Eigenvalue Decomposition Research Articles

Near-Field Synthesis of 1-D Shaped Patterns Through Spectral Factorization and Minimally-Redundant Array-Like Representations

Near-Field Synthesis of 1-D Shaped Patterns Through Spectral Factorization and Minimally-Redundant Array-Like Representations

A Fast Complexity Pursuit-Based Blind Source Separation Method for Structural Modal Identification

Complexity pursuit (CP) is an effective method for solving blind source separation (BSS) problems. Combined with the gradient descent algorithm, the complexity pursuit-gradient descent (CP-GD) algorithm can extract the structural modal contributions from vibration responses without measuring the inputs. However, the computational efficiency of CP-GD algorithm is very low since all data should be used in each iteration step of the gradient descent. This paper introduces the fast complexity pursuit-gradient descent (FCP-GD) algorithm, optimizing the traditional complexity calculation formula and using the subspace search method for calculating the de-mixing vectors, which significantly saves computation time. Furthermore, a comparative study between the FCP-GD and temporal predictability-generalized eigenvalue decomposition (TP-GED) algorithms was conducted using two-degree-of-freedom and three-degree-of-freedom systems and revealed an equivalence of these two algorithms. Finally, numerical simulations and experiments on multi-degree-of-freedom free vibration systems confirmed the effectiveness, efficiency, and robustness of the FCP-GD algorithm.

Quantum-Enhanced Eigenface Algorithm for Face Verification

Facial biometrics play a crucial role in identity verification, yet classical approaches face challenges related to computational complexity and security vulnerabilities. This paper explores the integration of quantum computing with eigenface-based face verification to enhance efficiency and security. By utilizing the advantages of Quantum Principal Component Analysis (QPCA), we achieve exponential speedups in eigenvalue decomposition, significantly reducing the computational burden of high-dimensional facial data processing. Our hybrid classical-quantum approach optimizes quantum state encoding and similarity measurement via the Swap Test techniques. Experimental results demonstrate improved verification accuracy and scalability compared to classical eigenfaces, particularly for large databases. Despite current hardware constraints, our findings establish a foundational framework for quantum-enhanced biometric systems. This work highlights the potential of quantum computing in facial recognition, and prepares the way for more efficient, secure, and scalable biometric authentication systems.

Factor analysis of electrocardiographic findings, anthropometric measures, and age in patients with chronic kidney disease

Chronic kidney disease (CKD) is a significant global health concern, frequently associated with cardiovascular complications resulting from autonomic nervous system dysfunction, which can be detected using electrocardiography (ECG). This study employed factor analysis to investigate the association between anthropometric measures, age, and ECG findings in patients with CKD. We conducted a cross-sectional study to evaluate the ECG findings of 25 male participants (aged 36 – 80 years) with stage 5 CKD who were randomly selected from the Nephrology Unit of a hospital in the Amazon region. All participants underwent anthropometric and blood pressure assessment before the ECG recording at a sampling rate of 1,000 Hz. Then, the participants were positioned supine and asked to breathe normally for 3 min. To analyze the ECG data, a bootstrap method was used to estimate statistical parameters from 1,000 resampled datasets. A two-step process involving principal component (PC) extraction and varimax rotation was used for factor analysis. The covariance matrix of the normalized data underwent eigenvalue decomposition. The first three PCs captured 68.7% of the total variability observed in the original dataset. The PR interval (iPR), RR interval (iRR), and corrected QT (QTc) interval contributed 0.843, 0.836, and −0.822, respectively, to PC1; body mass index (BMI) and abdominal circumference (AC) contributed 0.910 and 0.947, respectively, to PC2; and age had the largest contribution of 0.938 to PC3. In conclusion, BMI, AC, and age can be simple and reliable clinical tools for detecting underlying CKD in primary care. ECG changes in iPR, iRR, and QTc are common in patients with CKD, thus highlighting the potential role of machine learning in the early detection of cardiovascular disease.

Nonfragile leader‐following consensus control for Lurie descriptor fractional‐order multi‐agent systems

AbstractIn this brief, the nonfragile leader‐following consensus control issue of Lurie descriptor fractional‐order multi‐agent systems (LDFOMASs) is discussed with the order in and , respectively. At first, a new model with one leader agent and follower agents is constructed, where dynamics of agents are described by Lurie descriptor fractional‐order systems. Then, by applying the approximation linearization, eigenvalue decomposition, and interval uncertainty technologies, convert the leader‐following consensus of LDFOMASs into the admissibility of descriptor fractional‐order interval systems. Moreover, sufficient conditions which guarantee LDFOMASs to realize the consensus are obtained through the designed nonfragile state and output feedback schemes, respectively. Lastly, two numerical examples are given to clarify the validity of results presented.

Adiabatic perturbation theory for mode analysis in optical waveguides with large index variations without eigen value decomposition.

Eigen mode and eigen value evaluation is crucial in optical waveguide design and is quite time consuming. It is known that the perturbation method can compute the modes efficiently for an index perturbed structure based on the known eigen modes and thereby avoids the time consuming eigen value decomposition process. However, the perturbation method can only be applied to the waveguide structures with small index variations, and hence it poses a significant constraint to the applicability of the method to the optical waveguide inverse design problem, which might have large index variations. In this paper, an adiabatic perturbation theory is proposed to tackle this problem. The eigen modes and the propagation constants of the optical waveguides are evaluated gradually by adding the index variation with respect to the previous step. While keeping the index variation to a small value within a step, the optical waveguide structure achieves a significant index change in total and thereby enables the application of the perturbation theory to the optical waveguide inverse design problem.

Isotropic spectrum optimization for enhancing image fidelity in super-resolution structured illumination microscopy.

Super-resolution structured illumination microscopy (SR-SIM) performs spectral expansion of high-frequency information encoded in stripe patterns. However, using a limited number of pattern orientations (typically three) results in a petal-like frequency spectrum, leading to structural and intensity fidelity degradation in reconstructed images. In this Letter, we propose an integrated spatial-frequency domain SIM reconstruction method that enables isotropic spectrum expansion, called ISO-SIM. ISO-SIM overcomes structural artifacts caused by an anisotropic spectral expansion in traditional SIM imaging. We demonstrate the feasibility and fidelity of ISO-SIM through simulations, Argolight slide, and live-cell imaging. ISO-SIM enhanced structural similarity and reduced the error in the mean intensity ratio at certain spatial frequencies compared to Wiener-SIM. We further applied ISO-SIM to live-cell quantitative FRET imaging. ISO-SIM-FRET ensured that the measured FRET efficiency matched the ground truth, with a 19% reduction in the standard deviation compared to Wiener-SIM-FRET, maintaining intensity fidelity and enhancing the accuracy of quantitative analysis while suppressing artifacts.

Stacking Model-Based Classifiers for Dealing With Multiple Sets of Noisy Labels.

Supervised learning in presence of multiple sets of noisy labels is a challenging task that is receiving increasing interest in the ever-evolving landscape of healthcare analytics. Such an issue arises when multiple annotators are tasked to manually label the same training samples, potentially giving rise to discrepancies in class assignments among the supplied labels with respect to the ground truth. Commonly, the labeling process is entrusted to a small group of domain experts, and different level of experience and subjectivity may result in noisy training labels. To solve the classification task leveraging on the availability of multiple data annotators, we introduce a novel ensemble methodology constructed combining model-based classifiers separately trained on single sets of noisy labels. Eigenvalue Decomposition Discriminant Analysis is employed for the definition of the base learners, and six distinct averaging strategies are proposed to combine them. Two solutions necessitate a priori information, such as the partial knowledge of the ground truth labels or the annotators' level of expertise. Differently, the remaining four approaches are entirely data-driven. A simulation study and an application on real data showcase the improved predictive performance of our proposal, while also demonstrating the ability of automatically inferring annotators' expertise level as a by-product of the learning process.

Rigidity Expander Graphs

Jordán and Tanigawa recently introduced the d-dimensional algebraic connectivity ad(G) of a graph G. This is a quantitative measure of the d-dimensional rigidity of G which generalizes the well-studied notion of spectral expansion of graphs. We present a new lower bound for ad(G) defined in terms of the spectral expansion of certain subgraphs of G associated with a partition of its vertices into d parts. In particular, we obtain a new sufficient condition for the rigidity of a graph G. As a first application, we prove the existence of an infinite family of k-regular d-rigidity-expander graphs for every d≥2 and k≥2d+1. Conjecturally, no such family of 2d-regular graphs exists. Second, we show that ad(Kn)≥12nd, which we conjecture to be essentially tight. In addition, we study the extremal values ad(G) attains if G is a minimally d-rigid graph.

Multisource Heterogeneous Data Fusion-Based Process Monitoring of the Reheating Furnace in Steel Production.

The reheating furnace is the key piece of equipment in the hot rolling process of steel production. In order to fully exploit all of the data recorded from the production process representing different information, this paper designs a process monitoring algorithm with multisource information fusion by integrating multiple information to comprehensively monitor the operating state of the reheating furnace. Multisource information fusion combines process variable data of the reheating furnace and heating process data of the slab. To overcome the challenge of fusion of heterogeneous data due to different sampling patterns, univariate time series and multivariate time series data are fused by a transformer. In the fusion scheme, univariate time series data are represented by bidirectional gated recurrent unit for one-dimensional temporal representation, multivariate time series data are represented by temporal convolutional network for two-dimensional temporal representation, and multivariate time series data are represented by eigenvalue decomposition for correlation representation between variables. To evaluate the performance of the proposed method, computational experiments based on actual data are carried out. In univariate and multivariate time series representations, the highest predictions are obtained for bidirectional gated recurrent unit and temporal convolutional network by comparison with different regression algorithms, respectively. By comparing with fusing different fusion objects and different fusion schemes, the proposed algorithm achieves the highest accuracy (91.33%), precision (91.46%), and recall (92.59%), proving the effectiveness of the fusion approach. The process monitoring performance is compared with multivariate statistical process monitoring algorithms, which achieve the highest accuracy (95%), precision (93.45%), and recall (97.08%).

SPECTRUM OF GENERALIZED CESARO OPERATOR ON THE LORENTZ SPACES

The aim of this paper is to investigate the boundedness and spectrum of generalized Ces\`{a}ro operators defined on Lorentz spaces over a finite interval and the positive half-line. When $\beta=1$, these operators coincide with the classical Ces\`{a}ro operator. In this paper, we extend the results obtained for Sobolev spaces in \cite{Lizama} to Lorentz spaces. The primary tools employed in this work are $C_0$-groups, $C_0$-semigroups, and their generators. $C_0$-groups and $C_0$-semigroups are used to demonstrate the boundedness of the generalized Ces\`{a}ro operator. Since the spectrum of the bounded linear operators is non-empty, we investigate the spectrum of the generalized Ces\`{a}ro operator. The generators of these $C_0$-groups and $C_0$-semigroups are utilized to analyze the spectral properties of the generalized Ces\`{a}ro operator.We study the spectra of the generators and determine the spectra of the generalized Ces\`{a}ro operators using the spectral mapping theorem. Additionally, we provide results on the point spectrum of generalized Ces\`{a}ro operators defined on Lorentz spaces over a finite interval.

Conservative closures of the Vlasov-Poisson equations based on symmetrically weighted Hermite spectral expansion

Conservative closures of the Vlasov-Poisson equations based on symmetrically weighted Hermite spectral expansion

Enhanced Phase Optimization Using Spectral Radius Constraints and Weighted Eigenvalue Decomposition for Distributed Scatterer InSAR

Enhanced Phase Optimization Using Spectral Radius Constraints and Weighted Eigenvalue Decomposition for Distributed Scatterer InSAR

A Novel Azimuth Channel Errors Estimation Algorithm Based on Characteristic Clusters Statistical Treatment

Azimuth multi-channel techniques show great promise in high-resolution, wide-swath synthetic aperture radar systems. However, in practical engineering applications, errors between channels can significantly affect the reconstruction of multi-channel echo data, leading to a degraded synthetic aperture radar image. To address this issue, this article derives the formula expression in the two-dimensional time domain after single-channel processing under the assumption of an insufficient azimuth sampling rate and proposes a novel algorithm based on the statistical treatment of characteristic clusters. In this algorithm, channel imaging is first performed separately; then, the image is divided into a predefined number of sub-images. The characteristic clusters and points within each sub-image are identified, and their positions, amplitude, and phase information are used to obtain the range synchronization errors, amplitude errors, and phase errors between channels. Compared with traditional methods, the proposed method does not require iteration or the complex eigenvalue decomposition of the covariance matrix. Furthermore, it can utilize existing imaging tools and software in single-channel synthetic aperture radar systems. The effectiveness of the proposed method is validated through simulation experiments and real-world data processing.

The role of structural connectivity on brain function through a Markov model of signal transmission.

Structure determines function. However, this universal theme in biology has been surprisingly difficult to observe in human brain neuroimaging data. Here, we link structure to function by hypothesizing that brain signals propagate as a Markovian process on an underlying structure. We focus on a metric called commute time: the average number of steps for a random walker to go from region A to B and then back to A. Commute times based on white matter tracts from diffusion MRI exhibit an average ± standard deviation Spearman correlation of -0.26 ± 0.08 with functional MRI connectivity data across 434 UK Biobank individuals and -0.24 ± 0.06 across 400 HCP Young Adult brain scans. The correlation increases to -0.36 ± 0.14 and to -0.32 ± 0.12 when the principal contributions of both commute time and functional connectivity are compared for both datasets. The observed weak but robust correlations provide evidence of a relationship, albeit restricted, between neuronal connectivity and brain function. The correlations are stronger by 33% compared to broadly used communication measures such as search information and communicability. The difference further widens to a factor of 5 when commute times are correlated to the principal mode of functional connectivity from its eigenvalue decomposition. Overall, the study points to the utility of commute time to account for the role of polysynaptic (indirect) connectivity underlying brain function.

Flat broadband frequency upconversion within a thin-film lithium niobate waveguide achieved by multi-objective genetic algorithm particle swarm optimization

In the field of nonlinear infrared frequency upconversion within a poled thin film lithium niobate (TFLN) waveguide for spectroscopy, there is a persistent demand for achieving a flat broadband response, characterized by the minimal variation in output intensity across the desired wavelength range. We propose a design method that significantly broadens the spectral bandwidth and enhances the response flatness through multi-objective genetic algorithm particle swarm optimization (GAPSO). This approach minimizes human intervention in the optimization process, thereby enhancing efficiency and accuracy compared to traditional methods that depend on preset parameters. Compared to the traditional chirped periodically poled TFLN waveguide-based infrared frequency upconversion scheme, a remarkable spectral bandwidth expansion from 180 nm to 312 nm (a 73% increase) and an improved flatness from 1.71 dB to 0.56 dB (a reduction of over 67%) is achieved. This work not only paves the way for a more efficient flat broadband infrared frequency upconversion scheme but also opens new avenues for advancements in nonlinear optical applications, such as telecommunications and sensing technologies.

Polynomial Characterizations of Distance‐Biregular Graphs

ABSTRACTFiol, Garriga, and Yebra introduced the notion of pseudo‐distance‐regular vertices, which they used to come up with a new characterization of distance‐regular graphs. Building on that work, Fiol and Garriga developed the spectral excess theorem for distance‐regular graphs. We extend both these characterizations to distance‐biregular graphs and show how these characterizations can be used to study bipartite graphs with distance‐regular halved graphs and graphs with the spectrum of a distance‐biregular graph.

Multiclass Classification Framework of Motor Imagery EEG by Riemannian Geometry Networks.

In motor imagery (MI) tasks for brain computer interfaces (BCIs), the spatial covariance matrix (SCM) of electroencephalogram (EEG) signals plays a critical role in accurate classification. Given that SCMs are symmetric positive definite (SPD), Riemannian geometry is widely utilized to extract classification features. However, calculating distances between SCMs is computationally intensive due to operations like eigenvalue decomposition, and classical optimization techniques, such as gradient descent, cannot be directly applied to Riemannian manifolds, making the computation of the Riemannian mean more complex and reliant on iterative methods or approximations. In this paper, we propose a novel multiclass classification framework that integrates Riemannian geometry and neural networks to mitigate these challenges. The framework comprises two modules: a Riemannian module with multiple branches and a classification module. During training, a fusion loss function is introduced to update the branch corresponding to the true label, while other branches are updated using different loss functions along with the classification module. Comprehensive experiments on four sets of MI EEG data demonstrate the efficiency and effectiveness of the proposed model.

Existence and Uniqueness Results for Generalized Non-local Hallaire-Luikov Moisture Transfer Equation

This article focuses on inverse problem for Hallaire-Luikov moisture transfer equation involving Hilfer fractional derivative in time. Hallaire-Luikov equation is used to study heat and mass transfer in capillary-porous bodies. Spectral expansion method is used to find the solution of the inverse problem. By imposing certain conditions on the functions involved and utilizing certain properties of multinomial Mittag-Leffler function, it is shown that the solution to the equation, known as the inverse problem, is regular and unique. Moreover, the inverse problem exhibits ill-posedness in the sense of Hadamard. The article ends with an example to demonstrate these theoretical findings.

On convergence of spectral expansions for the equation of a vibrating beam, at one end of which an elastically fixed inertial load is concentrated

On convergence of spectral expansions for the equation of a vibrating beam, at one end of which an elastically fixed inertial load is concentrated