Frobenius Norm Research Articles

GNSS Signal Compression Acquisition Algorithm Based on Sensing Matrix Optimization

Due to the sparsity of GNSS signal in the correlation domain, compressed sensing theory is considered to be a promising technology for GNSS signal acquisition. However, the detection probability of the traditional compression acquisition algorithm is low under low signal-to-noise ratio (SNR) conditions. This paper proposes a GNSS compression acquisition algorithm based on sensing matrix optimization. The Frobenius norm of the difference between Gram matrix and an approximate equiangular tight frame (ETF) matrix is taken as the objective function, and the modified conjugate gradient method is adopted to reduce the mutual coherence between the measurement matrix and the sparse basis. Theoretical analysis and simulation results show that the proposed algorithm can significantly improve the detection probability compared with the existing compression acquisition algorithms under the same SNR.

Weighted tensor Golub–Kahan–Tikhonov-type methods applied to image processing using a t-product

This paper discusses weighted tensor Golub–Kahan-type bidiagonalization processes using the t-product. This product was introduced in M.E. Kilmer and C.D. Martin (2011). A few steps of a bidiagonalization process with a weighted least squares norm are carried out to reduce a large-scale linear discrete ill-posed problem to a problem of small size. The weights are determined by symmetric positive definite (SPD) tensors. Tikhonov regularization is applied to the reduced problem. An algorithm for tensor Cholesky factorization of SPD tensors is presented. The data is a laterally oriented matrix or a general third order tensor. The use of a weighted Frobenius norm in the fidelity term of Tikhonov minimization problems is appropriate when the noise in the data has a known covariance matrix that is not the identity. We use the discrepancy principle to determine both the regularization parameter in Tikhonov regularization and the number of bidiagonalization steps. Applications to image and video restoration are considered.

Robust Spectral Clustering Incorporating Statistical Sub-Graph Affinity Model

Hyperspectral image (HSI) clustering is a challenging work due to its high complexity. Subspace clustering has been proven to successfully excavate the intrinsic relationships between data points, while traditional subspace clustering methods ignore the inherent structural information between data points. This study uses graph convolutional subspace clustering (GCSC) for robust HSI clustering. The model remaps the self-expression of the data to non-Euclidean domains, which can generate a robust graph embedding dictionary. The EKGCSC model can achieve a globally optimal closed-form solution by using a subspace clustering model with the Frobenius norm and a Gaussian kernel function, making it easier to implement, train, and apply. However, the presence of noise can have a noteworthy negative impact on the segmentation performance. To diminish the impact of image noise, the concept of sub-graph affinity is introduced, where each node in the primary graph is modeled as a sub-graph describing the neighborhood around the node. A statistical sub-graph affinity matrix is then constructed based on the statistical relationships between sub-graphs of connected nodes in the primary graph, thus counteracting the uncertainty image noise by using more information. The model used in this work was named statistical sub-graph affinity kernel graph convolutional subspace clustering (SSAKGCSC). Experiment results on Salinas, Indian Pines, Pavia Center, and Pavia University data sets showed that the SSAKGCSC model can achieve improved segmentation performance and better noise resistance ability.

Dual graph-regularized sparse concept factorization for clustering

The concept factorization algorithm has received widespread attention and achieved remarkable results in the field of clustering. However, when modeling this clustering algorithm, it is necessary to initialize two new low-dimensional matrices that are independent of the objective matrix and continuously approximate the objective matrix through alternating iterative updating, thus inevitably introducing some noise factors that are undesirable for the model. Especially in the objective function constructed by square loss, the noise factors have a more significant influence on the clustering performance. To solve this issue, a dual graph-regularized sparse concept factorization (DGSCF) algorithm is proposed in this paper. In addition to maintaining the geometric structure of the data using dual graph regularization, DGSCF adopts an optimization framework based on l1 and Frobenius norms, which enhance the ability of feature selection and sparsity to eliminate the influence of noise factors on the algorithm performance. The corresponding alternating iterative updating rules and convergence proof of the DGSCF are provided. Finally, experiments on eight public datasets show its effectiveness and superiority.

Hyperspectral Image Denoising via Group Sparsity Regularized Hybrid Spatio-Spectral Total Variation

In this paper, we propose a new hyperspectral image (HSI) denoising model with the group sparsity regularized hybrid spatio-spectral total variation (GHSSTV) and low-rank tensor decomposition, which is based on the analysis of structural sparsity of HSIs. First, the global correlations among all modes are explored by the Tucker decomposition, which applies low-rank constraints to the clean HSIs. To avoid over-smoothing, we propose GHSSTV regularization to ensure the group sparsity not only in the first-order gradient domain but also in the second-order ones along the spatio-spectral dimensions. Then, the sparse noise in HSI can be detected by the ℓ1 norm. Furthermore, strong Gaussian noise is simulated by the Frobenius norm. The alternating direction multiplier method (ADMM) algorithm is employed to effectively solve the GHSSTV model. Finally, experimental results from a series of simulations and real-world data suggest a superior performance of the GHSSTV method in HSIs denoising.

On the banded Toeplitz structured distance to symmetric positive semidefiniteness

This paper is concerned with the determination of a close real banded positive definite Toeplitz matrix in the Frobenius norm to a given square real banded matrix. While it is straightforward to determine the closest banded Toeplitz matrix to a given square matrix, the additional requirement of positive definiteness makes the problem difficult. We review available theoretical results and provide a simple approach to determine a banded positive definite Toeplitz matrix.

The effect of large-scale forcing on small-scale dynamics of incompressible turbulence

The subtle but crucial effects of large-scale forcing on the small-scale velocity-gradient (VG) dynamics is examined using direct numerical simulation (DNS) data of incompressible turbulence. The interplay among large-scale forcing, inertia, pressure and viscous effects is characterised as a function of local streamline geometry and VG magnitude (Frobenius norm). When conditioned on local topology, forcing: (i) counteracts inertial and viscous action in the strain-dominated nodal topologies; and (ii) balances pressure action in the rotation-dominated unstable focal topologies. Unexpectedly, forcing acts to reduce VG magnitudes in regions of strong dissipation. In these regions, forcing balances the non-local pressure effects whereas viscous action offsets the nonlinear inertial effects. In regions of very low dissipation, forcing combines with inertia and pressure effects to offset viscous action. With regard to the probability distribution of the normalised VG invariants, the primary role of forcing is to nullify certain features (dilatational probability currents) of inertia, pressure and viscous action. This results in the emergence of universal statistical features (solenoidal probability currents) that are determined largely by inertia–pressure–viscous interactions. These findings serve to enhance our understanding of small-scale processes in turbulence and guide VG model development.

On the non-symmetric semidefinite Procrustes problem

In this paper, we consider the non-symmetric positive semidefinite Procrustes (NSPSDP) problem: Given two matrices X,B∈Rn,m, find the matrix A∈Rn,n that minimizes the Frobenius norm of AX−B and which is such that A+AT is positive semidefinite. We generalize the semi-analytical approach for the symmetric positive semidefinite Procrustes problem, where A is required to be positive semidefinite, that was proposed by Gillis and Sharma (A semi-analytical approach for the positive semidefinite Procrustes problem, Linear Algebra Appl. 540, 112-137, 2018). As for the symmetric case, we first show that the NSPSDP problem can be reduced to a smaller NSPSDP problem that always has a unique solution and where the matrix X is diagonal and has full rank. Then, an efficient semi-analytical algorithm to solve the NSPSDP problem is proposed, solving the smaller and well-posed problem with a fast gradient method which guarantees a linear rate of convergence. This algorithm is also applicable to solve the complex NSPSDP problem, where X,B∈Cn,m, as we show that the complex NSPSDP problem can be written as an overparametrized real NSPSDP problem. The efficiency of the proposed algorithm is illustrated on several numerical examples.

GPU‐accelerated and mixed norm regularized online extreme learning machine

SummaryExtreme learning machine (ELM) is a prominent example of neural network with its fast training speed, and good prediction performance. An online version of ELM called online sequential extreme learning machine (OS‐ELM) has also been proposed for the sequential training. Combined with the need for regularization to prevent over‐fitting in addition to the large number of neurons required in the hidden layer, OS‐ELM demands huge amount of computation power for the large‐scale data. In this article, a mixed norm () regularized online machine learning algorithm (MRO‐ELM) that is based on alternating direction method of multipliers (ADMM) is proposed. A linear combination of the mixed norm and the Frobenius norm regularization is applied using the ADMM framework and update formulas are derived. Graphics processing unit (GPU) accelerated version of MRO‐ELM (GPU‐MRO‐ELM) is also proposed to reduce the training time by processing appropriate parts in parallel using the implemented custom kernels. In addition, a novel automatic hyper‐parameter tuning method is incorporated to GPU‐MRO‐ELM using progressive validation with GPU acceleration. The experimental results show that the MRO‐ELM algorithm and its GPU version outperform OS‐ELM in terms of training speed, and testing accuracy. Also, compared to the cross validation, the proposed automatic hyper‐parameter tuning demonstrates dramatical reduction in the tuning time.

Sparse Reduced Rank Huber Regression in High Dimensions

We propose a sparse reduced rank Huber regression for analyzing large and complex high-dimensional data with heavy-tailed random noise. The proposed method is based on a convex relaxation of a rank- and sparsity-constrained nonconvex optimization problem, which is then solved using a block coordinate descent and an alternating direction method of multipliers algorithm. We establish nonasymptotic estimation error bounds under both Frobenius and nuclear norms in the high-dimensional setting. This is a major contribution over existing results in reduced rank regression, which mainly focus on rank selection and prediction consistency. Our theoretical results quantify the tradeoff between heavy-tailedness of the random noise and statistical bias. For random noise with bounded ( 1 + δ ) th moment with δ ∈ ( 0 , 1 ) , the rate of convergence is a function of δ, and is slower than the sub-Gaussian-type deviation bounds; for random noise with bounded second moment, we obtain a rate of convergence as if sub-Gaussian noise were assumed. We illustrate the performance of the proposed method via extensive numerical studies and a data application. Supplementary materials for this article are available online.

Kronecker component with robust low-rank dictionary for image denoising

Robust Kronecker Component Analysis (RKCA) model is a newly proposed denoising method, which transforms the basic sparse representation model used for 2-D data into the tensor sparse representation used for 3-D data. This model is based on the idea that the dictionary in the basic sparse representation model can be separated into two separable dictionaries using the Kronecker Product and the mode-n product. Moreover, the original 3-D data can also be decomposed into a sparse matrix and two dictionaries by using Tucker factorization. It uses the Frobenius norm constraint to get two low-rank dictionaries. In this paper, we take the Nuclear norm into RKCA to better capture the low-rank property of the two dictionaries. Firstly, we design a novel denoising model named Kronecker Component with Low-Rank Dictionary (KCLD), which replaces the Frobenius norm by Nuclear norm in order to capture the low-rank property better. Further, we design a more effective denoising model named Kronecker Component with Robust Low-Rank Dictionary (KCRD) by combining the Frobenius norm and the Nuclear norm. KCRD fuses the advantages of Frobenius norm and the Nuclear norm, thus can get the better low-rank dictionary. Then, an augmented Lagrange multiplier method is used by convex relaxation of the split variable to optimize the two proposed model. Last, through the comparison of a large number of experiments, our models are more competitive and effective than other proposed denoising methods in different types of images.

Efficient direct position determination using Frobenius norm approximation

To eliminate the bottleneck of computation and communication capability in classic direct position determination, recently several fast approximation methods have been proposed. However, the estimation accuracy of these methods is not satisfactory. In this letter, the common theoretical principle behind these fast approximation methods is analysed in depth. It is demonstrated that the maximum eigenvalue of a matrix can be approximated by the matrix norm. The performance of several commonly used matrix norms is then compared, and the Frobenius norm is preferable because of its great potential inferred from the analysis in this letter. Numerical simulations indicate that, compared with conventional methods, the proposed method can achieve better estimation accuracy with less computational complexity.

An Iterative Algorithm for the Generalized Reflexive Solution Group of a System of Quaternion Matrix Equations

In the present paper, an iterative algorithm is proposed for solving the generalized (P,Q)-reflexive solution group of a system of quaternion matrix equations ∑l=1M(AlsXlBls+ClsXl˜Dls)=Fs,s=1,2,…,N. A generalized (P,Q)-reflexive solution group, as well as the least Frobenius norm generalized (P,Q)-reflexive solution group, can be derived by choosing appropriate initial matrices, respectively. Moreover, the optimal approximate generalized (P,Q)-reflexive solution group to a given matrix group can be derived by computing the least Frobenius norm generalized (P,Q)-reflexive solution group of a reestablished system of matrix equations. Finally, some numerical examples are given to illustrate the effectiveness of the algorithm.

A finite iterative algorithm for the general discrete-time periodic Sylvester matrix equations

The purpose of this paper is to present an iterative algorithm for solving the general discrete-time periodic Sylvester matrix equations. It is proved by theoretical analysis that this algorithm can get the exact solutions of the periodic Sylvester matrix equations in a finite number of steps in the absence of round-off errors. Furthermore, when the discrete-time periodic Sylvester matrix equations are consistent, we can obtain its unique minimal Frobenius norm solution by choosing appropriate initial periodic matrices. Finally, we use some numerical examples to illustrate the effectiveness of the proposed algorithm.

Defining user spectra to classify Ethereum users based on their behavior

PurposeIn this paper, we define the concept of user spectrum and adopt it to classify Ethereum users based on their behavior.Design/methodology/approachGiven a time period, our approach associates each user with a spectrum showing the trend of some behavioral features obtained from a social network-based representation of Ethereum. Each class of users has its own spectrum, obtained by averaging the spectra of its users. In order to evaluate the similarity between the spectrum of a class and the one of a user, we propose a tailored similarity measure obtained by adapting to this context some general measures provided in the past. Finally, we test our approach on a dataset of Ethereum transactions.FindingsWe define a social network-based model to represent Ethereum. We also define a spectrum for a user and a class of users (i.e., token contract, exchange, bancor and uniswap), consisting of suitable multivariate time series. Furthermore, we propose an approach to classify new users. The core of this approach is a metric capable of measuring the similarity degree between the spectrum of a user and the one of a class of users. This metric is obtained by adapting the Eros distance (i.e., Extended Frobenius Norm) to this scenario.Originality/valueThis paper introduces the concept of spectrum of a user and a class of users, which is new for blockchains. Differently from past models, which represented user behavior by means of univariate time series, the user spectrum here proposed exploits multivariate time series. Moreover, this paper shows that the original Eros distance does not return satisfactory results when applied to user and class spectra, and proposes a modified version of it, tailored to the reference scenario, which reaches a very high accuracy. Finally, it adopts spectra and the modified Eros distance to classify Ethereum users based on their past behavior. Currently, no multi-class automatic classification approach tailored to Ethereum exists yet, albeit some single-class ones have been recently proposed. Therefore, the only way to classify users in Ethereum are online services (e.g., Etherscan), where users are classified after a request from them. However, the fraction of users thus classified is low. To address this issue, we present an automatic approach for a multi-class classification of Ethereum users based on their past behavior.

Quantifying the Alignment of Graph and Features in Deep Learning.

We show that the classification performance of graph convolutional networks (GCNs) is related to the alignment between features, graph, and ground truth, which we quantify using a subspace alignment measure (SAM) corresponding to the Frobenius norm of the matrix of pairwise chordal distances between three subspaces associated with features, graph, and ground truth. The proposed measure is based on the principal angles between subspaces and has both spectral and geometrical interpretations. We showcase the relationship between the SAM and the classification performance through the study of limiting cases of GCNs and systematic randomizations of both features and graph structure applied to a constructive example and several examples of citation networks of different origins. The analysis also reveals the relative importance of the graph and features for classification purposes.

Bounding the Frobenius norm of a q-deformed commutator

For two n×n complex matrices A and B, we define the q-deformed commutator as [A,B]q:=AB−qBA for a real parameter q. In this paper, we investigate a generalization of the Böttcher-Wenzel inequality which gives the sharp upper bound of the (Frobenius) norm of the commutator. In our generalisation, we investigate sharp upper bounds on the q-deformed commutator. This generalization can be studied in two different scenarios: firstly bounds for general matrices, and secondly for traceless matrices. For both scenarios, partial answers and conjectures are given for positive and negative q. In particular, denoting the Frobenius norm by ||.||F, when A or B is normal, we prove the following inequality to be true and sharp: ||[A,B]q||F2≤(1+q2)||A||F2||B||F2 for positive q. Also, we conjecture that the same bound is true for positive q when A or B is traceless. For negative q, we conjecture other sharp upper bounds to be true for the generic scenarios and the scenario when A or B is traceless. All conjectures are supported with numerics and proved for n=2.

Optimal estimation for lower bound of the packing number

Lower bound estimation of the packing number plays an important role in matrix denoising. By using the metric entropy of the Grassmannian manifold, Cai, Ma and Wu provide a lower bound estimation of the packing number with the metric based on Frobenius norm, see Cai et al. (2013). In this paper, we extend their result to all metrics based on unitarily invariant norms and then give an example to show the optimality of our estimation. Finally, we discuss a potential application in low-rank matrix denoising.

A norm inequality for three matrices

We prove a Frobenius norm inequality for three matrices, analogous to the well-known Bottcher--Wenzel inequality. The situation is also similar: standard inequalities would yield an upper bound, which however can be reduced by means of further, detailed investigations.

Euclidean Distance Bounds for Linear Matrix Inequalities Analytic Centers Using a Novel Bound on the Lambert Function

Linear matrix inequalities (LMIs) are ubiquitous in modern control theory, as well as in a variety of other fields in science and engineering. Their analytic centers, i.e. the maximum determinant elements of the feasible set spanned by these LMIs, are the solution of many well-known problems in statistics, communications, geometry, and control and can be approximated to arbitrary precision by semidefinite programs (SDPs). The quality of these approximations is measured with respect to the difference in log-determinant of both the exact and the approximate solutions to these SDPs, a quantity that follows directly from the duality theory of semidefinite programming. However, in many applications the relevant parameters are functions of the entries of the LMI argument $X$. In these cases it is of interest to develop metrics that quantify the quality of approximate solutions based on the error of these parameters, something that the log-determinant error fails to capture due to the nonlinear interaction of all the matrix entries. In this work we develop upper bounds on the Frobenius norm error between suboptimal solutions $X_f$ and the exact optimizer $X_*$ of maximum determinant problems, a metric that provides a direct translation to the entrywise error of $X$ and thus to the relevant parameters of the application. We show that these bounds can be expressed through the use of the Lambert function $W(x)$, i.e., the solution of the equation $W(x) e^{W(x)}= x$, and derive novel bounds for one of its branches to generate efficient closed-form bounds on the Euclidean distance to the LMI analytic center. Finally, we test the quality of these bounds numerically in the context of interior point methods termination criteria.