July  2016, 12(3): 975-990. doi: 10.3934/jimo.2016.12.975

Bounds for the spectral radius of nonnegative tensors

1. 

School of Mathematics and Statistics, Yunnan University, Kunming, China, China

2. 

Institute of Mathematics and Information science, Baoji University of Arts and Sciences, Baoji, China

3. 

Mathematics Science College, Beijing Normal University, Beijing, China

Received  March 2014 Revised  April 2015 Published  September 2015

Lower bounds and upper bounds for the spectral radius of a nonnegative tensor are provided. And it is proved that these bounds are better than the corresponding bounds in [Y. Yang, Q. Yang, Further results for Perron-Frobenius Theorem for nonnegative tensors, SIAM. J. Matrix Anal. Appl. 31 (2010), 2517-2530].
Citation: Chaoqian Li, Yaqiang Wang, Jieyi Yi, Yaotang Li. Bounds for the spectral radius of nonnegative tensors. Journal of Industrial & Management Optimization, 2016, 12 (3) : 975-990. doi: 10.3934/jimo.2016.12.975
References:
[1]

D. Cartwright and B. Sturmfels, The number of eigenvalues of a tensor,, Linear Algebra and its Applications, 438 (2013), 942. doi: 10.1016/j.laa.2011.05.040. Google Scholar

[2]

K. C. Chang, K. Pearson and T. Zhang, Perron-Frobenius theorem for nonnegative tensors,, Commun. Math. Sci., 6 (2008), 507. Google Scholar

[3]

A. Cichocki, R. Zdunek, A. H. Phan and S. Amari, Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multiway Data Analysis and Blind Source Separation,, John Wiley and Sons Ltd, (2009). doi: 10.1002/9780470747278. Google Scholar

[4]

L. De Lathauwer and B. D. Moor, From matrix to tensor: Multilinear algebra and signal processing,, in Mathematics in Signal Processing IV (ed. J. McWhirter), (1998), 1. Google Scholar

[5]

L. De Lathauwer, B. D. Moor and J. Vandewalle, A multilinear singular value decomposition,, SIAM J. Matrix Anal. Appl., 21 (2000), 1253. doi: 10.1137/S0895479896305696. Google Scholar

[6]

L. De Lathauwer, B. D. Moor and J. Vandewalle, On the best rank-1 and rank-($R_1,R_2,\ldots ,R_N$) approximation of higer-order tensors,, SIAM J. Matrix Anal. Appl., 21 (2000), 1324. Google Scholar

[7]

E. Deutsch, Bounds for the perron root of a nonnegative irreducible partitioned matrix,, Pacific Journal of Mathematics, 92 (1981), 49. doi: 10.2140/pjm.1981.92.49. Google Scholar

[8]

S. Friedland, S. Gaubert and L. Han, Perron-Frobenius theorem for nonnegative multilinear forms and extensions,, Linear Algebra and its Applications, 438 (2013), 738. doi: 10.1016/j.laa.2011.02.042. Google Scholar

[9]

S. L. Hu, Z. H. Huang, C. Ling and L. Qi, On Determinants and Eigenvalue Theory of Tensors,, Journal of Symbolic Computation, 50 (2013), 508. doi: 10.1016/j.jsc.2012.10.001. Google Scholar

[10]

R. A. Horn and C. R. Johnson, Matrix Analysis,, Cambridge University Press, (1985). Google Scholar

[11]

E. Kofidis and P. A. Regalia, On the best rank-1 approximation of higher-order supersymmetric tensors,, SIAM Journal on Matrix Analysis and Applications, 23 (2002), 863. Google Scholar

[12]

T. G. Kolda and J. R. Mayo, Shifted power method for computing tensor eigenpairs,, SIAM Journal on Matrix Analysis and Applications, 32 (2011), 1095. doi: 10.1137/100801482. Google Scholar

[13]

W. Ledermann, Bounds for the greastest latent root of a positive matrix,, J. London Math. Soc., 25 (1950), 265. Google Scholar

[14]

C. Li, Y. Li and X. Kong, New eigenvalue inclusion sets for tensors,, Numer. Linear Algebra Appl., 21 (2014), 39. doi: 10.1002/nla.1858. Google Scholar

[15]

L. H. Lim, Singular values and eigenvalues 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

[16]

Y. Liu, G. Zhou and N. F. Ibrahim, An always convergent algorithm for the largest eigenvalue of an irreducible nonnegative tensor,, Journal of Computational and Applied Mathematics, 235 (2010), 286. doi: 10.1016/j.cam.2010.06.002. Google Scholar

[17]

M. Ng, L. Qi and G. Zhou, Finding the largest eigenvalue of a nonnegative tensor,, SIAM J. Matrix Anal. Appl., 31 (2009), 1090. Google Scholar

[18]

Q. Ni, L. Qi and F. Wang, An eigenvalue method for the positive definiteness identification problem,, IEEE Transactions on Automatic Control, 53 (2008), 1096. doi: 10.1109/TAC.2008.923679. Google Scholar

[19]

G. Ni, L. Qi, F. Wang and Y. Wang, The degree of the E-characteristic polynomial of an even order tensor,, J. Math. Anal. Appl., 329 (2007), 1218. doi: 10.1016/j.jmaa.2006.07.064. Google Scholar

[20]

C. L. Nikias and J. M. Mendel, Signal processing with higher-order spectra,, IEEE Signal Process. Mag., 10 (1993), 10. doi: 10.1109/79.221324. Google Scholar

[21]

L. Qi, Eigenvalues of a Supersymmetric Tensor and Positive Definiteness of an Even Degree Multivariate Form,, Department of Applied Mathematics, (2004). doi: 10.2307/2152750. Google Scholar

[22]

L. Qi, Eigenvalues of a real supersymmetric tensor,, Journal of Symbolic Computation, 40 (2005), 1302. doi: 10.1016/j.jsc.2005.05.007. Google Scholar

[23]

L. Qi, Eigenvalues and invariants of tensors,, Journal of Mathematical Analysis and Applications, 325 (2007), 1363. doi: 10.1016/j.jmaa.2006.02.071. Google Scholar

[24]

L. Qi, W. Sun and Y. Wang, Numerical multilinear algebra and its applications,, Front. Math. China, 2 (2007), 501. doi: 10.1007/s11464-007-0031-4. Google Scholar

[25]

L. Qi, F. Wang and Y. Wang, Z-eigenvalue methods for a global polynomial optimization problem,, Mathematical Programming, 118 (2009), 301. doi: 10.1007/s10107-007-0193-6. Google Scholar

[26]

L. Qi, Y. Wang and E. X. Wu, D-eigenvalues of diffusion kurtosis tensors,, Journal of Computational and Applied Mathematics, 221 (2008), 150. doi: 10.1016/j.cam.2007.10.012. Google Scholar

[27]

Y. Wang, L. Qi and X. Zhang, A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor,, Numerical Linear Algebra with Applications, 16 (2009), 589. doi: 10.1002/nla.633. Google Scholar

[28]

Z. Wang and W. Wu, Bounds for the greatest eigenvalue of positive tensors,, Journal of Industral and Management Optimization, 10 (2013), 1031. doi: 10.3934/jimo.2014.10.1031. Google Scholar

[29]

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

[30]

Q. Yang and Y. Yang, Further results for Perron-Frobenius theorem for nonnegative tensors II,, SIAM. J. Matrix Anal. Appl., 32 (2011), 1236. doi: 10.1137/100813671. Google Scholar

[31]

T. Zhang and G. H. Golub, Rank-1 approximation of higher-order tensors,, SIAM J. Matrix Anal. Appl., 23 (2001), 534. doi: 10.1137/S0895479899352045. Google Scholar

show all references

References:
[1]

D. Cartwright and B. Sturmfels, The number of eigenvalues of a tensor,, Linear Algebra and its Applications, 438 (2013), 942. doi: 10.1016/j.laa.2011.05.040. Google Scholar

[2]

K. C. Chang, K. Pearson and T. Zhang, Perron-Frobenius theorem for nonnegative tensors,, Commun. Math. Sci., 6 (2008), 507. Google Scholar

[3]

A. Cichocki, R. Zdunek, A. H. Phan and S. Amari, Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multiway Data Analysis and Blind Source Separation,, John Wiley and Sons Ltd, (2009). doi: 10.1002/9780470747278. Google Scholar

[4]

L. De Lathauwer and B. D. Moor, From matrix to tensor: Multilinear algebra and signal processing,, in Mathematics in Signal Processing IV (ed. J. McWhirter), (1998), 1. Google Scholar

[5]

L. De Lathauwer, B. D. Moor and J. Vandewalle, A multilinear singular value decomposition,, SIAM J. Matrix Anal. Appl., 21 (2000), 1253. doi: 10.1137/S0895479896305696. Google Scholar

[6]

L. De Lathauwer, B. D. Moor and J. Vandewalle, On the best rank-1 and rank-($R_1,R_2,\ldots ,R_N$) approximation of higer-order tensors,, SIAM J. Matrix Anal. Appl., 21 (2000), 1324. Google Scholar

[7]

E. Deutsch, Bounds for the perron root of a nonnegative irreducible partitioned matrix,, Pacific Journal of Mathematics, 92 (1981), 49. doi: 10.2140/pjm.1981.92.49. Google Scholar

[8]

S. Friedland, S. Gaubert and L. Han, Perron-Frobenius theorem for nonnegative multilinear forms and extensions,, Linear Algebra and its Applications, 438 (2013), 738. doi: 10.1016/j.laa.2011.02.042. Google Scholar

[9]

S. L. Hu, Z. H. Huang, C. Ling and L. Qi, On Determinants and Eigenvalue Theory of Tensors,, Journal of Symbolic Computation, 50 (2013), 508. doi: 10.1016/j.jsc.2012.10.001. Google Scholar

[10]

R. A. Horn and C. R. Johnson, Matrix Analysis,, Cambridge University Press, (1985). Google Scholar

[11]

E. Kofidis and P. A. Regalia, On the best rank-1 approximation of higher-order supersymmetric tensors,, SIAM Journal on Matrix Analysis and Applications, 23 (2002), 863. Google Scholar

[12]

T. G. Kolda and J. R. Mayo, Shifted power method for computing tensor eigenpairs,, SIAM Journal on Matrix Analysis and Applications, 32 (2011), 1095. doi: 10.1137/100801482. Google Scholar

[13]

W. Ledermann, Bounds for the greastest latent root of a positive matrix,, J. London Math. Soc., 25 (1950), 265. Google Scholar

[14]

C. Li, Y. Li and X. Kong, New eigenvalue inclusion sets for tensors,, Numer. Linear Algebra Appl., 21 (2014), 39. doi: 10.1002/nla.1858. Google Scholar

[15]

L. H. Lim, Singular values and eigenvalues 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

[16]

Y. Liu, G. Zhou and N. F. Ibrahim, An always convergent algorithm for the largest eigenvalue of an irreducible nonnegative tensor,, Journal of Computational and Applied Mathematics, 235 (2010), 286. doi: 10.1016/j.cam.2010.06.002. Google Scholar

[17]

M. Ng, L. Qi and G. Zhou, Finding the largest eigenvalue of a nonnegative tensor,, SIAM J. Matrix Anal. Appl., 31 (2009), 1090. Google Scholar

[18]

Q. Ni, L. Qi and F. Wang, An eigenvalue method for the positive definiteness identification problem,, IEEE Transactions on Automatic Control, 53 (2008), 1096. doi: 10.1109/TAC.2008.923679. Google Scholar

[19]

G. Ni, L. Qi, F. Wang and Y. Wang, The degree of the E-characteristic polynomial of an even order tensor,, J. Math. Anal. Appl., 329 (2007), 1218. doi: 10.1016/j.jmaa.2006.07.064. Google Scholar

[20]

C. L. Nikias and J. M. Mendel, Signal processing with higher-order spectra,, IEEE Signal Process. Mag., 10 (1993), 10. doi: 10.1109/79.221324. Google Scholar

[21]

L. Qi, Eigenvalues of a Supersymmetric Tensor and Positive Definiteness of an Even Degree Multivariate Form,, Department of Applied Mathematics, (2004). doi: 10.2307/2152750. Google Scholar

[22]

L. Qi, Eigenvalues of a real supersymmetric tensor,, Journal of Symbolic Computation, 40 (2005), 1302. doi: 10.1016/j.jsc.2005.05.007. Google Scholar

[23]

L. Qi, Eigenvalues and invariants of tensors,, Journal of Mathematical Analysis and Applications, 325 (2007), 1363. doi: 10.1016/j.jmaa.2006.02.071. Google Scholar

[24]

L. Qi, W. Sun and Y. Wang, Numerical multilinear algebra and its applications,, Front. Math. China, 2 (2007), 501. doi: 10.1007/s11464-007-0031-4. Google Scholar

[25]

L. Qi, F. Wang and Y. Wang, Z-eigenvalue methods for a global polynomial optimization problem,, Mathematical Programming, 118 (2009), 301. doi: 10.1007/s10107-007-0193-6. Google Scholar

[26]

L. Qi, Y. Wang and E. X. Wu, D-eigenvalues of diffusion kurtosis tensors,, Journal of Computational and Applied Mathematics, 221 (2008), 150. doi: 10.1016/j.cam.2007.10.012. Google Scholar

[27]

Y. Wang, L. Qi and X. Zhang, A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor,, Numerical Linear Algebra with Applications, 16 (2009), 589. doi: 10.1002/nla.633. Google Scholar

[28]

Z. Wang and W. Wu, Bounds for the greatest eigenvalue of positive tensors,, Journal of Industral and Management Optimization, 10 (2013), 1031. doi: 10.3934/jimo.2014.10.1031. Google Scholar

[29]

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

[30]

Q. Yang and Y. Yang, Further results for Perron-Frobenius theorem for nonnegative tensors II,, SIAM. J. Matrix Anal. Appl., 32 (2011), 1236. doi: 10.1137/100813671. Google Scholar

[31]

T. Zhang and G. H. Golub, Rank-1 approximation of higher-order tensors,, SIAM J. Matrix Anal. Appl., 23 (2001), 534. doi: 10.1137/S0895479899352045. Google Scholar

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