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Nonsingular $H$-tensor and its criteria
1. | School of Management Science, Qufu Normal University, Rizhao Shandong, 276800, China |
2. | Department of Mathematics and Statistics, Curtin University of Technology, West Australia, WA 6102 |
3. | Department of Mathematics and Statistics, Curtin University, Perth, Western Australia, 6102, Australia |
References:
[1] |
J. Brachat, P. Comon, B. Mourrain and E. Tsigaridas, Symmetric tensor decomposition, Linear Algebra Appl., 433 (2010), 1851-1872.
doi: 10.1016/j.laa.2010.06.046. |
[2] |
K. C. Chang, K. Pearson and T. Zhang, Perron-Frobenius theorem for nonnegative tensors, Commun. Math. Science, 6 (2008), 507-520.
doi: 10.4310/CMS.2008.v6.n2.a12. |
[3] |
H. B. Chen and L. Q. Qi, Positive definiteness and semi-definiteness of even order symmetric Cauchy tensors, J. Industrial and Management Optim., 11 (2015), 1263-1274.
doi: 10.3934/jimo.2015.11.1263. |
[4] |
A. Cichocki, R. Zdunek, A. H. Phan and S. Amari, Nonnegative Matrix and Tensor Factorizations, John Wiley & Sons, Ltd, 2009.
doi: 10.1002/9780470747278. |
[5] |
L. Cvetkovic, V. Kostic and R. S. Varga, A new Geršgorin-type eigenvalue inclusion set, Elec. Trans. Numer. Anal., 18 (2004), 73-80. |
[6] |
L. Cvetkovic and V. Kostic, New criteria for identifying $H$-matrices, J Comput. Appl. Math., 180 (2005), 265-278.
doi: 10.1016/j.cam.2004.10.017. |
[7] |
L. De Lathauwer, B. De Moor and J. Vandewalle, A multilinear singular value decomposition, SIAM J. Matrix Anal. Appl., 21 (2000), 1253-1278.
doi: 10.1137/S0895479896305696. |
[8] |
W. Y. Ding, L. Q. Qi and Y. M. Wei, $M$-tensors and nonsingular $M$-tensors, Linear Algebra Appl., 439 (2013), 3264-3278.
doi: 10.1016/j.laa.2013.08.038. |
[9] |
S. Gandy, B. Recht and I. Yamada, Tensor completion and low-$n$-rank tensor recovery via convex optimization, Inverse Problems, 27 (2011), 025010, 19pp.
doi: 10.1088/0266-5611/27/2/025010. |
[10] |
R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, 1985.
doi: 10.1017/CBO9780511810817. |
[11] |
S. L. Hu and L. Qi, Algebraic connectivity of an even uniform hypergraph, J. Combination Optim., 24 (2012), 564-579.
doi: 10.1007/s10878-011-9407-1. |
[12] |
S. L. Hu, Z. H. Huang, C. Ling and L. Q. Qi, On determinants and eigenvalue theory of tensors, J. Symbolic Comput., 50 (2013), 508-531.
doi: 10.1016/j.jsc.2012.10.001. |
[13] |
S. L. Hu and L. Q. Qi, Algebraic connectivity of an even uniform hypergraph, J. Combinatorial Optim., 24 (2012), 564-579.
doi: 10.1007/s10878-011-9407-1. |
[14] |
M. R. Kannan, N. Shaked-Monderer and A. Berman, Some properties of strong H-tensors and general H-tensors, Linear Algebra Appl., 476 (2015), 42-55.
doi: 10.1016/j.laa.2015.02.034. |
[15] |
E. Kofidis and P. A. Regalia, On the best rank-1 approximation of higher-order supersymmetric tensors, SIAM J. Matrix Anal. Appl., 23 (2002), 863-884.
doi: 10.1137/S0895479801387413. |
[16] |
T. G. Kolda and B. W. Bader, Tensor decompositions and applications, SIAM Review, 51 (2009), 455-500.
doi: 10.1137/07070111X. |
[17] |
T. G. Kolda and J. R. Mayo, Shifted power method for computing tensor eigenvalues, SIAM J. Matrix Anal. Appl, 32 (2011), 1095-1124.
doi: 10.1137/100801482. |
[18] |
C. Q. Li, F. Wang, J. X. Zhao, Y. Zhu and Y. T. Li, Criterions for the positive definiteness of real supersymmetric tensors, J. Comput. Appl. Math., 255 (2014), 1-14.
doi: 10.1016/j.cam.2013.04.022. |
[19] |
Y. Y. Liu and F. H. Shang, An efficient matrix factorization method for tensor completion, IEEE Signal Processing Letters, 20 (2013), 307-310.
doi: 10.1109/LSP.2013.2245416. |
[20] |
L. H. Lim, Singular value and and eigenvalue of tensors, a variational approach, in CAMSAP'05: Proceeding of the IEEE International Workshop on Computational Advances in multiSensor Adaptive Processing, 2005, 129-132. |
[21] |
J. Liu, P. Musialski, P. Wonka and J. P. Ye, Tensor completion for estimating missing values in visual data, IEEE Trans. on Pattern Anal. Machine Intelligence, 35 (2013), 208-220. |
[22] |
M. Moakher, On the averaging of symmetric positive-definite tensors, J. Elasticity, 82 (2006), 273-296.
doi: 10.1007/s10659-005-9035-z. |
[23] |
M. Ng, L. Qi and G. Zhou, Finding the largest eigenvalue of a nonnegative tensor, SIAM J. Matrix Anal. Appl., 31 (2009), 1090-1099.
doi: 10.1137/09074838X. |
[24] |
Q. Ni, L. Qi and F. Wang, An eigenvalue method for testing positive definiteness of a multivariate form, IEEE Trans. on Auto. Control, 53 (2008), 1096-1107.
doi: 10.1109/TAC.2008.923679. |
[25] |
C. L. Nikias and J. M. Mendel, Signal processing with higher-order spectra, IEEE Signal Processing Magazine, 10 (1993), 10-37.
doi: 10.1109/79.221324. |
[26] |
L. Oeding and G. Ottaviani, Eigenvectors of tensors and algorithms for Waring decomposition, J. Symbolic Comput., 54 (2013), 9-35.
doi: 10.1016/j.jsc.2012.11.005. |
[27] |
A. M. Ostrowski, Über die Determinaanten mit überwiegender Hauptdiagonale, Comment Math. Helv., 10 (1937), 69-96.
doi: 10.1007/BF01214284. |
[28] |
L. Q. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40 (2005), 1302-1324.
doi: 10.1016/j.jsc.2005.05.007. |
[29] |
L. Qi, J. Y. Shao and Q. Wang, Regular uniform hypergraphs, $s$-cycles, $s$-paths and their largest Laplacian H-eigenvalues, Linear Algebra Appl., 443 (2014), 215-227.
doi: 10.1016/j.laa.2013.11.008. |
[30] |
L. Qi, C. Xu and Y. Xu, Nonnegative tensor factorization, completely positive tensors and an Hierarchically elimination algorithm, SIAM J. Matrix Anal. Appl., 35 (2014), 1227-1241.
doi: 10.1137/13092232X. |
[31] |
Y. S. Song and L. Qi, Necessary and sufficient conditions for copositive tensors, Linear and Multilinear Algebra, 63 (2015), 120-131.
doi: 10.1080/03081087.2013.851198. |
[32] |
Y. Song and L. Q. Qi, Infinite and finite dimensional Hilbert tensors, Linear Algebra Appl., 451 (2014), 1-14.
doi: 10.1016/j.laa.2014.03.023. |
[33] |
Y. S. Song and L. Q. Qi, Properties of some classes of structured tensors, J. Optim. Theory Appl., 165 (2015), 854-873.
doi: 10.1007/s10957-014-0616-5. |
[34] |
Y. Yang and Q. Yang, Further results for Perron-Frobenius theorem for nonnegative tensors, SIAM J. Matrix Anal. Appl., 31 (2010), 2517-2530.
doi: 10.1137/090778766. |
[35] |
P. Yuan and L. You, Some remarks on $P,P_0, B$ and $B_0$ tensors, Linear Algebra Appl., 459 (2014), 511-521.
doi: 10.1016/j.laa.2014.07.043. |
[36] |
L. P. Zhang, L. Q. Qi and G. L. Zhou, $M$-tensors and some applications, SIAM J. Matrix Anal. Appl., 35 (2014), 437-452.
doi: 10.1137/130915339. |
[37] |
X. Z. Zhang, C. Ling and L. Qi, The best rank-1 approximation of a symmetric tensor and related spherical optimization problems, SIAM J. Matrix Anal. Appl., 33 (2012), 806-821.
doi: 10.1137/110835335. |
show all references
References:
[1] |
J. Brachat, P. Comon, B. Mourrain and E. Tsigaridas, Symmetric tensor decomposition, Linear Algebra Appl., 433 (2010), 1851-1872.
doi: 10.1016/j.laa.2010.06.046. |
[2] |
K. C. Chang, K. Pearson and T. Zhang, Perron-Frobenius theorem for nonnegative tensors, Commun. Math. Science, 6 (2008), 507-520.
doi: 10.4310/CMS.2008.v6.n2.a12. |
[3] |
H. B. Chen and L. Q. Qi, Positive definiteness and semi-definiteness of even order symmetric Cauchy tensors, J. Industrial and Management Optim., 11 (2015), 1263-1274.
doi: 10.3934/jimo.2015.11.1263. |
[4] |
A. Cichocki, R. Zdunek, A. H. Phan and S. Amari, Nonnegative Matrix and Tensor Factorizations, John Wiley & Sons, Ltd, 2009.
doi: 10.1002/9780470747278. |
[5] |
L. Cvetkovic, V. Kostic and R. S. Varga, A new Geršgorin-type eigenvalue inclusion set, Elec. Trans. Numer. Anal., 18 (2004), 73-80. |
[6] |
L. Cvetkovic and V. Kostic, New criteria for identifying $H$-matrices, J Comput. Appl. Math., 180 (2005), 265-278.
doi: 10.1016/j.cam.2004.10.017. |
[7] |
L. De Lathauwer, B. De Moor and J. Vandewalle, A multilinear singular value decomposition, SIAM J. Matrix Anal. Appl., 21 (2000), 1253-1278.
doi: 10.1137/S0895479896305696. |
[8] |
W. Y. Ding, L. Q. Qi and Y. M. Wei, $M$-tensors and nonsingular $M$-tensors, Linear Algebra Appl., 439 (2013), 3264-3278.
doi: 10.1016/j.laa.2013.08.038. |
[9] |
S. Gandy, B. Recht and I. Yamada, Tensor completion and low-$n$-rank tensor recovery via convex optimization, Inverse Problems, 27 (2011), 025010, 19pp.
doi: 10.1088/0266-5611/27/2/025010. |
[10] |
R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, 1985.
doi: 10.1017/CBO9780511810817. |
[11] |
S. L. Hu and L. Qi, Algebraic connectivity of an even uniform hypergraph, J. Combination Optim., 24 (2012), 564-579.
doi: 10.1007/s10878-011-9407-1. |
[12] |
S. L. Hu, Z. H. Huang, C. Ling and L. Q. Qi, On determinants and eigenvalue theory of tensors, J. Symbolic Comput., 50 (2013), 508-531.
doi: 10.1016/j.jsc.2012.10.001. |
[13] |
S. L. Hu and L. Q. Qi, Algebraic connectivity of an even uniform hypergraph, J. Combinatorial Optim., 24 (2012), 564-579.
doi: 10.1007/s10878-011-9407-1. |
[14] |
M. R. Kannan, N. Shaked-Monderer and A. Berman, Some properties of strong H-tensors and general H-tensors, Linear Algebra Appl., 476 (2015), 42-55.
doi: 10.1016/j.laa.2015.02.034. |
[15] |
E. Kofidis and P. A. Regalia, On the best rank-1 approximation of higher-order supersymmetric tensors, SIAM J. Matrix Anal. Appl., 23 (2002), 863-884.
doi: 10.1137/S0895479801387413. |
[16] |
T. G. Kolda and B. W. Bader, Tensor decompositions and applications, SIAM Review, 51 (2009), 455-500.
doi: 10.1137/07070111X. |
[17] |
T. G. Kolda and J. R. Mayo, Shifted power method for computing tensor eigenvalues, SIAM J. Matrix Anal. Appl, 32 (2011), 1095-1124.
doi: 10.1137/100801482. |
[18] |
C. Q. Li, F. Wang, J. X. Zhao, Y. Zhu and Y. T. Li, Criterions for the positive definiteness of real supersymmetric tensors, J. Comput. Appl. Math., 255 (2014), 1-14.
doi: 10.1016/j.cam.2013.04.022. |
[19] |
Y. Y. Liu and F. H. Shang, An efficient matrix factorization method for tensor completion, IEEE Signal Processing Letters, 20 (2013), 307-310.
doi: 10.1109/LSP.2013.2245416. |
[20] |
L. H. Lim, Singular value and and eigenvalue of tensors, a variational approach, in CAMSAP'05: Proceeding of the IEEE International Workshop on Computational Advances in multiSensor Adaptive Processing, 2005, 129-132. |
[21] |
J. Liu, P. Musialski, P. Wonka and J. P. Ye, Tensor completion for estimating missing values in visual data, IEEE Trans. on Pattern Anal. Machine Intelligence, 35 (2013), 208-220. |
[22] |
M. Moakher, On the averaging of symmetric positive-definite tensors, J. Elasticity, 82 (2006), 273-296.
doi: 10.1007/s10659-005-9035-z. |
[23] |
M. Ng, L. Qi and G. Zhou, Finding the largest eigenvalue of a nonnegative tensor, SIAM J. Matrix Anal. Appl., 31 (2009), 1090-1099.
doi: 10.1137/09074838X. |
[24] |
Q. Ni, L. Qi and F. Wang, An eigenvalue method for testing positive definiteness of a multivariate form, IEEE Trans. on Auto. Control, 53 (2008), 1096-1107.
doi: 10.1109/TAC.2008.923679. |
[25] |
C. L. Nikias and J. M. Mendel, Signal processing with higher-order spectra, IEEE Signal Processing Magazine, 10 (1993), 10-37.
doi: 10.1109/79.221324. |
[26] |
L. Oeding and G. Ottaviani, Eigenvectors of tensors and algorithms for Waring decomposition, J. Symbolic Comput., 54 (2013), 9-35.
doi: 10.1016/j.jsc.2012.11.005. |
[27] |
A. M. Ostrowski, Über die Determinaanten mit überwiegender Hauptdiagonale, Comment Math. Helv., 10 (1937), 69-96.
doi: 10.1007/BF01214284. |
[28] |
L. Q. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40 (2005), 1302-1324.
doi: 10.1016/j.jsc.2005.05.007. |
[29] |
L. Qi, J. Y. Shao and Q. Wang, Regular uniform hypergraphs, $s$-cycles, $s$-paths and their largest Laplacian H-eigenvalues, Linear Algebra Appl., 443 (2014), 215-227.
doi: 10.1016/j.laa.2013.11.008. |
[30] |
L. Qi, C. Xu and Y. Xu, Nonnegative tensor factorization, completely positive tensors and an Hierarchically elimination algorithm, SIAM J. Matrix Anal. Appl., 35 (2014), 1227-1241.
doi: 10.1137/13092232X. |
[31] |
Y. S. Song and L. Qi, Necessary and sufficient conditions for copositive tensors, Linear and Multilinear Algebra, 63 (2015), 120-131.
doi: 10.1080/03081087.2013.851198. |
[32] |
Y. Song and L. Q. Qi, Infinite and finite dimensional Hilbert tensors, Linear Algebra Appl., 451 (2014), 1-14.
doi: 10.1016/j.laa.2014.03.023. |
[33] |
Y. S. Song and L. Q. Qi, Properties of some classes of structured tensors, J. Optim. Theory Appl., 165 (2015), 854-873.
doi: 10.1007/s10957-014-0616-5. |
[34] |
Y. Yang and Q. Yang, Further results for Perron-Frobenius theorem for nonnegative tensors, SIAM J. Matrix Anal. Appl., 31 (2010), 2517-2530.
doi: 10.1137/090778766. |
[35] |
P. Yuan and L. You, Some remarks on $P,P_0, B$ and $B_0$ tensors, Linear Algebra Appl., 459 (2014), 511-521.
doi: 10.1016/j.laa.2014.07.043. |
[36] |
L. P. Zhang, L. Q. Qi and G. L. Zhou, $M$-tensors and some applications, SIAM J. Matrix Anal. Appl., 35 (2014), 437-452.
doi: 10.1137/130915339. |
[37] |
X. Z. Zhang, C. Ling and L. Qi, The best rank-1 approximation of a symmetric tensor and related spherical optimization problems, SIAM J. Matrix Anal. Appl., 33 (2012), 806-821.
doi: 10.1137/110835335. |
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