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Journal articles on the topic 'Frobenius-Norm'

Author: Grafiati

Published: 4 June 2021

Last updated: 14 February 2022

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1

Cheng, Che-Man, Seak-Weng Vong, and David Wenzel. "Commutators with maximal Frobenius norm." Linear Algebra and its Applications 432, no. 1 (January 2010): 292–306. http://dx.doi.org/10.1016/j.laa.2009.08.008.

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2

Peng, Yang. "Inequalities for the Frobenius norm." Journal of Mathematical Inequalities, no. 2 (2015): 493–98. http://dx.doi.org/10.7153/jmi-09-43.

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3

Peng, Xi, Canyi Lu, Zhang Yi, and Huajin Tang. "Connections Between Nuclear-Norm and Frobenius-Norm-Based Representations." IEEE Transactions on Neural Networks and Learning Systems 29, no. 1 (January 2018): 218–24. http://dx.doi.org/10.1109/tnnls.2016.2608834.

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4

Böttcher, Albrecht, and David Wenzel. "The Frobenius norm and the commutator." Linear Algebra and its Applications 429, no. 8-9 (October 2008): 1864–85. http://dx.doi.org/10.1016/j.laa.2008.05.020.

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5

Chellaboina, V., and W. M. Haddad. "Is the Frobenius matrix norm induced?" IEEE Transactions on Automatic Control 40, no. 12 (1995): 2137–39. http://dx.doi.org/10.1109/9.478340.

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6

Toutounian, F., D. Khojasteh Salkuyeh, and M. Mojarrab. "LSMR Iterative Method for General Coupled Matrix Equations." Journal of Applied Mathematics 2015 (2015): 1–12. http://dx.doi.org/10.1155/2015/562529.

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By extending the idea of LSMR method, we present an iterative method to solve the general coupled matrix equations∑k=1qAikXkBik=Ci,i=1,2,…,p, (including the generalized (coupled) Lyapunov and Sylvester matrix equations as special cases) over some constrained matrix groups(X1,X2,…,Xq), such as symmetric, generalized bisymmetric, and(R,S)-symmetric matrix groups. By this iterative method, for any initial matrix group(X1(0),X2(0),…,Xq(0)), a solution group(X1*,X2*,…,Xq*)can be obtained within finite iteration steps in absence of round-off errors, and the minimum Frobenius norm solution or the minimum Frobenius norm least-squares solution group can be derived when an appropriate initial iterative matrix group is chosen. In addition, the optimal approximation solution group to a given matrix group(X¯1,X¯2,…,X¯q)in the Frobenius norm can be obtained by finding the least Frobenius norm solution group of new general coupled matrix equations. Finally, numerical examples are given to illustrate the effectiveness of the presented method.

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7

Yin, Feng, and Guang-Xin Huang. "An Iterative Algorithm for the Least Squares Generalized Reflexive Solutions of the Matrix Equations." Abstract and Applied Analysis 2012 (2012): 1–18. http://dx.doi.org/10.1155/2012/857284.

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The generalized coupled Sylvester systems play a fundamental role in wide applications in several areas, such as stability theory, control theory, perturbation analysis, and some other fields of pure and applied mathematics. The iterative method is an important way to solve the generalized coupled Sylvester systems. In this paper, an iterative algorithm is constructed to solve the minimum Frobenius norm residual problem: min over generalized reflexive matrix . For any initial generalized reflexive matrix , by the iterative algorithm, the generalized reflexive solution can be obtained within finite iterative steps in the absence of round-off errors, and the unique least-norm generalized reflexive solution can also be derived when an appropriate initial iterative matrix is chosen. Furthermore, the unique optimal approximate solution to a given matrix in Frobenius norm can be derived by finding the least-norm generalized reflexive solution of a new corresponding minimum Frobenius norm residual problem: with , . Finally, several numerical examples are given to illustrate that our iterative algorithm is effective.

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8

Cui, Xiangzhao, Chun Li, Jine Zhao, Li Zeng, Defei Zhang, and Jianxin Pan. "Covariance structure regularization via Frobenius-norm discrepancy." Linear Algebra and its Applications 510 (December 2016): 124–45. http://dx.doi.org/10.1016/j.laa.2016.08.013.

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9

Huckle, T., and A. Kallischko. "Frobenius norm minimization and probing for preconditioning." International Journal of Computer Mathematics 84, no. 8 (August 2007): 1225–48. http://dx.doi.org/10.1080/00207160701396387.

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10

González, Luis, and Antonio Suárez. "Improving approximate inverses based on Frobenius norm minimization." Applied Mathematics and Computation 219, no. 17 (May 2013): 9363–71. http://dx.doi.org/10.1016/j.amc.2013.03.057.

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11

Weinmann, A. "Gain-scheduled H∞ controller with minimum Frobenius norm." e & i Elektrotechnik und Informationstechnik 121, no. 2 (February 2004): 72–76. http://dx.doi.org/10.1007/bf03054970.

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12

Baksalary, Oskar Maria, and Götz Trenkler. "On subspace distances determined by the Frobenius norm." Linear Algebra and its Applications 448 (May 2014): 245–63. http://dx.doi.org/10.1016/j.laa.2014.01.017.

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13

Zhang, Haixian, Zhang Yi, and Xi Peng. "fLRR: fast low‐rank representation using Frobenius‐norm." Electronics Letters 50, no. 13 (June 2014): 936–38. http://dx.doi.org/10.1049/el.2014.1396.

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14

Custódio, A. L., H. Rocha, and L. N. Vicente. "Incorporating minimum Frobenius norm models in direct search." Computational Optimization and Applications 46, no. 2 (August 6, 2009): 265–78. http://dx.doi.org/10.1007/s10589-009-9283-0.

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15

del Olmo, V., and R. Fuster. "Some iterative methods related to Frobenius norm minimization." Computers & Mathematics with Applications 22, no. 10 (1991): 121–26. http://dx.doi.org/10.1016/0898-1221(91)90199-e.

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16

ik Choi, Daes. "A remark on “Inequalities for the Frobenius norm”." Journal of Mathematical Inequalities, no. 3 (2016): 899–900. http://dx.doi.org/10.7153/jmi-10-73.

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17

Thoiyab, N. Mohamed, P. Muruganantham, Grienggrai Rajchakit, Nallappan Gunasekaran, Bundit Unyong, Usa Humphries, Pramet Kaewmesri, and Chee Peng Lim. "Global Stability Analysis of Neural Networks with Constant Time Delay via Frobenius Norm." Mathematical Problems in Engineering 2020 (October 12, 2020): 1–14. http://dx.doi.org/10.1155/2020/4321312.

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This paper deals with the global asymptotic robust stability (GARS) of neural networks (NNs) with constant time delay via Frobenius norm. The Frobenius norm result has been utilized to find a new sufficient condition for the existence, uniqueness, and GARS of equilibrium point of the NNs. Some suitable Lyapunov functional and the slope bounded functions have been employed to find the new sufficient condition for GARS of NNs. Finally, we give some comparative study of numerical examples for explaining the advantageous of the proposed result along with the existing GARS results in terms of network parameters.

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18

Merikoski, Jorma K., Pentti Haukkanen, Mika Mattila, and Timo Tossavainen. "On the spectral and Frobenius norm of a generalized Fibonacci r-circulant matrix." Special Matrices 6, no. 1 (January 1, 2018): 23–36. http://dx.doi.org/10.1515/spma-2018-0003.

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Abstract Consider the recursion g0 = a, g1 = b, gn = gn−1 + gn−2, n = 2, 3, . . . . We compute the Frobenius norm of the r-circulant matrix corresponding to g0, . . . , gn−1. We also give three lower bounds (with equality conditions) for the spectral norm of this matrix. For this purpose, we present three ways to estimate the spectral norm from below in general.

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19

Mieldzioc, Adam, Monika Mokrzycka, and Aneta Sawikowska. "Covariance regularization for metabolomic data on the drought resistance of barley." Biometrical Letters 56, no. 2 (December 1, 2019): 165–81. http://dx.doi.org/10.2478/bile-2019-0010.

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SummaryModern chromatography largely uses the technique of gas chromatography coupled with mass spectrometry (GC–MS). For a set of data concerning the drought resistance of barley, the problem of the characterization of a covariance structure is investigated with the use of two methods. The first is based on the Frobenius norm and the second on the entropy loss function. For the four considered covariance structures – compound symmetry, three-diagonal and penta-diagonal Toeplitz and autoregression of order one – the Frobenius norm indicates the compound symmetry matrix and autoregression of order one as the most relevant, whilst the entropy loss function gives a slight indication in favor of the compound symmetry structure.

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20

Li, Ning, Qing-Wen Wang, and Jing Jiang. "An Efficient Algorithm for the Reflexive Solution of the Quaternion Matrix EquationAXB+CXHD=F." Journal of Applied Mathematics 2013 (2013): 1–14. http://dx.doi.org/10.1155/2013/217540.

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We propose an iterative algorithm for solving the reflexive solution of the quaternion matrix equationAXB+CXHD=F. When the matrix equation is consistent over reflexive matrixX, a reflexive solution can be obtained within finite iteration steps in the absence of roundoff errors. By the proposed iterative algorithm, the least Frobenius norm reflexive solution of the matrix equation can be derived when an appropriate initial iterative matrix is chosen. Furthermore, the optimal approximate reflexive solution to a given reflexive matrixX0can be derived by finding the least Frobenius norm reflexive solution of a new corresponding quaternion matrix equation. Finally, two numerical examples are given to illustrate the efficiency of the proposed methods.

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21

Zhou, Zhongli, and Guangxin Huang. "An Iterative Algorithm for the Reflexive Solution of the General Coupled Matrix Equations." Scientific World Journal 2013 (2013): 1–15. http://dx.doi.org/10.1155/2013/952974.

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The general coupled matrix equations (including the generalized coupled Sylvester matrix equations as special cases) have numerous applications in control and system theory. In this paper, an iterative algorithm is constructed to solve the general coupled matrix equations over reflexive matrix solution. When the general coupled matrix equations are consistent over reflexive matrices, the reflexive solution can be determined automatically by the iterative algorithm within finite iterative steps in the absence of round-off errors. The least Frobenius norm reflexive solution of the general coupled matrix equations can be derived when an appropriate initial matrix is chosen. Furthermore, the unique optimal approximation reflexive solution to a given matrix group in Frobenius norm can be derived by finding the least-norm reflexive solution of the corresponding general coupled matrix equations. A numerical example is given to illustrate the effectiveness of the proposed iterative algorithm.

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22

Jin, Ming, Youming Li, and Qi Zeng. "Modified covariance Frobenius norm detector for cognitive radio systems." IEICE Electronics Express 8, no. 10 (2011): 762–66. http://dx.doi.org/10.1587/elex.8.762.

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23

Yuan, Shi-Fang, Yi-Bin Yu, Ming-Zhao Li, and Hua Jiang. "A direct method to Frobenius norm-based matrix regression." International Journal of Computer Mathematics 97, no. 9 (September 26, 2019): 1767–80. http://dx.doi.org/10.1080/00207160.2019.1668558.

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24

Zou, Limin. "On a conjecture concerning the Frobenius norm of matrices." Linear and Multilinear Algebra 60, no. 1 (January 2012): 27–31. http://dx.doi.org/10.1080/03081087.2010.518145.

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25

Rodriguez-Lujan, Luis, Pedro Larrañaga, and Concha Bielza. "Frobenius Norm Regularization for the Multivariate Von Mises Distribution." International Journal of Intelligent Systems 32, no. 2 (July 9, 2016): 153–76. http://dx.doi.org/10.1002/int.21834.

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26

Dokmanić, Ivan, and Rémi Gribonval. "Concentration of the Frobenius Norm of Generalized Matrix Inverses." SIAM Journal on Matrix Analysis and Applications 40, no. 1 (January 2019): 92–121. http://dx.doi.org/10.1137/17m1145409.

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27

Kuzma, Bojan, and Tatjana Petek. "A note on Frobenius norm preservers of Jordan product." Operators and Matrices, no. 4 (2013): 915–25. http://dx.doi.org/10.7153/oam-07-51.

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28

Liu, Wei-Hui, Ze-Jia Xie, and Xiao-Qing Jin. "Frobenius norm inequalities of commutators based on different products." Operators and Matrices, no. 2 (2021): 645–57. http://dx.doi.org/10.7153/oam-2021-15-43.

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29

Cai, Yun. "Minimization of the difference of Nuclear and Frobenius norms for noisy low rank matrix recovery." International Journal of Wavelets, Multiresolution and Information Processing 18, no. 02 (September 19, 2019): 1950056. http://dx.doi.org/10.1142/s0219691319500565.

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This paper considers recovery of matrices that are low rank or approximately low rank from linear measurements corrupted with additive noise. We study minimization of the difference of Nuclear and Frobenius norms (abbreviated as [Formula: see text] norm) as a nonconvex and Lipschitz continuous metric for solving this noisy low rank matrix recovery problem. We mainly study two types of bounded observation noisy low rank matrix recovery problems, including the [Formula: see text]-norm bounded noise and the Dantizg Selector noise. Based on the matrix restricted isometry property (abbreviated as M-RIP), we prove that this [Formula: see text] norm-based minimization method can stably recover a (approximately) low rank matrix in the two types bounded noisy low rank matrix recovery problems. In addition, we use the truncated difference of Nuclear and Frobenius norms (denoted as the truncated [Formula: see text] norm) to recover a low rank matrix when the observation noise is the Dantizg Selector noise. We give the stable recovery result for this truncated [Formula: see text] norm minimization in Dantizg Selector noise case when the linear measurement map satisfies the M-RIP condition.

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30

Jiang, Zhaolin, Jinjiang Yao, and Fuliang Lu. "On Skew Circulant Type Matrices Involving Any Continuous Fibonacci Numbers." Abstract and Applied Analysis 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/483021.

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Circulant and skew circulant matrices have become an important tool in networks engineering. In this paper, we consider skew circulant type matrices with any continuous Fibonacci numbers. We discuss the invertibility of the skew circulant type matrices and present explicit determinants and inverse matrices of them by constructing the transformation matrices. Furthermore, the maximum column sum matrix norm, the spectral norm, the Euclidean (or Frobenius) norm, and the maximum row sum matrix norm and bounds for the spread of these matrices are given, respectively.

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31

Jiang, Zhaolin, and Yunlan Wei. "Skew Circulant Type Matrices Involving the Sum of Fibonacci and Lucas Numbers." Abstract and Applied Analysis 2015 (2015): 1–9. http://dx.doi.org/10.1155/2015/951340.

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Skew circulant and circulant matrices have been an ideal research area and hot issue for solving various differential equations. In this paper, the skew circulant type matrices with the sum of Fibonacci and Lucas numbers are discussed. The invertibility of the skew circulant type matrices is considered. The determinant and the inverse matrices are presented. Furthermore, the maximum column sum matrix norm, the spectral norm, the Euclidean (or Frobenius) norm, the maximum row sum matrix norm, and bounds for the spread of these matrices are given, respectively.

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32

Yuan, Haoliang. "Robust patch-based sparse representation for hyperspectral image classification." International Journal of Wavelets, Multiresolution and Information Processing 15, no. 03 (March 20, 2017): 1750028. http://dx.doi.org/10.1142/s021969131750028x.

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Sparse representation classification (SRC) has been successfully applied into hyperspectral image (HSI). A test sample (pixel) can be linearly represented by a few training samples of the training set. The class label of the test sample is then decided by the reconstruction residuals. To incorporate the spatial information to improve the classification performance, a patch matrix, which includes a spatial neighborhood set, is used to replace the original pixel. Generally, the objective function of the reconstruction residuals is represented as Frobenius-norm, which actually treats the elements in the reconstruction residuals in the same way. However, when a patch locates in the image edge, the samples in the patch may belong to different classes. Frobenius-norm is not suitable to compute the reconstruction residuals. In this paper, we propose a robust patch-based sparse representation classification (RPSRC) based on [Formula: see text]-norm. An iteration algorithm is given to compute RPSRC efficiently. Extensive experimental results on two real-life HSI datasets demonstrate the effectiveness of RPSRC.

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33

Yin, Feng, and Guang-Xin Huang. "An Iterative Algorithm for the Generalized Reflexive Solutions of the Generalized Coupled Sylvester Matrix Equations." Journal of Applied Mathematics 2012 (2012): 1–28. http://dx.doi.org/10.1155/2012/152805.

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An iterative algorithm is constructed to solve the generalized coupled Sylvester matrix equations(AXB-CYD,EXF-GYH)=(M,N), which includes Sylvester and Lyapunov matrix equations as special cases, over generalized reflexive matricesXandY. When the matrix equations are consistent, for any initial generalized reflexive matrix pair[X1,Y1], the generalized reflexive solutions can be obtained by the iterative algorithm within finite iterative steps in the absence of round-off errors, and the least Frobenius norm generalized reflexive solutions can be obtained by choosing a special kind of initial matrix pair. The unique optimal approximation generalized reflexive solution pair[X̂,Ŷ]to a given matrix pair[X0,Y0]in Frobenius norm can be derived by finding the least-norm generalized reflexive solution pair[X̃*,Ỹ*]of a new corresponding generalized coupled Sylvester matrix equation pair(AX̃B-CỸD,EX̃F-GỸH)=(M̃,Ñ), whereM̃=M-AX0B+CY0D,Ñ=N-EX0F+GY0H. Several numerical examples are given to show the effectiveness of the presented iterative algorithm.

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34

Zheng, Hao, and Lei Pan. "An Improved Block 2DPCA Face Recognition Algorithm with L1-Norm." Advanced Materials Research 457-458 (January 2012): 1077–82. http://dx.doi.org/10.4028/www.scientific.net/amr.457-458.1077.

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Most of the existing Block 2DPCA algorithms are based on Frobenius norm, which makes them sensitive to outliers. In this paper, we propose a new improved Block 2DPCA algorithm with L1-norm, which is robust to outliers. The experimental results of FERET face database indicated that the recognition performance of new algorithm is superior to Block 2DPCA, more robust than Block 2DPCA.

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35

Liu, Aijing, Guoliang Chen, and Xiangyun Zhang. "A New Method for the Bisymmetric Minimum Norm Solution of the Consistent Matrix EquationsA1XB1=C1,A2XB2=C2." Journal of Applied Mathematics 2013 (2013): 1–6. http://dx.doi.org/10.1155/2013/125687.

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We propose a new iterative method to find the bisymmetric minimum norm solution of a pair of consistent matrix equationsA1XB1=C1,A2XB2=C2. The algorithm can obtain the bisymmetric solution with minimum Frobenius norm in finite iteration steps in the absence of round-off errors. Our algorithm is faster and more stable than Algorithm 2.1 by Cai et al. (2010).

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36

Huang, Yan, Guisheng Liao, Yijian Xiang, Zhen Zhang, Jie Li, and Arye Nehorai. "Reweighted Nuclear Norm and Reweighted Frobenius Norm Minimizations for Narrowband RFI Suppression on SAR System." IEEE Transactions on Geoscience and Remote Sensing 57, no. 8 (August 2019): 5949–62. http://dx.doi.org/10.1109/tgrs.2019.2903579.

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37

Knirsch, Hanna, Markus Petz, and Gerlind Plonka. "Optimal rank-1 Hankel approximation of matrices: Frobenius norm and spectral norm and Cadzow's algorithm." Linear Algebra and its Applications 629 (November 2021): 1–39. http://dx.doi.org/10.1016/j.laa.2021.07.004.

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38

Sun, Ji-guang. "Condition numbers of algebraic Riccati equations in the Frobenius norm." Linear Algebra and its Applications 350, no. 1-3 (July 2002): 237–61. http://dx.doi.org/10.1016/s0024-3795(02)00294-x.

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39

Djouadi, S. M. "Comments on "Is the Frobenius Matrix Norm Induced?" [with reply]." IEEE Transactions on Automatic Control 48, no. 3 (March 2003): 518–19. http://dx.doi.org/10.1109/tac.2003.809159.

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40

Chellaboina, V., and W. M. Haddad. "Authors' reply - Comments on "Is the frobenius matrix norm induced?"." IEEE Transactions on Automatic Control 48, no. 3 (March 2003): 519–20. http://dx.doi.org/10.1109/tac.2003.809161.

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41

Ding, J., and N. Rhee. "BV-norm convergence of interpolation approximations for Frobenius-Perron operators." Journal of Physics: Conference Series 96 (February 1, 2008): 012049. http://dx.doi.org/10.1088/1742-6596/96/1/012049.

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42

Han, Lixing, Michael Neumann, and Michael Tsatsomeros. "Spectral Radii of Fixed Frobenius Norm Perturbations of Nonnegative Matrices." SIAM Journal on Matrix Analysis and Applications 21, no. 1 (January 1999): 79–92. http://dx.doi.org/10.1137/s0895479897318241.

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43

Wu, Yan-Dong, and Xu-Qing Liu. "A short note on the Frobenius norm of the commutator." Mathematical Notes 87, no. 5-6 (June 2010): 903–7. http://dx.doi.org/10.1134/s0001434610050305.

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44

Hu, Juju, Shuqin Liu, and Yinghua Ji. "Measurement of quantum correlation by Frobenius norm in non-Markovian system." International Journal of Quantum Information 16, no. 03 (April 2018): 1850022. http://dx.doi.org/10.1142/s0219749918500223.

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In order to measure the quantum correlation of a bipartite state quickly, an easy method is to construct a test matrix through the commutations among the blocks of its density matrix. Then, the Frobenius norm of the test matrix can be used to measure the quantum correlation. In this paper, we apply the measurement by Frobenius norm ([Formula: see text] to the dynamics evolution of the non-Markovian quantum system and compare it with the typical quantum discord ([Formula: see text] proposed by Ollivier and Zurek. The research results show that [Formula: see text] can indeed measure the quantum correlation of a bipartite state as same as [Formula: see text]. Further studies find that there are still differences between the two measurements: in some regions, when [Formula: see text] is zero, [Formula: see text] is not zero. It indicates that [Formula: see text] is more detailed than [Formula: see text] to measure quantum correlation of a bipartite state.

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45

Sergeev, Ivan. "Generalizations of 2-Dimensional Diagonal Quantum Channels with Constant Frobenius Norm." Reports on Mathematical Physics 83, no. 3 (June 2019): 349–72. http://dx.doi.org/10.1016/s0034-4877(19)30055-2.

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46

You, Sung Hyun, Choon Ki Ahn, Yuriy S. Shmaliy, and Shunyi Zhao. "Minimum Weighted Frobenius Norm Discrete-Time FIR Filter With Embedded Unbiasedness." IEEE Transactions on Circuits and Systems II: Express Briefs 65, no. 9 (September 2018): 1284–88. http://dx.doi.org/10.1109/tcsii.2018.2810812.

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47

Heldring, A., E. Ubeda, and J. M. Rius. "Stochastic Estimation of the Frobenius Norm in the ACA Convergence Criterion." IEEE Transactions on Antennas and Propagation 63, no. 3 (March 2015): 1155–58. http://dx.doi.org/10.1109/tap.2014.2386306.

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48

Guglielmi, N., M. Gürbüzbalaban, T. Mitchell, and M. L. Overton. "Approximating the Real Structured Stability Radius with Frobenius-Norm Bounded Perturbations." SIAM Journal on Matrix Analysis and Applications 38, no. 4 (January 2017): 1323–53. http://dx.doi.org/10.1137/16m1110169.

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49

Enshaei, Sharareh, Wah June Leong, and Mahboubeh Farid. "Diagonal quasi-Newton method via variational principle under generalized Frobenius norm." Optimization Methods and Software 31, no. 6 (July 7, 2016): 1258–71. http://dx.doi.org/10.1080/10556788.2016.1196205.

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50

Lem, Kong Hoong. "Truncated singular value decomposition in ripped photo recovery." ITM Web of Conferences 36 (2021): 04008. http://dx.doi.org/10.1051/itmconf/20213604008.

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Singular value decomposition (SVD) is one of the most useful matrix decompositions in linear algebra. Here, a novel application of SVD in recovering ripped photos was exploited. Recovery was done by applying truncated SVD iteratively. Performance was evaluated using the Frobenius norm. Results from a few experimental photos were decent.

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