this paper, we deal with extremal eigenvalues of the adjacency matrices of complete signed graphs. The complete signed graphs with maximal index (i.e. the largest eigenvalue) with n vertices and m≤⌊n2/4⌋ negative edges have been already determined. We address the remaining case by characterizing those with m>⌊n2/4⌋ negative edges. We also identify the unique signed graph with maximal index among complete signed graphs whose negative edges induce a tree of diameter at least d for any given d. This extends a recent result by Li, Lin, and Meng [Discrete Math. 346 (2023), 113250] who established the same result for d=2. Finally, we prove that the smallest minimum eigenvalue of complete signed graphs with n vertices whose negative edges induce a tree is −n2−1−1+O(1n).

Crafting a Digital Divide Index for Assessing Agricultural Disparities Among Farmers: An Application of AHP Process

a new measurement tool for assessing the digital divide disparity among farmers in e agriculture was devised during 2023-24. The index comprises four dimensions, namely ICT ownership, awareness, accessibility, and usage, respectively. For dimension selection, the researchers employed the Analytic Hierarchy Process (AHP). The maximum eigen value (ë max) of the pairwise comparison matrix was 4.0049, serving as the overall weight for the decision-making process. In the item selection process, Likert-type scale procedures were used to assess statements for final indicator selection. An interview schedule with 45 indicators was administered to 60 farmers in a non-sampling area, with five ambiguous indicators subsequently removed. The digital divide index customized for farmers demonstrated strong reliability, with a split-half reliability coefficient of 0.846 (Spearman-Brown formula) and a Pearson correlation coefficient of 0.733. Internal consistency was good, as indicated by a Cronbach’s Alpha of 0.813, and the test displayed strong validity with a score of 0.919. Overall, these findings confirm the reliability and validity of the test in measuring the intended construct. Ultimately, the 38 items were consolidated into a standardized farmers’ digital divide index for assessing agricultural disparities in the farming community.

Overcoming Catastrophic Forgetting in Continual Learning by Exploring Eigenvalues of Hessian Matrix.

Neural networks tend to suffer performance deterioration on previous tasks when they are applied to multiple tasks sequentially without access to previous data. The problem is commonly known as catastrophic forgetting, a significant challenge in continual learning (CL). To overcome the catastrophic forgetting, regularization-based CL methods construct a regularization-based term, which can be considered as the approximation loss function of previous tasks, to penalize the update of parameters. However, the rigorous theoretical analysis of regularization-based methods is limited. Therefore, we theoretically analyze the forgetting and the convergence properties of regularization-based methods. The theoretical results demonstrate that the upper bound of the forgetting has a relationship with the maximum eigenvalue of the Hessian matrix. Hence, to decrease the upper bound of the forgetting, we propose eiGenvalues ExplorAtion Regularization-based (GEAR) method, which explores the geometric properties of the approximation loss of prior tasks regarding the maximum eigenvalue. Extensive experimental results demonstrate that our method mitigates catastrophic forgetting and outperforms existing regularization-based methods.

Smoothing algorithm for the maximal eigenvalue of non-defective positive matrices

&lt;abstract&gt;&lt;p&gt;This paper introduced a smoothing algorithm for calculating the maximal eigenvalue of non-defective positive matrices. Two special matrices were constructed to provide monotonically increasing lower-bound estimates and monotonically decreasing upper-bound estimates of the maximal eigenvalue. The monotonicity and convergence of these estimations was also proven. Finally, the effectiveness of the algorithm was demonstrated with numerical examples.&lt;/p&gt;&lt;/abstract&gt;

Distributed platooning control of automated vehicles subject to replay attacks based on proportional integral observers

Secure platooning control plays an important role in enhancing the cooperative driving safety of automated vehicles subject to various security vulnerabilities. This paper focuses on the distributed secure control issue of automated vehicles affected by replay attacks. A proportional-integral-observer (PIO) with predetermined forgetting parameters is first constructed to acquire the dynamical information of vehicles. Then, a time-varying parameter and two positive scalars are employed to describe the temporal behavior of replay attacks. In light of such a scheme and the common properties of Laplace matrices, the closed-loop system with PIO-based controllers is transformed into a switched and time-delayed one. Furthermore, some sufficient conditions are derived to achieve the desired platooning performance by the view of the Lyapunov stability theory. The controller gains are analytically determined by resorting to the solution of certain matrix inequalities only dependent on maximum and minimum eigenvalues of communication topologies. Finally, a simulation example is provided to illustrate the effectiveness of the proposed control strategy.

Trial-by-trial Variability of TMS-EEG in Healthy Controls and Patients with Depressive Disorder.

Depressive disorder has been known to be associated with high variability in resting-state electroencephalography (EEG) signals. However, this phenomenon is often ignored in stimulus-related brain activities. This study proposed a new method to explore the EEG variability evoked by transcranial magnetic stimulation (TMS, TMS-EEG) in depressive disorder (DE) patients. The TMS-EEG data were collected from 34 DE patients and 36 healthy controls (HC). The maximum eigenvalue of the real binary correlation matrix, calculated between different trials using cross-correlation and surrogate methods, was extracted to assess trial-by-trial variability (TTV) of TMS-EEG. The new method was found to more sensitive and reliable than the standard deviation method. DE patients exhibited significantly smaller TTV in Gamma band and greater TTV in Delta band than HC. Furthermore, the HAMD-17 scores were negatively correlated with TTV values in Gamma band. This study represented the first investigation into the TTV in TMS-EEG data and revealed abnormal values in DE patients. Those findings enhance our understanding of TMS-EEG technology and provide valuable insights for studying the characteristics of DE.

Weakly-Supervised Contrastive Learning for Unsupervised Object Discovery.

Unsupervised object discovery (UOD) refers to the task of discriminating the whole region of objects from the background within a scene without relying on labeled datasets, which benefits the task of bounding-box-level localization and pixel-level segmentation. This task is promising due to its ability to discover objects in a generic manner. We roughly categorize existing techniques into two main directions, namely the generative solutions based on image resynthesis, and the clustering methods based on self-supervised models. We have observed that the former heavily relies on the quality of image reconstruction, while the latter shows limitations in effectively modeling semantic correlations. To directly target at object discovery, we focus on the latter approach and propose a novel solution by incorporating weakly-supervised contrastive learning (WCL) to enhance semantic information exploration. We design a semantic-guided self-supervised learning model to extract high-level semantic features from images, which is achieved by fine-tuning the feature encoder of a self-supervised model, namely DINO, via WCL. Subsequently, we introduce Principal Component Analysis (PCA) to localize object regions. The principal projection direction, corresponding to the maximal eigenvalue, serves as an indicator of the object region(s). Extensive experiments on benchmark unsupervised object discovery datasets demonstrate the effectiveness of our proposed solution. The source code and experimental results are publicly available via our project page at https://github.com/npucvr/WSCUOD.git.

NOVEL APPROACH FOR EIGENVALUE PROBLEMS USING THE MONTE CARLO METHOD

In this paper, we have studied various types of Monte Carlo methods along with the Power method to evaluate the maximum and minimum eigenvalue of a linear system of equations. We have studied how the accuracy of the maximum eigenvalue depends on the parameters, l (moves in Markova chain), ℵ (no of Markova chain), ℘ (accelerating parameter), and a parameter m (the power applied on the resolving matrix). We have applied these methods to the randomly chosen symmetric matrices. We have also made comparisons for the different matrices of different orders depending on the parameters by using the Monte Carlo methods. We are using Matlab 2020R for the calculation.

Examining the Causality between Financial Instruments in Terms of Portfolio Management

In portfolio theory, the selection of financial instruments to be included in the portfolio is of great importance. For this purpose, the direction and strength of the relationship between the financial instruments to be included in the portfolio have also become important. From this point of view, the main objective of this study is to analyze the interrelationships among financial instruments. In this study, the causality relationship between gold, BIST-100 index, bitcoin and exchange rate was analyzed. The data of the financial instruments used in the study consist of 208 observations as weekly values for the years 2019-2022. Before applying causality analysis, Johensen cointegration test was performed to test whether there is a cointegration relationship between the variables. According to the results of the cointegration test, both the trace statistic and the maximum eigenvalue result indicate that there is one cointegration between the variables. According to the Toda-Yamamoto causality test, it was concluded that there is a bidirectional causality relationship between the exchange rate and bitcoin price and that the exchange rate is the cause of the BIST-100 index.

Mean-square output consensus for heterogeneous multi-agent systems over nonidentical packet loss channels

In this paper, we study the cooperative output consensus problem of discrete-time heterogeneous multi-agent systems over nonidentical channel fading networks. Different from the traditional output consensus problem, the communication channels among agents are perturbed by stochastic multiplicative noises, which can also describe the packet loss phenomenon. The perturbed channels can result in a significant challenge for consensus analysis since their effect cannot be separated linearly from the communication topology. Based on the distributed feedforward control method, a novel distributed dynamic output feedback controller is developed to ensure output synchronization for each agent in the mean-square sense. To facilitate the consensus analysis, we utilize the relationship between the graph Laplacian matrix and the incidence matrix and matrix transformation technique, and find sufficient solvability conditions relying on each agent’s dynamics, the maximum eigenvalue of communication graph and the packet loss probability for the scenarios of undirected graph and directed graph, respectively. Finally, simulation results are given to verify the theoretical analysis.

Cultivation of students’ craftsmanship identity and ability based on the theory of directed percolation

Abstract In this paper, data mining is performed based on the directed percolation theory influence mining algorithm, analyzing the maximum influence node mining proportion threshold and calculating the maximum eigenvalue of the non-return matrix. The convergence conditions of the iterative process of immobile points are analyzed to obtain the percolation threshold of the information dissemination process of directed percolation, and the initial vector is selected among the node-associated features to form the iterative formula to mine the maximum percolation theoretical influence. According to the results, students' focus on craftsmanship has risen significantly, and the percentage of students who consider persistent concentration spirit important has increased to 75%. The directed percolation theory's craftsmanship spirit is crucial for developing students' vocational skills and comprehensive literacy.

Relaxation strategy for the block Landweber method

The block Landweber iteration is a general method for the solution of linear systems which is widely applied for image reconstructions. The convergence behavior of the block Landweber iteration is of both theoretical and practical importance. When the linear system is consistent, we derive a neat iterative representation and find an optimal relaxation coefficient by minimizing the 2-norm of the newly derived iteration matrix in the range of the conjugate transpose of the corresponding block matrix. We also establish the accelerating convergence relaxation strategies based on the maximum eigenvalue of each block matrix. When the linear system is inconsistent, we normalize each block matrix by the 2-norms of the each weighted block matrix at a uniform scale to give the relaxation strategies. The numerical simulations show that the relaxation strategies proposed for the block Landweber iteration in both consistent and inconsistent cases in this paper have the better reconstruction performances. Compared with the Landweber iteration, the block Landweber iteration has superiority in reconstruction results.

Free Metaplectic Wigner Distribution: Definition and Heisenberg’s Uncertainty Principles

Inspired by a definition of the closed-form instantaneous cross-correlation Wigner distribution (Zhang in IEEE Trans Signal Process 67(21):5527–5542, 2019), we generalize the notion of Wigner distribution to the so-called free metaplectic Wigner distribution (FMWD) through three free metaplectic transforms, in order to tackle a challenge in high-dimensional complex features information processing. We provide some representative special cases for this general form, including the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> -dimensional nonseparable affine characteristic Wigner distribution, kernel function Wigner distribution, convolution representation Wigner distribution and instantaneous cross-correlation Wigner distribution. We establish the standard Heisenberg’s uncertainty principles (HUPs) of the real-valued function for the FMWD. We also establish the standard HUPs of the complex-valued function for some specific (i.e., the orthogonal, the orthonormal, the minimum eigenvalue commutative and the maximum eigenvalue commutative) FMWDs. In view of the one-dimensional case of our results, we solve a burning question regarding the limit of time-frequency superresolution triggered by the linear canonical transform free parameters.

Sensorless Control Design of High-Speed Electric Drives in Discrete-Time Domain for Mild-Hybrid Turboprop Aircraft Applications

High-speed Permanent Magnet Synchronous Machine (PMSM) sensorless control design is not without challenges, especially when the ratio of power electronic modulation frequency to electrical fundamental frequency (M-F ratio) is relatively low. The Model Reference Adaptive System (MRAS) is a mature sensorless control method and has been widely used in industrial drives. There, an observer is commonly used to estimate rotor speed and angle using machine voltages and currents. The estimated angle is then further implemented for voltage and current frame transformations. Thus, the MRAS sensorless control is indeed a strong-coupled, multi-variable and nonlinear system. This paper proposes a detailed MRAS sensorless control system design method in discrete-time domain for high speed drives with relatively low M-F ratio with good system stability confidence. A detailed discrete local-linearized small-signal model is developed and state-space evolution matrix of current closed-loop system has been derived. The system stability can be defined and analysed with its maximum eigenvalue and the controller design is to make sure that the maximum eigenvalue is falling within the stable region. This proposed design method has been validated on a dual three-phase PMSM by simulation and experiment results with selected stable design point.

Consensus of fractional-order multi-agent systems with sampled position data

In this paper, the consensus of fractional-order multi-agent systems with the order α satisfying α∈(0,1] is investigated. A novel distributed control protocol is constructed, in which only the past sampled position data of neighbors are utilized. Then, based on Laplace transform, Mittag-Leffler function, and matrix theory, necessary and sufficient criteria depending on the order, coupling gains, sampling period, and communication topology, are established to attain consensus for the systems. Meanwhile, the intervals of coupling gains and sampling period associated with the maximum eigenvalue of Laplacian matrix over an undirected graph are presented to attain consensus. Finally, the availability of theoretical results is verified by numerical simulations.

Recursive ensemble canonical variate analysis for online incipient fault detection in dynamic processes

Data-driven fault detection has made significant advancements. However, detecting incipient faults is still a challenging problem for traditional data-driven methods, because it has low amplitude and is easily masked by measurement noise. In this paper, we propose an approach for detecting incipient faults named recursive ensemble canonical variate analysis (RECVA) for dynamic processes. First, inspired by ensemble learning, multiple CVA-based incipient fault models are obtained based on sampling from the training data set using bootstrap. It generates two matrices based on the principal component and the residual subspace. Then, two sensitive detection metrics are created using the maximum eigenvalue of a one-step sliding window over each row of the two aforementioned matrices. Based on the first-order perturbation (FOP) theory, the eigenvalue can be recursively updated. Utilizing the two detection metrics, RECVA can effectively detect incipient faults, which cannot be detected by conventional statistics. The effectiveness of RECVA is fully validated with the incipient fault data collected from the continuous stirred tank heater (CSTH) and the Tennessee Eastman process (TEP) simulation benchmark. Compared with traditional PCA, CVA, and other incipient fault detection methods based on divergence, simulation demonstrated that incipient faults can be effectively detected at an early stage, which is very important for avoiding serious failures in the industrial processes, resulting in economic losses and casualties.

Modeling and small-signal stability analysis of doubly-fed induction generator integrated system

Owing to their stability, doubly-fed induction generator (DFIG) integrated systems have gained considerable interest and are the most widely implemented type of wind turbines and due to the increasing escalation of the wind generation penetration rate in power systems. In this study, we investigate a DFIG integrated system comprising four modules: (1) a wind turbine that considers the maximum power point tracking and pitch-angle control, (2) induction generator, (3) rotor/ grid-side converter with the corresponding control strategy, and (4) AC power grid. The detailed small-signal modeling of the entire system is performed by linearizing the dynamic characteristic equation at the steady-state value. Furthermore, a dichotomy method is proposed based on the maximum eigenvalue real part function to obtain the critical value of the parameters. Root-locus analysis is employed to analyze the impact of changes in the phase-locked loop, short-circuit ratio, and blade inertia on the system stability. Lastly, the accuracy of the small-signal model and the real and imaginary parts of the calculated dominant poles in the theoretical analysis are verified using PSCAD/EMTDC.