Search Results (57)

This paper conducts a rigorous study on the spectral properties and operator-space distances of perturbed Dirichlet–Neumann tridiagonal (PDNT) Toeplitz matrices, with emphasis on their asymptotic behaviors. We establish explicit closed-form solutions for the eigenvalues and associated eigenvectors, highlighting their fundamental importance for characterizing matrix stability in the presence of perturbations. By exploiting the structural characteristics of PDNT Toeplitz matrices, we obtain closed-form expressions quantifying the distance to normality, the deviation from normality. Full article

In this paper, we aim to develop an efficient algorithm for the solving Tensor Robust Principal Component Analysis (TRPCA) problem, which focuses on obtaining a low-rank approximation of a tensor by separating sparse and impulse noise. A common approach is to minimize the convex surrogate of the tensor rank by shrinking its singular values. Due to the existence of various definitions of tensor ranks and their corresponding convex surrogates, numerous studies have explored optimal solutions under different formulations. However, many of these approaches suffer from computational inefficiency primarily due to the repeated use of tensor singular value decomposition in each iteration. To address this issue, we propose a novel TRPCA algorithm that introduces a new convex relaxation for the tensor norm and computes low-rank approximation more efficiently. Specifically, we adopt the tensor average rank and tensor nuclear norm, and further relax the tensor nuclear norm into a sum of the tensor Frobenius norms of the factor tensors. By alternating updates of the truncated factor tensors, our algorithm achieves efficient use of computational resources. Experimental results demonstrate that our algorithm achieves significantly faster performance than existing reference methods known for efficient computation while maintaining high accuracy in recovering low-rank tensors for applications such as color image recovery and background subtraction. Full article

Traditional antenna arrays for direction-of-arrival (DOA) estimation typically require numerous elements to achieve target performance, increasing system complexity and cost. Reconfigurable intelligent surfaces (RISs) offer a promising alternative, yet their performance critically depends on phase coding design. To address this, we propose a phase coding design method for RIS-aided DOA estimation with a single receiving channel. First, we establish a system model where averaged received signals construct a power-based formulation. This transforms DOA estimation into a compressed sensing-based sparse recovery problem, with the RIS far-field power radiation pattern serving as the measurement matrix. Then, we derive the decoupled expression of the measurement matrix, which consists of the phase coding matrix, propagation phase shifts, and array steering matrix. The phase coding design is then formulated as a Frobenius norm minimization problem, approximating the Gram matrix of the equivalent measurement matrix to an identity matrix. Accordingly, the phase coding design problem is reformulated as a Frobenius norm minimization problem, where the Gram matrix of the equivalent measurement matrix is approximated to an identity matrix. The phase coding is deterministically constructed as the product of a unitary matrix and a partial Hadamard matrix. Simulations demonstrate that the proposed phase coding design outperforms random phase coding in terms of angular estimation accuracy, resolution probability, and the requirement of coding sequences. Full article

Graph attention networks are pivotal for modeling non-Euclidean data, yet they face dual challenges: training oscillations induced by projection-based high-dimensional constraints and gradient anomalies due to poor adaptation to heterophilic structure. To address these issues, we propose LDC-GAT (Lyapunov-Stable Graph Attention Network with Dynamic Filtering and Constraint-Aware Optimization), which jointly optimizes both forward and backward propagation processes. In the forward path, we introduce Dynamic Residual Graph Filtering, which integrates a tunable self-loop coefficient to balance neighborhood aggregation and self-feature retention. This filtering mechanism, constrained by a lower bound on Dirichlet energy, improves multi-head attention via multi-scale fusion and mitigates overfitting. In the backward path, we design the Fro-FWNAdam, a gradient descent algorithm guided by a learning-rate-aware perceptron. An explicit Frobenius norm bound on weights is derived from Lyapunov theory to form the basis of the perceptron. This stability-aware optimizer is embedded within a Frank–Wolfe framework with Nesterov acceleration, yielding a projection-free constrained optimization strategy that stabilizes training dynamics. Experiments on six benchmark datasets show that LDC-GAT outperforms GAT by 10.54% in classification accuracy, which demonstrates strong robustness on heterophilic graphs. Full article

Multi-source heterogeneous data has been widely adopted in developing artificial intelligence systems in recent years. In real-world scenarios, raw multi-source data are generally unlabeled and inherently contain multi-view noise and feature redundancy, leading to extensive research on unsupervised multi-view feature selection. However, existing approaches mainly utilize the local adjacency relationships and the $L_{2,1}$-norm to guide the feature selection process, which may lead to instability in performance. To address these problems, this paper proposes a Consensus Guided Multi-view Unsupervised Feature Selection with Hybrid Regularization (CGMvFS). Specifically, CGMvFS integrates multiple view-specific basic partitions into a unified consensus matrix, which is constructed to guide the feature selection process by preserving comprehensive pairwise constraints across diverse views. A hybrid regularization strategy incorporating the $L_{2,1}$-norm and the Frobenius norm is introduced into the feature selection objective function, which not only promotes feature sparsity but also effectively prevents overfitting, thereby improving the stability of the model. Extensive empirical evaluations demonstrate that CGMVFS outperforms state-of-the-art comparison approaches across diverse multi-view datasets. Full article

This paper proposes a class of randomized Kaczmarz and Gauss–Seidel-type methods for solving the matrix equation $AXB=C$, where the matrices A and B may be either full-rank or rank deficient and the system may be consistent or inconsistent. These iterative methods offer high computational efficiency and low memory requirements, as they avoid costly matrix–matrix multiplications. We rigorously establish theoretical convergence guarantees, proving that the generated sequences converge to the minimal Frobenius-norm solution (for consistent systems) or the minimal Frobenius-norm least squares solution (for inconsistent systems). Numerical experiments demonstrate the superiority of these methods over conventional matrix multiplication-based iterative approaches, particularly for high-dimensional problems. Full article

This paper investigates Bitcoin’s resilience against the U.S. dollar—widely recognized as the global reserve currency—by applying a multi-method wavelet analysis framework to daily price data of Bitcoin, the USD strength index (DXY), the euro, and other assets ranging from August 2015 to June 2024. Quantitative measures—particularly the Frobenius norm of wavelet coherence and an exponential decay phase-weighting scheme—reveal that Bitcoin’s out-of-phase relationship with the dollar is lower and more sporadic than that of mainstream assets, indicating it is not tightly governed by dollar fluctuations. Even after controlling for the euro’s dominant influence in the DXY, BTC continues to show weaker coupling than mainstream assets—reinforcing the idea that it may serve as a partial hedge against dollar-driven volatility. These results support the hypothesis that Bitcoin may serve as a resilient store of value and hedge against dollar-driven market volatility, placing Bitcoin within the broader debate on global monetary frameworks. As global monetary conditions evolve, the resilience of Bitcoin (BTC) relative to the world’s leading reserve currency—the U.S. dollar—has significant implications for both investors and policymakers. Full article

This paper presents a simplified and computationally feasible multivariate extension. A correlation matrix is constructed using pairwise Spearman’s footrule correlation coefficients, and these coefficients are shown to jointly converge to a multivariate normal distribution. A global test statistic based on the Frobenius norm of this matrix asymptotically follows a weighted sum of chi-square distributions. Simulation studies and two real-world applications (a sensory analysis of French Jura wines and the characterization of plant leaf specimens) demonstrate the practical utility of the proposed method, bridging the gap between theoretical rigor and practical implementation in multivariate nonparametric inference. Full article

As programming education becomes increasingly complex, grading student code has become a challenging task. Traditional methods, such as dynamic and static analysis, offer foundational approaches but often fail to provide granular insights, leading to inconsistencies in grading and feedback. This study addresses the limitations of these methods by integrating contrastive learning with explainable AI techniques to assess SQL code submissions. We employed contrastive learning to differentiate between student and correct SQL solutions, projecting them into a high-dimensional latent space, and used the Frobenius norm to measure the distance between these representations. This distance was used to predict the percentage of points deducted from each student’s solution. To enhance interpretability, we implemented feature ablation and integrated gradients, which provide insights into the specific tokens in student code that impact the grading outcomes. Our findings indicate that this approach improves the accuracy, consistency, and transparency of automated grading, aligning more closely with human grading standards. The results suggest that this framework could be a valuable tool for automated programming assessment systems, offering clear, actionable feedback and making machine learning models in educational contexts more interpretable and effective. Full article

The node representation learning capability of Graph Convolutional Networks (GCNs) is fundamentally constrained by dynamic instability during feature propagation, yet existing research lacks systematic theoretical analysis of stability control mechanisms. This paper proposes a Stability-Optimized Graph Convolutional Network (SO-GCN) that enhances training stability and feature expressiveness in shallow architectures through continuous–discrete dual-domain stability constraints. By constructing continuous dynamical equations for GCNs and rigorously proving conditional stability under arbitrary parameter dimensions using nonlinear operator theory, we establish theoretical foundations. A Precision Weight Parameter Mechanism is introduced to determine critical Frobenius norm thresholds through feature contraction rates, optimized via differentiable penalty terms. Simultaneously, a Dynamic Step-size Adjustment Mechanism regulates propagation steps based on spectral properties of instantaneous Jacobian matrices and forward Euler discretization. Experimental results demonstrate SO-GCN’s superiority: 1.1–10.7% accuracy improvement on homophilic graphs (Cora/CiteSeer) and 11.22–12.09% enhancement on heterophilic graphs (Texas/Chameleon) compared to conventional GCN. Hilbert–Schmidt Independence Criterion (HSIC) analysis reveals SO-GCN’s superior inter-layer feature independence maintenance across 2–7 layers. This study establishes a novel theoretical paradigm for graph network stability analysis, with practical implications for optimizing shallow architectures in real-world applications. Full article

In the control-based approach of medical treatment of various illnesses such as diabetes mellitus, certain angiogenic cancers, or in anesthesia, the starting point used to be some “patient model” on the basis of which the appropriate administration of the drugs can be designed. The identification of the “patient model’s parameters” is always a hard and sometimes unsolvable mathematical task. Furthermore, these parameters have wide variability between patients. In principle, either robust or adaptive techniques can be used to tackle the problem of modeling imprecisions. In this paper, the potential application of a variant of Fixed Point Iteration-Based Adaptive Controllers was investigated in model-based control. The main point was the introduction of a “parameter estimation error significance metric” through the use of which the individual model parameter estimation can be avoided, and even the consequences of the deficiencies of the approximate model as a whole can be estimated. The adaptive controller forces the system to track the prescribed nominal trajectory; therefore, it brings about the “actual control situation” in which the consequences of the estimation errors are of interest. One component of the adaptive control is a “rotational block” that creates a multidimensional orthogonal (rotation) matrix that rotates arrays of identical Frobenius norms into each other. Since in a recent publication under review it was proved that the angle of the necessary rotation satisfies the mathematical criteria of metrics in a metric space, even in quite complicated nonlinear and multidimensional cases, this simple value can serve as a metric for this purpose. To exemplify the method, an under-actuated nonlinear system of 2 degree of freedom and relative order 4 was controlled by a special adaptive backstepping controller that was designed on a purely kinematic basis. From this point of view, it has a strong relationship with the PID controllers. This simple model was rich enough to exemplify parameters that require precise identification because their error produces quite significant consequences, and other parameters that do not require very precise identification. It was found that the method provided the dynamic models with reliable parameter sensitivity estimation metrics. Full article

An optimizer plays a decisive role in the efficiency and effectiveness of model training in deep learning. Although Adam and its variants are widely used, the impact of model complexity on training is not considered, which leads to instability or slow convergence when a complex model is trained. To address this issue, we propose an AMC (Adam with Model Complexity) optimizer, which dynamically adjusts the learning rate by incorporating model complexity, thereby improving training stability and convergence speed. AMC uses the Frobenius norm of the model to measure its complexity, automatically decreasing the learning rate of complex models and increasing the learning rate of simple models, thus optimizing the training process. We provide a theoretical analysis to demonstrate the relationship between model complexity and learning rate, as well as the convergence and convergence bounds of AMC. Experiments on multiple benchmark datasets show that, compared to several widely used optimizers, AMC exhibits better stability and faster convergence, especially in the training of complex models. Full article

In this paper, we explore the least-norm solution to the classical matrix equation $AXB=C$ over the dual quaternion algebra, where A, B, and C are given matrices, while X remains the unknown matrix. We begin by transforming the definition of the Frobenius norm for dual quaternion matrices into an equivalent form. Using this new expression, we investigate the least-norm solution to the equation $AXB=C$ under solvability conditions. Additionally, we examine the minimum real part of the norm solution in cases where a least-norm solution does not exist. Finally, we provide two numerical examples to illustrate the main findings of our study. Full article

We devise two algorithms for approximating solutions of PSDisation, a problem in actuarial science and finance, to find the nearest valid correlation matrix that is positive semidefinite (PSD). The first method converts the PSDisation problem with a positive semidefinite constraint and other linear constraints into iterative Linear Programmings (LPs) or Quadratic Programmings (QPs). The LPs or QPs in our formulation give an upper bound of the optimal solution of the original problem, which can be improved during each iteration. The biggest advantage of this iterative method is its great flexibility when working with different choices of norms or with user-defined constraints. Second, a gradient descent method is designed specifically for PSDisation under the Frobenius norm to measure how close the two metrices are. Experiments on randomly generated data show that this method enjoys better resilience to noise while maintaining good accuracy. For example, in our experiments with noised data, the iterative quadratic programming algorithm performs best in more than 41% to 67% of the samples when the standard deviation of noise is 0.02, and the gradient descent method performs best in more than 70% of the samples when the standard deviation of noise is 0.2. Examples of applications in finance, as well as in the machine learning field, are given. Computational results are presented followed by discussion on future improvements. Full article

The present research investigates the global asymptotic stability of bidirectional associative memory (BAM) neural networks using distinct sufficient conditions. The primary objective of this study is to establish new generalized criteria for the global asymptotic robust stability of time-delayed BAM neural networks at the equilibrium point, utilizing the Frobenius norm and the positive symmetrical approach. The new sufficient conditions are derived with the help of the Lyapunov–Krasovskii functional and the Frobenius norm, which are important in deep learning for a variety of reasons. The derived conditions are not influenced by the system parameter delays of the BAM neural network. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed conclusions regarding network parameters. Full article