Nonsingular H-tensor and its criteria

Previous Article
JIMO Home
This Issue
Next Article
Semidefinite programming via image space analysis

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

Abstract
Related Papers

$H$-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. 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. 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. 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, (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. Google Scholar
[6]	L. Cvetkovic and V. Kostic, New criteria for identifying $H$-matrices,, J Comput. Appl. Math., 180* (2005), 265. 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. 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. 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). 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. 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. 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. 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. 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. doi: 10.1137/S0895479801387413. Google Scholar
[16]	T. G. Kolda and B. W. Bader, Tensor decompositions and applications,, SIAM Review, 51 (2009), 455. 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. 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. 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. 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. 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. Google Scholar
[22]	M. Moakher, On the averaging of symmetric positive-definite tensors,, J. Elasticity, 82 (2006), 273. 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. 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. 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. 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. 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. doi: 10.1007/BF01214284. Google Scholar
[28]	L. Q. Qi, Eigenvalues of a real supersymmetric tensor,, J. Symbolic Comput., 40 (2005), 1302. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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, (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. Google Scholar
[6]	L. Cvetkovic and V. Kostic, New criteria for identifying $H$-matrices,, J Comput. Appl. Math., 180* (2005), 265. 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. 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. 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). 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. 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. 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. 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. 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. doi: 10.1137/S0895479801387413. Google Scholar
[16]	T. G. Kolda and B. W. Bader, Tensor decompositions and applications,, SIAM Review, 51 (2009), 455. 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. 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. 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. 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. 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. Google Scholar
[22]	M. Moakher, On the averaging of symmetric positive-definite tensors,, J. Elasticity, 82 (2006), 273. 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. 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. 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. 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. 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. doi: 10.1007/BF01214284. Google Scholar
[28]	L. Q. Qi, Eigenvalues of a real supersymmetric tensor,, J. Symbolic Comput., 40 (2005), 1302. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1137/110835335. Google Scholar

[1]	H. M. Hastings, S. Silberger, M. T. Weiss, Y. Wu. A twisted tensor product on symbolic dynamical systems and the Ashley's problem. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 549-558. doi: 10.3934/dcds.2003.9.549
[2]	Jinghong Liu, Yinsuo Jia. Gradient superconvergence post-processing of the tensor-product quadratic pentahedral finite element. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 495-504. doi: 10.3934/dcdsb.2015.20.495
[3]	Jan Boman, Vladimir Sharafutdinov. Stability estimates in tensor tomography. Inverse Problems & Imaging, 2018, 12 (5) : 1245-1262. doi: 10.3934/ipi.2018052
[4]	Mengmeng Zheng, Ying Zhang, Zheng-Hai Huang. Global error bounds for the tensor complementarity problem with a P-tensor. Journal of Industrial & Management Optimization, 2019, 15 (2) : 933-946. doi: 10.3934/jimo.2018078
[5]	Nicolas Van Goethem. The Frank tensor as a boundary condition in intrinsic linearized elasticity. Journal of Geometric Mechanics, 2016, 8 (4) : 391-411. doi: 10.3934/jgm.2016013
[6]	Henry O. Jacobs, Hiroaki Yoshimura. Tensor products of Dirac structures and interconnection in Lagrangian mechanics. Journal of Geometric Mechanics, 2014, 6 (1) : 67-98. doi: 10.3934/jgm.2014.6.67
[7]	François Monard. Efficient tensor tomography in fan-beam coordinates. Inverse Problems & Imaging, 2016, 10 (2) : 433-459. doi: 10.3934/ipi.2016007
[8]	Kaili Zhang, Haibin Chen, Pengfei Zhao. A potential reduction method for tensor complementarity problems. Journal of Industrial & Management Optimization, 2019, 15 (2) : 429-443. doi: 10.3934/jimo.2018049
[9]	François Monard. Efficient tensor tomography in fan-beam coordinates. Ⅱ: Attenuated transforms. Inverse Problems & Imaging, 2018, 12 (2) : 433-460. doi: 10.3934/ipi.2018019
[10]	Zhong Wan, Chunhua Yang. New approach to global minimization of normal multivariate polynomial based on tensor. Journal of Industrial & Management Optimization, 2008, 4 (2) : 271-285. doi: 10.3934/jimo.2008.4.271
[11]	Yangyang Xu, Ruru Hao, Wotao Yin, Zhixun Su. Parallel matrix factorization for low-rank tensor completion. Inverse Problems & Imaging, 2015, 9 (2) : 601-624. doi: 10.3934/ipi.2015.9.601
[12]	Michael K. Ng, Chi-Pan Tam, Fan Wang. Multi-view foreground segmentation via fourth order tensor learning. Inverse Problems & Imaging, 2013, 7 (3) : 885-906. doi: 10.3934/ipi.2013.7.885
[13]	Michael Anderson, Atsushi Katsuda, Yaroslav Kurylev, Matti Lassas and Michael Taylor. Metric tensor estimates, geometric convergence, and inverse boundary problems. Electronic Research Announcements, 2003, 9: 69-79.
[14]	Chen Ling, Liqun Qi. Some results on $l^k$-eigenvalues of tensor and related spectral radius. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 381-388. doi: 10.3934/naco.2011.1.381
[15]	Jop Briët, Assaf Naor, Oded Regev. Locally decodable codes and the failure of cotype for projective tensor products. Electronic Research Announcements, 2012, 19: 120-130. doi: 10.3934/era.2012.19.120
[16]	Grégory Faye, Pascal Chossat. A spatialized model of visual texture perception using the structure tensor formalism. Networks & Heterogeneous Media, 2013, 8 (1) : 211-260. doi: 10.3934/nhm.2013.8.211
[17]	Venkateswaran P. Krishnan, Plamen Stefanov. A support theorem for the geodesic ray transform of symmetric tensor fields. Inverse Problems & Imaging, 2009, 3 (3) : 453-464. doi: 10.3934/ipi.2009.3.453
[18]	Xiaoyu Zheng, Peter Palffy-Muhoray. One order parameter tensor mean field theory for biaxial liquid crystals. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 475-490. doi: 10.3934/dcdsb.2011.15.475
[19]	Steve Rosencrans, Xuefeng Wang, Shan Zhao. Estimating eigenvalues of an anisotropic thermal tensor from transient thermal probe measurements. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5441-5455. doi: 10.3934/dcds.2013.33.5441
[20]	Yanfei Wang, Dmitry Lukyanenko, Anatoly Yagola. Magnetic parameters inversion method with full tensor gradient data. Inverse Problems & Imaging, 2019, 13 (4) : 745-754. doi: 10.3934/ipi.2019034

2018 Impact Factor: 1.025

American Institute of Mathematical Sciences