Gershgorin Research Articles

Multistability of state-dependent switched neural networks with Mexican-hat-type activation functions

This paper investigates the issue of multistability in state-dependent switched neural networks (SSNNs) with Mexican-hat-type activation functions (AFs). It establishes the coexistence and stability of multiple equilibrium points (EPs). Initially, the state space is partitioned based on the geometric characteristics of the Mexican-hat-type AF, enabling to determine the positions of the EPs. Secondly, the coexistence of 9h17h25h33h4 EPs for n-neurons SSNNs under specific sufficient conditions is proved with the Brouwer’s fixed-point theorem. Next, by using diagonally dominant matrix theory and Gershgorin circle theorem, it is proven that there are 5h14h23h32h4 asymptotically stable EPs under some conditions, where h1,h2,h3 and h4 are nonnegative integers satisfying 0≤h1+h2+h3+h4≤n. Therefore, we can obtain that SSNNs can have larger storage capacity by selecting the appropriate parameters hi,i=1,2,3,4. Finally, the correctness of the results in this paper is verified through two numerical examples.

Robustness of Turing models and gene regulatory networks with a sweet spot.

Traditional linear stability analysis based on matrix diagonalization is a computationally intensive process for high-dimensional systems of differential equations, posing substantial limitations for the exploration of Turing systems of pattern formation where an additional wave-number parameter needs to be investigated. In this paper, we introduce an efficient and intuitive technique that leverages Gershgorin's theorem to determine upper limits on regions of parameter space and the wave number beyond which Turing instabilities cannot occur. This method offers a streamlined avenue for exploring the phase diagrams of other complex multi-parametric models, such as those found in gene regulatory networks in systems biology. Due to its suitability for the asymptotic limit of infinitely large systems, it predicts the existence of a sweet spot in network size for maximal Jacobian stability.

A Novel Clustering Algorithm Integrating Gershgorin Circle Theorem and Nonmaximum Suppression for Neural Spike Data Analysis

(1) Problem Statement: The development of clustering algorithms for neural recordings has significantly evolved, reaching a mature stage with predominant approaches including partitional, hierarchical, probabilistic, fuzzy logic, density-based, and learning-based clustering. Despite this evolution, there remains a need for innovative clustering algorithms that can efficiently analyze neural spike data, particularly in handling diverse and noise-contaminated neural recordings. (2) Methodology: This paper introduces a novel clustering algorithm named Gershgorin—nonmaximum suppression (G–NMS), which incorporates the principles of the Gershgorin circle theorem, and a deep learning post-processing method known as nonmaximum suppression. The performance of G–NMS was thoroughly evaluated through extensive testing on two publicly available, synthetic neural datasets. The evaluation involved five distinct groups of experiments, totaling eleven individual experiments, to compare G–NMS against six established clustering algorithms. (3) Results: The results highlight the superior performance of G–NMS in three out of five group experiments, achieving high average accuracy with minimal standard deviation (SD). Specifically, in Dataset 1, experiment S1 (various SNRs) recorded an accuracy of 99.94 ± 0.01, while Dataset 2 showed accuracies of 99.68 ± 0.15 in experiment E1 (Easy 1) and 99.27 ± 0.35 in experiment E2 (Easy 2). Despite a slight decrease in average accuracy in the remaining two experiments, D1 (Difficult 1) and D2 (Difficult 2) from Dataset 2, compared to the top-performing clustering algorithms in these categories, G–NMS maintained lower SD, indicating consistent performance. Additionally, G–NMS demonstrated robustness and efficiency across various noise-contaminated neural recordings, ranging from low to high signal-to-noise ratios. (4) Conclusions: G–NMS’s integration of deep learning techniques and eigenvalue inclusion theorems has proven highly effective, marking a significant advancement in the clustering domain. Its superior performance, characterized by high accuracy and low variability, opens new avenues for the development of high-performing clustering algorithms, contributing significantly to the body of research in this field.

Gershgorin circle theorem-based feature extraction for biomedical signal analysis.

Recently, graph theory has become a promising tool for biomedical signal analysis, wherein the signals are transformed into a graph network and represented as either adjacency or Laplacian matrices. However, as the size of the time series increases, the dimensions of transformed matrices also expand, leading to a significant rise in computational demand for analysis. Therefore, there is a critical need for efficient feature extraction methods demanding low computational time. This paper introduces a new feature extraction technique based on the Gershgorin Circle theorem applied to biomedical signals, termed Gershgorin Circle Feature Extraction (GCFE). The study makes use of two publicly available datasets: one including synthetic neural recordings, and the other consisting of EEG seizure data. In addition, the efficacy of GCFE is compared with two distinct visibility graphs and tested against seven other feature extraction methods. In the GCFE method, the features are extracted from a special modified weighted Laplacian matrix from the visibility graphs. This method was applied to classify three different types of neural spikes from one dataset, and to distinguish between seizure and non-seizure events in another. The application of GCFE resulted in superior performance when compared to seven other algorithms, achieving a positive average accuracy difference of 2.67% across all experimental datasets. This indicates that GCFE consistently outperformed the other methods in terms of accuracy. Furthermore, the GCFE method was more computationally-efficient than the other feature extraction techniques. The GCFE method can also be employed in real-time biomedical signal classification where the visibility graphs are utilized such as EKG signal classification.

Overcoming Dimensionality Constraints: A Gershgorin Circle Theorem-Based Feature Extraction for Weighted Laplacian Matrices in Computer Vision Applications.

In graph theory, the weighted Laplacian matrix is the most utilized technique to interpret the local and global properties of a complex graph structure within computer vision applications. However, with increasing graph nodes, the Laplacian matrix's dimensionality also increases accordingly. Therefore, there is always the "curse of dimensionality"; In response to this challenge, this paper introduces a new approach to reducing the dimensionality of the weighted Laplacian matrix by utilizing the Gershgorin circle theorem by transforming the weighted Laplacian matrix into a strictly diagonal domain and then estimating rough eigenvalue inclusion of a matrix. The estimated inclusions are represented as reduced features, termed GC features; The proposed Gershgorin circle feature extraction (GCFE) method was evaluated using three publicly accessible computer vision datasets, varying image patch sizes, and three different graph types. The GCFE method was compared with eight distinct studies. The GCFE demonstrated a notable positive Z-score compared to other feature extraction methods such as I-PCA, kernel PCA, and spectral embedding. Specifically, it achieved an average Z-score of 6.953 with the 2D grid graph type and 4.473 with the pairwise graph type, particularly on the E_Balanced dataset. Furthermore, it was observed that while the accuracy of most major feature extraction methods declined with smaller image patch sizes, the GCFE maintained consistent accuracy across all tested image patch sizes. When the GCFE method was applied to the E_MNSIT dataset using the K-NN graph type, the GCFE method confirmed its consistent accuracy performance, evidenced by a low standard deviation (SD) of 0.305. This performance was notably lower compared to other methods like Isomap, which had an SD of 1.665, and LLE, which had an SD of 1.325; The GCFE outperformed most feature extraction methods in terms of classification accuracy and computational efficiency. The GCFE method also requires fewer training parameters for deep-learning models than the traditional weighted Laplacian method, establishing its potential for more effective and efficient feature extraction in computer vision tasks.

Graph Laplace Regularization-based pressure sensor placement strategy for leak localization in the water distribution networks under joint hydraulic and topological feature spaces

Urban water distribution networks (WDNs) have wide range and intricate topology, which include leakage, pipe burst and other abnormal states during production and operation. With the continuous development of the Internet of Things (IoT) technology in recent years, the means of monitoring the WDNs by using wireless sensor network technology has gradually received attention and extensive research. Most of the existing researches select the deployment location of sensors according to the hydraulic state of the WDNs, but the connectivity and topology between the nodes of the WDNs are not fully considered and analyzed. In this study, a new method that can integrate the topological features and hydraulic model information of the WDN is proposed to solve the problem of optimal sensor placement. First, the method preprocesses the covariance matrix of the pressure sensitivity matrix of the water distribution network by a diffusion kernel-based data prefiltering method and obtains the new network topology weights and its Laplacian matrix under the constraints of the network topology through a data-based graphical Laplacian learning method. Then, the sensor placement problem is transformed into a matrix minimum eigenvalue constraint problem by the Graph Laplace Regularization (GLR)-based method, and finally the selection of sensor nodes is accomplished by the method based on Gershgorin Disc Alignment (GDA). The proposed strategy is tested on a passive Hanoi network, an active Net 3 network, and a larger network, PA2, and is compared with some existing methods. The results show that the proposed solution achieves good performance in three different leak localization methods.

LERE: Learning-Based Low-Rank Matrix Recovery with Rank Estimation

A fundamental task in the realms of computer vision, Low-Rank Matrix Recovery (LRMR) focuses on the inherent low-rank structure precise recovery from incomplete data and/or corrupted measurements given that the rank is a known prior or accurately estimated. However, it remains challenging for existing rank estimation methods to accurately estimate the rank of an ill-conditioned matrix. Also, existing LRMR optimization methods are heavily dependent on the chosen parameters, and are therefore difficult to adapt to different situations. Addressing these issues, A novel LEarning-based low-rank matrix recovery with Rank Estimation (LERE) is proposed. More specifically, considering the characteristics of the Gerschgorin disk's center and radius, a new heuristic decision rule in the Gerschgorin Disk Theorem is significantly enhanced and the low-rank boundary can be exactly located, which leads to a marked improvement in the accuracy of rank estimation. According to the estimated rank, we select row and column sub-matrices from the observation matrix by uniformly random sampling. A 17-iteration feedforward-recurrent-mixed neural network is then adapted to learn the parameters in the sub-matrix recovery processing. Finally, by the correlation of the row sub-matrix and column sub-matrix, LERE successfully recovers the underlying low-rank matrix. Overall, LERE is more efficient and robust than existing LRMR methods. Experimental results demonstrate that LERE surpasses state-of-the-art (SOTA) methods. The code for this work is accessible at https://github.com/zhengqinxu/LERE.

Efficient interpolation of molecular properties across chemical compound space with low-dimensional descriptors

We demonstrate accurate data-starved models of molecular properties for interpolation in chemical compound spaces with low-dimensional descriptors. Our starting point is based on three-dimensional, universal, physical descriptors derived from the properties of the distributions of the eigenvalues of Coulomb matrices. To account for the shape and composition of molecules, we combine these descriptors with six-dimensional features informed by the Gershgorin circle theorem. We use the nine-dimensional descriptors thus obtained for Gaussian process regression based on kernels with variable functional form, leading to extremely efficient, low-dimensional interpolation models. The resulting models trained with 100 molecules are able to predict the product of entropy and temperature (S × T) and zero point vibrational energy (ZPVE) with the absolute error under 1 kcal mol−1 for >78 % and under 1.3 kcal mol−1 for >92 % of molecules in the test data. The test data comprises 20 000 molecules with complexity varying from three atoms to 29 atoms and the ranges of S × T and ZPVE covering 36 kcal mol−1 and 161 kcal mol−1, respectively. We also illustrate that the descriptors based on the Gershgorin circle theorem yield more accurate models of molecular entropy than those based on graph neural networks that explicitly account for the atomic connectivity of molecules.

Small-Signal Stability Analysis and Improvement in Multi-Converter Grid-Tied System Based on Gerschgorin Disc Theorem

The integration of a large number of voltage source converters (VSCs) into the power grid decreases the small-signal stability of the power system. When several VSCs with different control parameters are simultaneously connected to the power grid to form a multi-converter grid-tied system, the potential destabilizing factors increase. Thus, parameter optimization for stability-weakest parameters that have the greatest impact on the system stability becomes more significant in addressing small-signal stability issues. This paper first proposes a stability evaluation function based on the Gerschgorin disc theorem, which can assess the stability of the multi-converter grid-tied system. Then a parameter sensitivity method is proposed to identify the stability-weakest parameters. Finally, an iterative calculation-based parameter optimization method is developed to regulate the identified stability-weakest parameters. Hence, the parameter optimization technique in this research can improve the system stability without requiring eigenvalue solutions and has the merit of low computational complexity. Simulation results based on both the MATLAB/Simulink (2023a) and the RT-LAB (OPAL-RT 5700) platform of a multi-converter grid-tied system validate the correctness of the theoretical analysis and the effectiveness of the parameter optimization method.

A Stability Study by Routh-Hurwitz Criterion and Gershgorin Circles for Covid-19

A Stability Study by Routh-Hurwitz Criterion and Gershgorin Circles for Covid-19

Performance Comparison of MIMO Gain/Phase Analysis Methods for Sub-Synchronous Instability

Recently, there has been great interest in the development of stability criteria for grid-connected power converters and generators using frequency-domain impedance models. We consider two recently introduced stability analysis methods and compare their small-signal stability performance for a DFIG wind farm connected to a series-compensated grid. The first method, known as the Low-Complexity Global Nyquist Condition, uses Gershgorin disks derived from the loop gain matrix to provide a stability criterion. The second method, introduced by the authors, uses a combined matrix gain and phase stability analysis. The two methods are compared by using them to predict the onset of instability in a series-compensated wind farm under a tripped line. The results reveal that the gain/phase based approach achieves less conservatism than the Nyquist-based method.

The conditions for the analog of QED photons in semi-classical periodically driven systems

The Floquet and quantum electrodynamics (QED) Hamiltonians are widely used in various contexts for light-matter interactions. While they exhibit structural similarity, the QED Hamiltonian has a bounded spectrum while the Floquet Hamiltonian does not. Thus, it remains uncertain if they share the same or similar spectra, even at high energy with a substantial average photon count. Using the Gershgorin circle theorem, we bound analytically the difference between the spectra of the QED and Floquet Hamiltonians. We establish a common spectrum by imposing the constraints of high photon numbers and narrow photon statistics. Following the analytic proof, we numerically demonstrate this bound’s implications on a model Xe atom previously used in high harmonic generation, showing correspondence between Floquet and QED photons.

A multivariate Riesz basis of ReLU neural networks

We consider the trigonometric-like system of piecewise linear functions introduced recently by Daubechies, DeVore, Foucart, Hanin, and Petrova. We provide an alternative proof that this system forms a Riesz basis of L2([0,1]) based on the Gershgorin theorem. We also generalize this system to higher dimensions d>1 by a construction, which avoids using (tensor) products. As a consequence, the functions from the new Riesz basis of L2([0,1]d) can be easily represented by neural networks. Moreover, the Riesz constants of this system are independent of d, making it an attractive building block regarding future multivariate analysis of neural networks.

A Gershgorin Circle Theorem Inspired Controller for Renewable Energy Source Integrated Power Systems and Multimicrogrids

Increasing load demands, variations in the operating conditions, continuous integration of the renewable energy sources constitute a multifarious challenge for the researchers to maintain the quality in performances of a load frequency controller. To contemplate such issues, a Gershgorin circle theorem inspired controller assimilated with the particle swarm optimization is proposed for multiarea power systems. The proposed method ensures system stability while abating the frequency variations in the operational areas and the power deviations in the interconnected tie-lines. The present analysis is substantiated through an interconnected photovoltaic (PV) integrated thermal power systems. Furthermore, the competence of the proposed method against increasing complexities is validated by considering an interconnected multimicrogrid systems consisting of renewable and nonrenewable energy sources. The performances of the proposed controller are demonstrated considering fixed and time-varying load disturbances, and fluctuations in renewable energy sources. Results corroborate that the proposed method is successful in maintaining the system stability while mitigating the frequency and the power deviations in all such operating conditions. A comparative analysis of the proposed method with prevalent control techniques (e.g., PI and PID-based controllers, tilt–integral–derivative, integral–tilt–derivative) and optimization algorithms (e.g., genetic algorithm, gravitational search algorithm, imperialist competitive algorithm) is presented to highlight the substantial performance improvements in the considered systems.

Additive autoregressive models for matrix valued time series

In this article, we develop additive autoregressive models (Add‐ARM) for the time series data with matrix valued predictors. The proposed models assume separable row, column and lag effects of the matrix variables, attaining stronger interpretability when compared with existing bilinear matrix autoregressive models. We utilize the Gershgorin's circle theorem to impose some certain conditions on the parameter matrices, which make the underlying process strictly stationary. We also introduce the alternating least squares estimation method to solve the involved equality constrained optimization problems. Asymptotic distributions of the parameter estimators are derived. In addition, we employ hypothesis tests to run diagnostics on the parameter matrices. The performance of the proposed models and methods is further demonstrated through simulations and real data analysis.

Image Focus Measure Based on Polynomial Coefficients and Reduced Gerschgorin Circle Approach

In this paper, a simple and efficient no-reference image quality assessment technique based on polynomial coefficient and reduced Gerschgorin circle is presented. The dominant features and dynamics of the test images are captured using the polynomial coefficient. Moreover, the image focus measure is modeled using the concept of reduced Gerschgorin's higher circle bound. The presented image focus measure is tested on different synthetic images datasets such as LIVE, TID2008, CSIQ, Cornel and IVC. Further, the proposed approach is also examined on real-time images captured by the camera in different conditions. It is observed that the proposed approach demonstrates important properties of focus measures such as unimodality and contrast invariance. Moreover, the presented technique is robust under the different varying noisy conditions.

A Small-Signal Stability Criterion for DC Microgrid Based on Extended-Gershgorin Theorem

Recent years have witnessed emerging oscillation stability issues in DC microgrid via power electronic converters. Considering the confidentiality of the parameters and structure of power electronic devices, the impedance method is widely used to study the small signal stability. However, the stability assessment based on traditional impedance model and Gershgorin Theorem have some shortcomings for multiple converters DC microgrid. This may lead to the problem that the stability analysis results are not accurate enough or the criteria cannot be applied. In this article, the system is modeled as a medium and high dimensional negative feedback system. Further, a sufficient stability criterion based on extended Gershgorin Theorem is developed through reshaping the range of eigenvalues estimation for the non-diagonal advantage system. Finally, the validity of the criterion is verified by a 6 converters system. Test results indicate the proposed method has the potential to assess the high-order system stability with low computational complexity. Moreover, its low conservatism as a sufficient criterion is helpful to guide the system design.

On the iterative diagonalization of matrices in quantum chemistry: Reconciling preconditioner design with Brillouin-Wigner perturbation theory.

Iterative diagonalization of large matrices to search for a subset of eigenvalues that may be of interest has become routine throughout the field of quantum chemistry. Lanczos and Davidson algorithms hold a monopoly, in particular, owing to their excellent performance on diagonally dominant matrices. However, if the eigenvalues happen to be clustered inside overlapping Gershgorin disks, the convergence rate of both strategies can be noticeably degraded. In this work, we show how Davidson, Jacobi-Davidson, Lanczos, and preconditioned Lanczos correction vectors can be formulated using the reduced partitioning procedure, which takes advantage of the inherent flexibility promoted by Brillouin-Wigner perturbation (BW-PT) theory's resolvent operator. In doing so, we establish a connection between various preconditioning definitions and the BW-PT resolvent operator. Using Natural Localized Molecular Orbitals (NLMOs) to construct Configuration Interaction Singles (CIS) matrices, we study the impact the preconditioner choice has on the convergence rate for these comparatively dense matrices. We find that an attractive by-product of preconditioning the Lanczos algorithm is that the preconditioned variant only needs 21%-35% and 54%-61% of matrix-vector operations to extract the lowest energy solution of several Hartree-Fock- and NLMO-based CIS matrices, respectively. On the other hand, the standard Davidson preconditioning definition seems to be generally optimal in terms of requisite matrix-vector operations.

Identification of Homogeneous Subgroups from Resting-State fMRI Data

The identification of homogeneous subgroups of patients with psychiatric disorders can play an important role in achieving personalized medicine and is essential to provide insights for understanding neuropsychological mechanisms of various mental disorders. The functional connectivity profiles obtained from functional magnetic resonance imaging (fMRI) data have been shown to be unique to each individual, similar to fingerprints; however, their use in characterizing psychiatric disorders in a clinically useful way is still being studied. In this work, we propose a framework that makes use of functional activity maps for subgroup identification using the Gershgorin disc theorem. The proposed pipeline is designed to analyze a large-scale multi-subject fMRI dataset with a fully data-driven method, a new constrained independent component analysis algorithm based on entropy bound minimization (c-EBM), followed by an eigenspectrum analysis approach. A set of resting-state network (RSN) templates is generated from an independent dataset and used as constraints for c-EBM. The constraints present a foundation for subgroup identification by establishing a connection across the subjects and aligning subject-wise separate ICA analyses. The proposed pipeline was applied to a dataset comprising 464 psychiatric patients and discovered meaningful subgroups. Subjects within the identified subgroups share similar activation patterns in certain brain areas. The identified subgroups show significant group differences in multiple meaningful brain areas including dorsolateral prefrontal cortex and anterior cingulate cortex. Three sets of cognitive test scores were used to verify the identified subgroups, and most of them showed significant differences across subgroups, which provides further confirmation of the identified subgroups. In summary, this work represents an important step forward in using neuroimaging data to characterize mental disorders.

Inseparable Gershgorin discs and the existence of conjugate complex eigenvalues of real matrices

We investigate the converse of the known fact that if the Gershgorin discs of a real n-by-n matrix may be separated by positive diagonal similarity, then the eigenvalues are real. In the 2-by-2 case, with appropriate signs for the off-diagonal entries, we find that the converse is correct, which raises several questions. First, in the 3-by-3 case, the converse is not generally correct, but, empirically, it is frequently true. Then, in the n-by-n case, n ≥ 3 , we find that if all the 2-by-2 principal submatrices have inseparable discs (‘strongly inseparable discs’), the full matrix must have a nontrivial pair of conjugate complex eigenvalues (i.e. cannot have all real eigenvalues). This hypothesis cannot generally be weakened.