Search

By employing a weighted Frobenius norm with a positive definite matrix ω , we introduce natural generalizations of the famous Böttcher-Wenzel (BW) inequality. Based on the combination of the weighted Frobenius norm ▪ and the standard Frobenius norm ▪, there are exactly five possible generalizations, labeled (i) through (v), for the bounds on the norms of the commutator [ A , B ] : = A B − B A. In this paper, we establish the tight bounds for cases (iii) and (v), and propose conjectures regarding the tight bounds for cases (i) and (ii). Additionally, the tight bound for case (iv) is derived as a corollary of case (i). All these bounds (i)-(v) serve as generalizations of the BW inequality. The conjectured bounds for cases (i) and (ii) (and thus also (iv)) are numerically supported for matrices up to size n = 15. Proofs are provided for n = 2 and certain special cases. Interestingly, we find applications of these bounds in quantum physics, particularly in the contexts of the uncertainty relation and open quantum dynamics. [ABSTRACT FROM AUTHOR]

This paper provides additive perturbation bounds for the generalized core inverse $ A^{d,\dag } $ A d , † of a complex matrix A. Given perturbation matrix E, we first present upper bounds for $ \frac {\|(A+E)^{d,\dag }-A^{d,\dag }\|_{2}}{\|A^{d,\dag }\|_{2}} $ ‖ (A + E) d , † − A d , † ‖ 2 ‖ A d , † ‖ 2 under the condition $ \mathcal {R}(E)\subseteq \mathcal {R}(A^k), \mathcal {N}(A^{k}A^{\dag })\subseteq \mathcal {N}(E), {\rm where}\ k=\mathrm {ind}(A) $ R (E) ⊆ R (A k) , N (A k A †) ⊆ N (E) , where k = ind (A) , and then under the condition $ (A+E)(A+E)^{d,\dag }=AA^{d,\dag } $ (A + E) (A + E) d , † = A A d , † . We then explore an integral formula for the generalized core inverse of a perturbed matrix associated with a semi-stable matrix. Furthermore, a perturbation bound for $ \|(A+E)^{d,\dag }-A^{d,\dag }\|_{F} $ ‖ (A + E) d , † − A d , † ‖ F is obtained without conditions. Sufficient conditions for the continuity of the generalized core inverse can be obtained as a result. Finally, numerical examples are presented to illustrate the derived bounds. [ABSTRACT FROM AUTHOR]

The matrix-valued Allen-Cahn (MAC) equation was first introduced as a model problem of finding the stationary points of an energy for orthogonal matrix-valued functions and has attracted much attention in recent years. It is well known that the MAC equation satisfies the maximum bound principle (MBP) with respect to either the matrix 2-norm or the Frobenius norm, which plays a key role in understanding the physical meaning and the wellposedness of the model. To preserve this property, we extend the explicit integrating factor Runge-Kutta (IFRK) method to the MAC equation. Moreover, we construct a new three-stage third-order and a new four-stage fourth-order IFRK schemes based on the classical Runge-Kutta schemes. Under a reasonable time-step constraint, we prove the MBP preservation of the IFRK method with respect to the matrix 2-norm, based on which, we further establish their optimal error estimates in the matrix 2-norm. Although numerical results indicate that the IFRK method preserves the MBP with respect to the Frobenius norm, a detailed analysis shows that it is hard to prove this preservation by using the same approach for the case of 2-norm. Several numerical experiments are carried out to test the convergence of the IFRK schemes and to verify the MBP preservation with respect to the matrix 2-norm and the Frobenius norm, respectively. Energy stability is also observed, which clearly indicates the orthogonality of the stationary solution of the MAC equation. In addition, we simulate the coarsening dynamics to verify the motion law of the interface for different initial conditions. [ABSTRACT FROM AUTHOR]

For d ≥ 2 , let X be a random vector having a Bingham distribution on S d - 1 , the unit sphere centered at the origin in R d , and let Σ denote the symmetric matrix parameter of the distribution. Let Ψ (Σ) be the normalizing constant of the distribution and let ∇ Ψ d (Σ) be the matrix of first-order partial derivatives of Ψ (Σ) with respect to the entries of Σ . We derive complete asymptotic expansions for Ψ (Σ) and ∇ Ψ d (Σ) , as d → ∞ ; these expansions are obtained subject to the growth condition that ‖ Σ ‖ , the Frobenius norm of Σ , satisfies ‖ Σ ‖ ≤ γ 0 d r / 2 for all d, where γ 0 > 0 and r ∈ [ 0 , 1) . Consequently, we obtain for the covariance matrix of X an asymptotic expansion up to terms of arbitrary degree in Σ . Using a range of values of d that have appeared in a variety of applications of high-dimensional spherical data analysis, we tabulate the bounds on the remainder terms in the expansions of Ψ (Σ) and ∇ Ψ d (Σ) and we demonstrate the rapid convergence of the bounds to zero as r decreases. [ABSTRACT FROM AUTHOR]

For each pair of matrices A and B with the same order, let ‖ A − B ‖ F denote their Frobenius distance. This paper deals mainly with the Frobenius distances from projections to an idempotent matrix. For every idempotent Q ∈ C n × n , a projection m (Q) called the matched projection can be induced. It is proved that m (Q) is the unique projection whose Frobenius distance away from Q takes the minimum value among all the Frobenius distances from projections to Q , while I n − m (Q) is the unique projection whose Frobenius distance away from Q takes the maximum value. Furthermore, it is proved that for every number α between the minimum value and the maximum value, there exists a projection P whose Frobenius distance away from Q takes the value α. Based on the above characterization of the minimum distance, some Frobenius norm upper bounds and lower bounds of ‖ P − Q ‖ F are derived under the condition of P Q = Q on a projection P and an idempotent Q. [ABSTRACT FROM AUTHOR]

Fully capturing valid complementary information in multi-view data enhances the connection between similar data points and weakens the correlation between different data point categories. In this paper, we propose a new multi-view clustering via dual-norm and Hilbert-Schmidt independence criterion (HSIC) induction (MCDHSIC) approach, which can enhance the complementarity, reduce the redundancy between multi-view representations, and improve the accuracy of the clustering results. This model uses the HSIC as the diversity regularization term to capture the nonlinear relationship between different views. In addition, l1-norm and Frobenius norm constraints are imposed to obtain a subspace representation matrix with inter-class sparsity and intra-class consistency. Moreover, we also designed a valuable approach to optimizing the proposed model and theoretically analyzing the convergence of the MCDHSIC method. The results of extensive experiments conducted on five challenging data sets show that the proposed method objectively achieves a highly competent performance compared with several other state-of-the-art multi-view clustering methods. [ABSTRACT FROM AUTHOR]

Time synchronization is a critical prerequisite for unmanned aerial vehicle ad hoc networks (UANETs) to facilitate navigation and positioning, formation control, and data fusion. However, given the dynamic changes in UANETs, improving the convergence speeds of distributed consensus time synchronization algorithms with only local information poses a major challenge. To address this challenge, this study first establishes a convex model on the basis of graph theory and relevant theories of random matrices to approximate the original problem. Subsequently, three acceleration schemes for consensus algorithms are derived by minimizing the Frobenius norm of the iteration matrix. Additionally, this study provides a new upper bound for constant communication weights and discusses the limitations of existing metrics used to measure the convergence speeds of consensus algorithms. Finally, the proposed schemes are compared with existing ones through simulation. Our results indicate that the three proposed schemes can achieve faster convergence while maintaining high-precision synchronization in scenarios with static or known topological structures of networks. In scenarios where the topological structure of a UANET is time-varying and unknown, the scheme proposed in this paper achieves the fastest convergence speed.

It is difficult to assess the quality of a fitted model in unsupervised learning problems. Latent variable models, such as variational autoencoders and Gaussian mixture models, are often trained with likelihood-based approaches. In the scope of Goodhart's law, when a metric becomes a target it ceases to be a good metric and therefore we should not use likelihood to assess the quality of the fit of these models. The solution we propose is a new metric for model comparison or regularization that relies on moments. The key idea is to study the difference between the data moments and the model moments using a matrix norm, such as the Frobenius norm. We first show how to use this new metric for model comparison and then for regularization. We show that our proposed metric is faster to compute and has a smaller variance than the commonly used procedure of drawing samples from the fitted distribution. We conclude this article with a demonstration of both applications and we discuss our findings and future work. [ABSTRACT FROM AUTHOR]

In this paper, we present two lower bounds for the smallest singular value of nonsingular matrices. Moreover, we illustrate with numerical examples that these bounds are better than the existing bounds. [ABSTRACT FROM AUTHOR]

In this paper, we prove a norm inequality for three real 2×2 matrices conjectured by L. László [3] recently, which is a generalization of the famous Böttcher-Wenzel inequality. [ABSTRACT FROM AUTHOR]

In this paper, we investigate Zermelo's navigation problem for some special rotation surfaces. In this respect, we find some Randers-type metrics for these rotation surfaces. Furthermore, we get the H-distortion for the metric induced by surfaces.

Matrix commutator and anticommutator play an important role in mathematics, mathematical physic, and quantum physic. The commutator and anticommutator of two n × n complex matrices A and B are defined by [A,B] = AB - BA and (A,B) = AB + BA, respectively. Cauchy-Toeplitz matrix and exchange matrix are two of the special matrices and they have excellent properties. In this study, we mainly focus on Frobenius norm of the commutator of Cauchy-Toeplitz matrix and exchange matrix. Moreover, we give upper and lower bounds for the Frobenius norm of the commutator of Cauchy-Toeplitz matrix and exchange matrix. [ABSTRACT FROM AUTHOR]

In this paper, we investigate Zermelo's navigation problem for some special rotation surfaces. In this respect, we find some Randers-type metrics for these rotation surfaces. Furthermore, we get the H-distortion for the metric induced by surfaces. [ABSTRACT FROM AUTHOR]

We extend a result proved by Dirr and Wimmer [IEEE Trans. Automat. Control 52(2007)] for polynomials to the matrix valued analytic functions and thereby obtain generalizations of some well-known results concerning the zero free regions of a class of analytic functions. [ABSTRACT FROM AUTHOR]

In this paper, the structured distance in the Frobenius norm of a real irreducible tridiagonal 2-Toeplitz matrix T to normality is determined. In the first part of the paper, we introduced the normal form a real tridiagonal 2-Toeplitz matrix. The eigenvalues of a real tridiagonal 2-Toeplitz matrix are known. In the second part of this paper, we discuss eigenvalues of a normal real tridiagonal 2-Toeplitz matrix. From this, we used to investigate the structured distance to the closest normal matrix and departure from normality. In the last part of this paper, the numerical range of normal real tridiagonal 2-Toeplitz matrices is presented and the special case of normal real tridiagonal 2-Toeplitz matrices also is considered. [ABSTRACT FROM AUTHOR]

One of the fundamental problems in multilinear algebra, the minimum ratio between the spectral and Frobenius norms of tensors, has received considerable attention in recent years. While most values are unknown for real and complex tensors, the asymptotic order of magnitude and tight lower bounds have been established. However, little is known about nonnegative tensors. In this paper, we present an almost complete picture of the ratio for nonnegative tensors. In particular, we provide a tight lower bound that can be achieved by a wide class of nonnegative tensors under a simple necessary and sufficient condition, which helps to characterize the extreme tensors and obtain results such as the asymptotic order of magnitude. We show that the ratio for symmetric tensors is no more than that for general tensors multiplied by a constant depending only on the order of tensors, hence determining the asymptotic order of magnitude for real, complex, and nonnegative symmetric tensors. We also find that the ratio is in general different from the minimum ratio between the Frobenius and nuclear norms for nonnegative tensors, a sharp contrast to the case for real tensors and complex tensors. [ABSTRACT FROM AUTHOR]

In this work, by introducing a scalar matrix αI, we transform the complex symmetric indefinite linear systems (W+iT)x=b into a block two-by-two complex equations equivalently, and propose an efficient relaxed shift-splitting (ERSS) preconditioner. By adopting the relaxation technique, the ERSS preconditioner is not only a computational advantage but also closer to the original two-by-two of complex coefficient matrix. The eigenvalue distributions of the preconditioned matrix are analysed. An efficient and practical formula for computing the parameter value α is also derived by computing the Frobenius norm of symmetric indefinite matrix T. Numerical examples on a few model problems are illustrated to verify the performances of the ERSS preconditioner.

In this work, we are concerned with (R, S) – conjugate solutions to coupled Sylvester complex matrix equations with conjugate of two unknowns. When the considered two matrix equations are consistent, it is demonstrated that the solutions can be obtained by utilizing this iterative algorithm for any initial arbitrary [Formula: see text] – conjugate matrices [Formula: see text]. A necessary and sufficient condition is established to guarantee that the proposed method converges to the [Formula: see text] – conjugate solutions. Finally, two numerical examples are provided to demonstrate the efficiency of the described iterative technique.

In this paper, we introduce a one-parameter generalization of the famous Böttcher-Wenzel (BW) inequality in terms of a q -deformed commutator. For n × n matrices A and B , we consider the inequality Re 〈 [ B , A ] , [ B , A ] q 〉 ≤ c (q) ‖ A ‖ 2 ‖ B ‖ 2 , where 〈 A , B 〉 = tr (A ⁎ B) is the Hilbert-Schmidt inner product, ‖ A ‖ is the Frobenius norm, [ A , B ] = A B − B A is the commutator, and [ A , B ] q = A B − q B A is the q -deformed commutator. We prove that when n = 2 , or when A is normal with any size n , the optimal bound is given by c (q) = (1 + q) + 2 (1 + q 2) 2. We conjecture that this is also true for any matrices, and this conjecture is perfectly supported for n up to 15 by numerical optimization. When q = 1 , this inequality is exactly the BW inequality. When q = 0 , this inequality leads the sharp bound for the r -function which is recently derived for the application to universal constraints of relaxation rates in open quantum dynamics. [ABSTRACT FROM AUTHOR]

In this work, we are concerned with (R, S) – conjugate solutions to coupled Sylvester complex matrix equations with conjugate of two unknowns. When the considered two matrix equations are consistent, it is demonstrated that the solutions can be obtained by utilizing this iterative algorithm for any initial arbitrary (R , S) – conjugate matrices V 1 , W 1 . A necessary and sufficient condition is established to guarantee that the proposed method converges to the (R , S) – conjugate solutions. Finally, two numerical examples are provided to demonstrate the efficiency of the described iterative technique. [ABSTRACT FROM AUTHOR]

In this paper, the multiplicative and additive perturbation bounds of core inverse under the Frobenius norm are presented. Moreover, we also get the additive perturbation bound of core-EP inverse under the Frobenius norm. Finally, numerical examples are given to illustrate that the multiplicative perturbation bound of core inverse is superior than the additive perturbation bound of core inverse under certain circumstances. [ABSTRACT FROM AUTHOR]

The sensitivity of the solution of the periodic continuous Sylvester equation AkXkDk − BkXk⊕1Ck = Ek, k = 0,1,...,K − 1, depends on the separation of two periodic matrix pairs { (A k , B k) } k = 0 K − 1 and { (C k , D k) } k = 0 K − 1 . In this paper, we derive upper and lower bounds of this separation in terms of periodic generalized Schur decompositions. The theoretical results are verified by a numerical example. [ABSTRACT FROM AUTHOR]

For two n × n complex matrices A and B , we define the q -deformed commutator as [ A , B ] q : = A B − q B A 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 | | F 2 ≤ (1 + q 2) | | A | | F 2 | | B | | F 2 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. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Signal Processing is the property of Journal of Signal Processing and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we investigate the genuine entanglement in tripartite systems based on partial transposition and the norm of correlation tensors of the density matrices. We first derive an analytical sufficient criterion to detect genuine entanglement of tripartite qubit states combining with the partial transposition of the density matrices. Then, we use the norm of correlation tensors to study genuine entanglement for tripartite qudit quantum states and obtain a genuine entanglement criterion by constructing certain matrices. With detailed examples, our results are seen to be able to detect more genuine tripartite entangled states than previous studies. [ABSTRACT FROM AUTHOR]

A low complexity scheme is developed to design the sparse linear phase FIR filters based on the improved iteratively re-weighted orthogonal matching pursuit(OMP) algorithm. The scheme employs the Frobenius norm to simplify searching for the desired sparse FIR filter at each iteration. Furthermore, a log function is introduced to enhance the weight updating process at each iteration. The simulation results demonstrate that our scheme can effectively generate the sparse linear phase FIR filters which satisfy the various given design specifications. [ABSTRACT FROM AUTHOR]

In this paper, we obtain two new lower bounds for the smallest singular value of nonsingular matrices which is better than the bound presented by Zou [1], Lin and Xie [2] under certain circumstances. [ABSTRACT FROM AUTHOR]

Motivated by a spectral analysis of the generator of a completely positive trace-preserving semigroup, we analyze the real functional A , B ∈ M n (C) → r (A , B) = 1 2 (〈 [ B , A ] , B A 〉 + 〈 [ B , A ⁎ ] , B A ⁎ 〉) ∈ R where 〈 A , B 〉 : = tr (A ⁎ B) is the Hilbert-Schmidt inner product, and [ A , B ] : = A B − B A is the commutator. In particular we discuss upper and lower bounds of the form c − ‖ A ‖ 2 ‖ B ‖ 2 ≤ r (A , B) ≤ c + ‖ A ‖ 2 ‖ B ‖ 2 where ‖ A ‖ is the Frobenius norm. We prove that the optimal upper and lower bounds are given by c ± = 1 ± 2 2. If A is restricted to be traceless, the bounds are further improved to be c ± = 1 ± 2 (1 − 1 n) 2. Interestingly, these upper bounds, especially the latter one, provide new constraints on relaxation rates for the quantum dynamical semigroup tighter than previously known constraints in the literature. A relation with the Böttcher-Wenzel inequality is also discussed. [ABSTRACT FROM AUTHOR]

We characterize optimal rank-1 matrix approximations with Hankel or Toeplitz structure with regard to two different norms, the Frobenius norm and the spectral norm, in a new way. More precisely, we show that these rank-1 matrix approximation problems can be solved by maximizing special rational functions. Our approach enables us to show that the optimal solutions with respect to these two norms have completely different structure and only coincide in the trivial case when the singular value decomposition already provides an optimal rank-1 approximation with the desired Hankel or Toeplitz structure. We also prove that the Cadzow algorithm for structured low-rank approximations always converges to a fixed point in the rank-1 case. However, it usually does not converge to the optimal solution, neither with regard to the Frobenius norm nor the spectral norm. [ABSTRACT FROM AUTHOR]

Modern investigation techniques (e.g., metabolomic, proteomic, lipidomic, genomic, transcriptomic, phenotypic), allow to collect high-dimensional data, where the number of observations is smaller than the number of features. In such cases, for statistical analyzing, standard methods cannot be applied or lead to ill-conditioned estimators of the covariance matrix. To analyze the data, we need an estimator of the covariance matrix with good properties (e.g., positive definiteness), and therefore covariance matrix identification is crucial. The paper presents an approach to determine the block-structured estimator of the covariance matrix based on an example of metabolomic data on the drought resistance of barley. This method can be used in many fields of science, e.g., in agriculture, medicine, food and nutritional sciences, toxicology, functional genomics and nutrigenomics. [ABSTRACT FROM AUTHOR]

Let A be an n × n complex matrix. The spread of the matrix A, denotes by sp(A), is the largest distance between any two eigenvalues of A. It was introduced by L. Mirsky. In the present work we give new lower and upper bounds for the spread of A. Some of these bounds are explicit functions of the entries of A, others involve the rows and the columns of A. Bounds for the spread of the power of A are obtained too. Block matrices arise in many aspects of matrix theory, we present some bounds for the spread of the general 2 × 2 block matrix. [ABSTRACT FROM AUTHOR]

In this work we give new upper and lower bounds for the numerical radius of a complex square matrix A using the entries and the trace of A. [ABSTRACT FROM AUTHOR]

A correction, along with an improvement, is made on a recent joint work of the second author and the third author. An optimal perturbation bound is also clarified for certain 2 × 2 Hermitian matrices. [ABSTRACT FROM AUTHOR]

Since the nuclear norm may lead to suboptimal solutions to rank minimization problems, many nonconvex surrogates have been proposed to provide better rank function approximations. In this paper, we use the ratio of the nuclear norm and the Frobenius norm, denoted as N/F, as a new nonconvex surrogate of the rank function. The N/F can provide a close approximation for the matrix rank. In particular, the N/F is the same as the rank for rank 1 matrices. Furthermore, compared to other popular nonconvex approximations, the N/F model has no extra parameters to tune. Theoretically, we establish the exact recovery condition and the stable recovery condition for the N/F minimization problem with or without noise, respectively. Computationally, we use the alternating direction method of multipliers with a specific variable-splitting scheme to solve the N/F minimization problem. What is more, we incorporate a box constraint in the N/F model, and propose the ADMM-box to solve it. The convergence analysis of ADMM and ADMM-box are also given. Extensive experiments on both synthetic data and color images verify the effectiveness of our proposed methods. [ABSTRACT FROM AUTHOR]

We review a general parameterization of order-3 magic squares due to Lucas and sequentially compound it to produce parameterized magic square matrices of order n = 3 ℓ (ℓ = 2 , 3 , ...). Expressions are found for the singular values of the compound matrices. From the singular values, the Frobenius norm, and consideration of the balanced ternary numeral system, we determine numerical values for the parameters in a compound Lucas square so that it is natural (having elements 0 , 1 , ... , n 2 − 1). A less general parameterization due to Frierson is related to Lucas' parameterization and our results specialize to it, complementing previous results. [ABSTRACT FROM AUTHOR]

Motivated by inferring cellular signaling networks using noisy flow cytometry data, we develop procedures to draw inference for Bayesian networks based on error-prone data. Two methods for inferring causal relationships between nodes in a network are proposed based on penalized estimation methods that account for measurement error and encourage sparsity. We discuss consistency of the proposed network estimators and develop an approach for selecting the tuning parameter in the penalized estimation methods. Empirical studies are carried out to compare the proposed methods with a naive method that ignores measurement error. Finally, we apply these methods to infer signaling networks using single cell flow cytometry data. [ABSTRACT FROM AUTHOR]

The precision of the camera calibration is one of the key factors that affect attitude measurement accuracy in many computer vision tasks. This study proposes a new calibration approach for binocular cameras. Firstly, based on singular value decomposition, the best transformation matrix to the essential matrix is approximated as the initial guess, which is solved in using the Frobenius norm. Secondly, the initial guess is refined through maximum likelihood estimation. A new calculating expression is derived for computing the relative position matrix of the binocular cameras. The Levenberg–Marquardt algorithm is then implemented to refine the initial guess. Large sets of synthesised and real point correspondences were tested to demonstrate the validity of the proposed method. Extensive experiments demonstrated that the proposed method outperforms the state‐of‐the‐art methods. The error rate of the proposed method was 0.5% for the length test and about 1% for the angle test at a range of 1 m. This method can advance three‐dimensional (3D) computer vision one additional step from laboratory environments to real‐world use. [ABSTRACT FROM AUTHOR]