Frobenius Norm Research Articles

Lost in the Shuffle: Testing Power in the Presence of Errorful Network Vertex Labels

Two-sample network hypothesis testing is an important inference task with applications across diverse fields such as medicine, neuroscience, and sociology. Many of these testing methodologies operate under the implicit assumption that the vertex correspondence across networks is a priori known. This assumption is often untrue, and the power of the subsequent test can degrade when there are misaligned/label-shuffled vertices across networks. This power loss due to shuffling is theoretically explored in the context of random dot product and stochastic block model networks for a pair of hypothesis tests based on Frobenius norm differences between estimated edge probability matrices or between adjacency matrices. The loss in testing power is further reinforced by numerous simulations and experiments, both in the stochastic block model and in the random dot product graph model, where the power loss across multiple recently proposed tests in the literature is considered. Lastly, the impact that shuffling can have in real-data testing is demonstrated in a pair of examples from neuroscience and from social network analysis.

Exploring Bounded Component Analysis Using an ℓ∞ Norm Criterion

In this paper we propose a new criterion for the Blind Source Separation (BSS) of antisparse bounded sources, based on the sum of the ℓ∞-norm of the sources. Based on the observation that the mixing process of bounded sources with any mixing matrix with unitary Frobenius norm will increase the ℓ∞-norm of the sources, unless it is the identity matrix, the minimization of the sum of the ℓ∞-norm of the sources can be used for the estimation of a separation matrix. To that, a Principle Component Analysis technique followed by a Givens Rotations based optimization method can be used for the separation of independent bounded sources. Also, the Givens Rotations based optimization method can be used for the separation of correlated bounded sources mixed by a rotation matrix. We theoretically analyze the proposed criterion and assess its performance through numerical simulations involving three distinct types of bounded signals. Our theoretical and experimental findings underscore the efficacy of the ℓ∞ norm as a suitable contrast function for antisparse bounded sources, showcasing its superior performance relative to a state-of-the-art algorithm.

Suppression of the Tissue Component With the Total Least‐Squares Algorithm to Improve Second Harmonic Imaging of Ultrasound Contrast Agents

ABSTRACTThe second harmonic (SH) of ultrasound contrast agents (UCAs) is widely used in contrast‐enhanced ultrasound imaging; however, is affected by the nonlinearity of surrounding tissue. Suppression of the tissue component based on the total least‐squares (STLS) algorithm is proposed to enhance the SH imaging of UCAs. The image blocks of pulse‐inversion‐based SH images before and after UCA injections are set as the reference and input of the total least‐squares model, respectively. The optimal coefficients of the model are obtained by minimizing the Frobenius norm of perturbations in the input and output signals. After processing all image blocks, the complete SH image of UCAs is obtained by subtracting the optimal output of the model (i.e., the estimated tissue SH image) from the SH image after UCA injection. Simulation and in vivo experiments confirm that the STLS approach offers clearer capillaries. For in vivo experiments, the STLS‐based contrast‐to‐tissue ratios and contrasts increase by 26.90% and 56.27%, as well as 26.99% and 56.43%, respectively, compared with those based on bubble‐echo deconvolution and pulse inversion bubble‐wavelet imaging methods. The STLS approach enhances the SH imaging of UCAs by effectively suppressing more tissue SH components, having the potential to provide more accurate diagnostic information for clinical applications.

Refining uniform approximation algorithm for low-rank Chebyshev embeddings

Abstract Nowadays, low-rank approximations are a critical component of many numerical procedures. Traditionally the problem of low-rank approximation of matrices is solved in unitary invariant norms such as Frobenius or spectral norm due to the existence of efficient methods for constructing approximations. However, recent results discover the potential of low-rank approximations in the Chebyshev norm, which naturally arises in many applications. In this paper, we investigate the problem of uniform approximation of vectors, which is the main component in the low-rank approximations of matrices in the Chebyshev norm. The principal novelty of this paper is the accelerated algorithm for solving uniform approximation problems. We also analyze the iterative procedure of the proposed algorithm and demonstrate that it has a geometric convergence rate. Finally, we provide an extensive numerical evaluation, which demonstrates the effectiveness of the proposed procedures.

Coherent Set Identification Via Direct Low Rank Maximum Likelihood Estimation

We analyze connections between two low rank modeling approaches from the last decade for treating dynamical data. The first one is the coherence problem (or coherent set approach), where groups of states are sought that evolve under the action of a stochastic transition matrix in a way maximally distinguishable from other groups. The second one is a low rank factorization approach for stochastic matrices, called direct Bayesian model reduction (DBMR), which estimates the low rank factors directly from observed data. We show that DBMR results in a low rank model that is a projection of the full model, and exploit this insight to infer bounds on a quantitative measure of coherence within the reduced model. Both approaches can be formulated as optimization problems, and we also prove a bound between their respective objectives. On a broader scope, this work relates the two classical loss functions of nonnegative matrix factorization, namely the Frobenius norm and the generalized Kullback–Leibler divergence, and suggests new links between likelihood-based and projection-based estimation of probabilistic models.

Interval-Valued Random Matrices

This paper introduces a novel approach that combines symbolic data analysis with matrix theory through the concept of interval-valued random matrices. This framework is designed to address the complexities of real-world data, offering enhanced statistical modeling techniques particularly suited for large and complex datasets where traditional methods may be inadequate. We develop both frequentist and Bayesian methods for the statistical inference of interval-valued random matrices, providing a comprehensive analytical framework. We conduct extensive simulations to compare the performance of these methods, demonstrating that Bayesian estimators outperform maximum likelihood estimators under the Frobenius norm loss function. The practical utility of our approach is further illustrated through an application to climatology and temperature data, highlighting the advantages of interval-valued random matrices in real-world scenarios.

Graph Regularized Sparse L2,1 Semi‐Nonnegative Matrix Factorization for Data Reduction

ABSTRACTNon‐negative Matrix Factorization (NMF) is an effective algorithm for multivariate data analysis, including applications to feature selection, pattern recognition, and computer vision. Its variant, Semi‐Nonnegative Matrix Factorization (SNF), extends the ability of NMF to render parts‐based data representations to include mixed‐sign data. Graph Regularized SNF builds upon this paradigm by adding a graph regularization term to preserve the local geometrical structure of the data space. Despite their successes, SNF‐related algorithms to date still suffer from instability caused by the Frobenius norm due to the effects of outliers and noise. In this paper, we present a new SNF algorithm that utilizes the noise‐insensitive norm. We provide monotonic convergence analysis of the SNF algorithm. In addition, we conduct numerical experiments on three benchmark mixed‐sign datasets as well as several randomized mixed‐sign matrices to demonstrate the performance superiority of SNF over conventional SNF algorithms under the influence of Gaussian noise at different levels.

Quantum radiation reaction: Analytical approximations and obtaining the spectrum from moments

We derive analytical χ≪1 approximations for spin-dependent quantum radiation reaction for locally constant and locally monochromatic fields. We show how to factor out fast spin oscillations and obtain the degree of polarization in the plane orthogonal to the magnetic field from the Frobenius norm of the Mueller matrix. We show that spin effects lead to a transseries in χ, with powers χk, logarithms (lnχ)k, and oscillating terms, cos(…/χ) and sin(…/χ). In our approach we can obtain each moment, ⟨(kP)m⟩, of the light-front longitudinal momentum independently of the other moments and without considering the spectrum. We show how to obtain a low-energy expansion of the spectrum from the moments by treating m as a continuous, complex parameter and performing an inverse Mellin transform. We also show how to obtain the spectrum, without making a low-energy approximation, from a handful of moments using the principle of maximum entropy. Published by the American Physical Society 2024

A method of estimating degree of influence from wind for improving sound source localization in windy place

In this paper, we propose a method for visualizing the influence of wind on sound propagating in a room where wind is blowing, and rating the degree of influence. Since non-uniform airflow affects sound propagation and gives time-variant characteristics to the transfer function, a method is needed to understand the influence of wind. Therefore we propose a method that measures impulse responses multiple times indoors, determines a similarity matrix between impulse responses, and visualizes it as a heatmap. We evaluate the proposed method in four types of average wind velocity (0.0 m/s, 2.7 m/s, 3.3 m/s, 3.9 m/s). The result of the evaluation showed that the greater the influence of the wind, the higher the value of diagonal elements of the similarity matrix than off-diagonal elements, and the tendency for diagonal elements to be emphasized. We also confirm that the rating based on the Frobenius norm of the similarity matrix takes a lower value as the influence of the wind is greater.

Prediction of Turbulent Boundary Layer Flow Dynamics with Transformers

Time-marching of turbulent flow fields is computationally expensive using traditional Computational Fluid Dynamics (CFD) solvers. Machine Learning (ML) techniques can be used as an acceleration strategy to offload a few time-marching steps of a CFD solver. In this study, the Transformer (TR) architecture, which has been widely used in the Natural Language Processing (NLP) community for prediction and generative tasks, is utilized to predict future velocity flow fields in an actuated Turbulent Boundary Layer (TBL) flow. A unique data pre-processing step is proposed to reduce the dimensionality of the velocity fields, allowing the processing of full velocity fields of the actuated TBL flow while taking advantage of distributed training in a High Performance Computing (HPC) environment. The trained model is tested at various prediction times using the Dynamic Mode Decomposition (DMD) method. It is found that under five future prediction time steps with the TR, the model is able to achieve a relative Frobenius norm error of less than 5%, compared to fields predicted with a Large Eddy Simulation (LES). Finally, a computational study shows that the TR achieves a significant speed-up, offering computational savings approximately 53 times greater than those of the baseline LES solver. This study demonstrates one of the first applications of TRs on actuated TBL flow intended towards reducing the computational effort of time-marching. The application of this model is envisioned in a coupled manner with the LES solver to provide few time-marching steps, which will accelerate the overall computational process.

Two-Dimensional Direction Finding for L-Shaped Coprime Array via Minimization of the Ratio of the Nuclear Norm and the Frobenius Norm

More recently, the ability of the coprime array to yield large array apertures and high degrees of freedom in comparison with the uniform linear array (ULA) has drawn an enormous amount of attention. In light of this, we propose a low-rank matrix completion algorithm via minimization of the ratio of the nuclear norm and the Frobenius norm (N/F) to solve the two-dimensional (2D) direction finding problem for the L-shaped coprime array (LsCA). Specifically, we first interpolate the virtual co-array signal related to the cross-correlation matrix (CCM) and utilize the interpolated virtual signal for Toeplitz matrix reconstruction. Then, the N/F method is employed to perform low-rank matrix completion on the reconstructed matrix. Finally, exploiting the conjugate symmetry characteristics of the completed matrix, we further develop a direction-finding algorithm that enables 2D angle estimation. Remarkably, the 2D angles are able to be automatically paired by the proposed algorithm. Numerical simulation findings demonstrate that the proposed N/F algorithm generates excellent angular resolution and computational complexity. Furthermore, this algorithm yields better estimation accuracy compared to the competing algorithms.

SPLHRNMTF: robust orthogonal non-negative matrix tri-factorization with self-paced learning and dual hypergraph regularization for predicting miRNA-disease associations

MicroRNAs (miRNAs) have been demonstrated to be closely related to human diseases. Studying the potential associations between miRNAs and diseases contributes to our understanding of disease pathogenic mechanisms. As traditional biological experiments are costly and time-consuming, computational models can be considered as effective complementary tools. In this study, we propose a novel model of robust orthogonal non-negative matrix tri-factorization (NMTF) with self-paced learning and dual hypergraph regularization, named SPLHRNMTF, to predict miRNA-disease associations. More specifically, SPLHRNMTF first uses a non-linear fusion method to obtain miRNA and disease comprehensive similarity. Subsequently, the improved miRNA-disease association matrix is reformulated based on weighted k-nearest neighbor profiles to correct false-negative associations. In addition, we utilize L2,1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_{2,1}$$\\end{document} norm to replace Frobenius norm to calculate residual error, alleviating the impact of noise and outliers on prediction performance. Then, we integrate self-paced learning into NMTF to alleviate the model from falling into bad local optimal solutions by gradually including samples from easy to complex. Finally, hypergraph regularization is introduced to capture high-order complex relations from hypergraphs related to miRNAs and diseases. In 5-fold cross-validation five times experiments, SPLHRNMTF obtains higher average AUC values than other baseline models. Moreover, the case studies on breast neoplasms and lung neoplasms further demonstrate the accuracy of SPLHRNMTF. Meanwhile, the potential associations discovered are of biological significance.

Elastic deep multi-view autoencoder with diversity embedding

Current research on multi-view clustering (MVC) is pushing the boundaries of knowledge, allowing the extraction of valuable insights from various points of view. Recently, many researchers in this field have turned to deep learning techniques, particularly autoencoders (AE). These methods are highly effective at recognizing nonlinear and complex structures, making them a popular choice in this area. However, while they effectively handle Gaussian noise through the use of the Frobenius norm, they exhibit sensitivity to outlier data and Laplacian noise. Additionally, existing multi-view methodologies tend to usually rely on shared information across multi-view data, this means they often miss out on the diverse and complementary information each view can provide, leading in an inadequate utilization of the information available. In this paper, we present an innovative solution to address these challenges. Our novel Elastic Deep Multi-view Autoencoder with Diversity Embedding (EDMVAE-DE) method incorporates an Elastic Deep AE to enhance its robustness. Furthermore, we integrate an exclusivity constraint term to enhance the diversity of specific representations across distinct views, increasing both modeling consistency and diversity within a unified framework. To address problems arising from incorrectly classified neighbors, we also incorporate a graph constraint term. Our experiments, performed on various multi-view datasets, clearly show that EDMVAE-DE achieves the best performance in its class1.

Structural Damage Detection under Ambient Excitation Using Symbolic Three-Order Square Matrix Formed by Specific-Interval-Sampled Time-Domain Signals.

In the structural health monitoring of vibration systems, varying excitation always affects the accuracy of damage identification. The proposed symbolic three-order square matrix damage detection method with the matrix norm as a damage indicator can solve the difficult problem of damage identification under ambient excitation. The new sampling pattern extracts data from signals in the time domain at specific intervals based on the structural properties with the help of the autocorrelation coefficient. Then, the data extracted are converted into symbols and arranged into a three-order square matrix, and the Frobenius norm of the matrix is used for structural damage identification as a reliable damage indicator. In this process, the transmissibility function is employed to eliminate the effects of varying excitation. First, the method was verified by a cracked simply supported beam-a simulated Abaqus model. Then, a wooden truss bridge in the laboratory and an actual engineering scenario under ambient excitation together demonstrated the effectiveness and accuracy of the damage identification method and proved the proposed method to be robust to different types of damage under ambient excitation. Compared with other related methods, this method is more intuitive and efficient.

Quaternion Nuclear Norm Minus Frobenius Norm Minimization for color image reconstruction

Color image restoration methods typically represent images as vectors in Euclidean space or combinations of three monochrome channels. However, they often overlook the correlation between these channels, leading to color distortion and artifacts in the reconstructed image. To address this, we present Quaternion Nuclear Norm Minus Frobenius Norm Minimization (QNMF), a novel approach for color image reconstruction. QNMF utilizes quaternion algebra to capture the relationships among RGB channels comprehensively. By employing a regularization technique that involves nuclear norm minus Frobenius norm, QNMF approximates the underlying low-rank structure of quaternion-encoded color images. Theoretical proofs are provided to ensure the method’s mathematical integrity. Demonstrating versatility and efficacy, the QNMF regularizer excels in various color low-level vision tasks, including denoising, deblurring, inpainting, and random impulse noise removal, achieving state-of-the-art results.

The BiCG Algorithm for Solving the Minimal Frobenius Norm Solution of Generalized Sylvester Tensor Equation over the Quaternions

In this paper, we develop an effective iterative algorithm to solve a generalized Sylvester tensor equation over quaternions which includes several well-studied matrix/tensor equations as special cases. We discuss the convergence of this algorithm within a finite number of iterations, assuming negligible round-off errors for any initial tensor. Moreover, we demonstrate the unique minimal Frobenius norm solution achievable by selecting specific types of initial tensors. Additionally, numerical examples are presented to illustrate the practicality and validity of our proposed algorithm. These examples include demonstrating the algorithm’s effectiveness in addressing three-dimensional microscopic heat transport and color video restoration problems.

F-Norm-Based Soft LDA Algorithm for Fault Detection in Chemical Production Processes.

In chemical production processes, outliers are inevitable. Many existing feature extraction algorithms are overly sensitive to outliers and excessively focus on secondary features while ignoring the key features in the data. To address this problem, the Frobenius norm based soft linear discriminant analysis algorithm (FBSLA) is proposed in this paper. Specifically, FBSLA uses the Frobenius norm instead of its square as a metric to enhance the robustness of the algorithm. Furthermore, a nonreduced dimensionality projection matrix is introduced to make the training data features more obvious. Additionally, soft constraint is adopted instead of the traditional hard constraint to reduce the sensitivity caused by outliers. To validate the effectiveness of FBSLA, in this paper, experiments are conducted on the Tennessee Eastman Process and the Penicillin Fermentation Process data sets. According to experimental results, FBSLA significantly outperforms other state-of-the-art algorithms in terms of fault detection accuracy.

Robust multi-view clustering via structure regularization concept factorization

Recently, many concept factorization-based multi-view clustering methods have been proposed and achieved promising results on text multi-view data. However, existing methods are limited in the following two aspects. (1) The Frobenius norm used in these methods is sensitive to noise and outliers; (2) These methods ignore the global structural information of the data. To address the above problems, we propose a robust concept factorization framework for multi-view clustering, which not only improves the robustness but also fully exploits the available information of multi-view data. Specifically, L2,1-norm is used to evaluate the error of the factorization, thus eliminating the effect of outliers and improving robustness. In addition, to retain more structure information of the original data, the global and local structure information are taken into consideration simultaneously, which makes the learned low-dimensional matrix more discriminative. Further, to make use of the complementary information of the different views, we introduce an adaptive weight learning strategy to assign weights for different views. An iterative updating algorithm is proposed to solve the proposed optimization problem. We compare the proposed method with state-of-the-art alternative methods on benchmark multi-view data sets. The extensive experimental results show the effectiveness and superiority of the proposed method.

Consistent and specific multi-view multi-label learning with correlation information

In multi-view multi-label (MVML) learning, each sample is represented by several heterogeneous distinct feature representations while associated with a set of class labels simultaneously. To achieve MVML learning, most of the existing methods contribute to the recovery of a consistent subspace, i.e., a shared feature representation, among multiple views. Nevertheless, each view has its inherent specific properties used in the discrimination process of labels. These methods lose sight in the specific information exploitation, and therefore are easily trapped in a sub-optimal result. In this study, we present an optimization framework CSVL to solve the learning problem. The main technical contribution in CSVL is a formulation for MVML learning while consistent subspace across views, specific subspace for each view, and the correlations among labels are taken into account. Specifically, consistent subspace is recovered by imposing a low-rank constraint among multiple views, and specific subspace of each view is extra generated with Frobenius norm. To further improve model generalization capability, we preserve both feature manifolds from multiple views and label correlations from multiple labels. Extensive experiments on 7 benchmark datasets show that our proposal CSVL has the advantages in MVML learning.

Nuclear Norm Minus Frobenius Norm Minimization with Rank Residual Constraint for Image Denoising

Nuclear Norm Minus Frobenius Norm Minimization with Rank Residual Constraint for Image Denoising