American Institute of Mathematical Sciences

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October  2016, 12(4): 1173-1186. doi: 10.3934/jimo.2016.12.1173

Nonsingular $H$-tensor and its criteria

Yiju Wang 1, , Guanglu Zhou 2, and  Louis Caccetta 3,

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

Received  February 2015 Revised  March 2015 Published  January 2016

$H$-tensor is a new developed concept in tensor analysis and it is an extension of $H$-matrix and $M$-tensor. Based on the spectral theory of nonnegative tensors, several equivalent conditions of nonsingular $H$-tensors are established in the literature. However, these conditions can not be used as a criteria to identify nonsingular $H$-tensors as they are hard to verify. In this paper, based on the diagonal product dominance and $S$ diagonal product dominance of a tensor, we establish some new implementable criteria in identifying nonsingular $H$-tensors. The positive definiteness of nonsingular $H$-tensors with positive diagonal entries is also discussed in this paper. The obtained results extend the corresponding conclusions for nonsingular $H$-matrices and improve the existing results for nonsingular $H$-tensors.
Keywords: $M$-tensor, $H$-tensor, diagonal product dominance., nonsingularity.
Mathematics Subject Classification: 15A69, 12E05, 12E10, 65F1.
Citation: Yiju Wang, Guanglu Zhou, Louis Caccetta. Nonsingular $H$-tensor and its criteria. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1173-1186. doi: 10.3934/jimo.2016.12.1173
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.  Google Scholar

[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.  Google Scholar

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L. Cvetkovic, V. Kostic and R. S. Varga, A new Geršgorin-type eigenvalue inclusion set, Elec. Trans. Numer. Anal., 18 (2004), 73-80.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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. Google Scholar

[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. Google Scholar

[22]

M. Moakher, On the averaging of symmetric positive-definite tensors, J. Elasticity, 82 (2006), 273-296. doi: 10.1007/s10659-005-9035-z.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[27]

A. M. Ostrowski, Über die Determinaanten mit überwiegender Hauptdiagonale, Comment Math. Helv., 10 (1937), 69-96. doi: 10.1007/BF01214284.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

L. Cvetkovic, V. Kostic and R. S. Varga, A new Geršgorin-type eigenvalue inclusion set, Elec. Trans. Numer. Anal., 18 (2004), 73-80.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, 1985. doi: 10.1017/CBO9780511810817.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

T. G. Kolda and B. W. Bader, Tensor decompositions and applications, SIAM Review, 51 (2009), 455-500. doi: 10.1137/07070111X.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[22]

M. Moakher, On the averaging of symmetric positive-definite tensors, J. Elasticity, 82 (2006), 273-296. doi: 10.1007/s10659-005-9035-z.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

A. M. Ostrowski, Über die Determinaanten mit überwiegender Hauptdiagonale, Comment Math. Helv., 10 (1937), 69-96. doi: 10.1007/BF01214284.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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