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 and 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), 942-952. doi: 10.1016/j.laa.2011.05.040.

[2]

K. C. Chang, K. Pearson and T. Zhang, Perron-Frobenius theorem for nonnegative tensors, Commun. Math. Sci., 6 (2008) 507-520.

[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, Chichester, 2009. doi: 10.1002/9780470747278.

[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), Selected papers presented at 4th IMA Int. Conf. on Mathematics in Signal Processing, Oxford University Press, Oxford, UK, 1998, 1-15.

[5]

L. De Lathauwer, B. D. Moor and J. Vandewalle, A multilinear singular value decomposition, SIAM J. Matrix Anal. Appl., 21 (2000), 1253-1278. doi: 10.1137/S0895479896305696.

[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), 1324-1342.

[7]

E. Deutsch, Bounds for the perron root of a nonnegative irreducible partitioned matrix, Pacific Journal of Mathematics, 92 (1981), 49-56. doi: 10.2140/pjm.1981.92.49.

[8]

S. Friedland, S. Gaubert and L. Han, Perron-Frobenius theorem for nonnegative multilinear forms and extensions, Linear Algebra and its Applications, 438 (2013), 738-749. doi: 10.1016/j.laa.2011.02.042.

[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), 508-531. doi: 10.1016/j.jsc.2012.10.001.

[10]

R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985.

[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), 863-884.

[12]

T. G. Kolda and J. R. Mayo, Shifted power method for computing tensor eigenpairs, SIAM Journal on Matrix Analysis and Applications, 32 (2011), 1095-1124. doi: 10.1137/100801482.

[13]

W. Ledermann, Bounds for the greastest latent root of a positive matrix, J. London Math. Soc., 25 (1950), 265-268.

[14]

C. Li, Y. Li and X. Kong, New eigenvalue inclusion sets for tensors, Numer. Linear Algebra Appl., 21 (2014), 39-50. doi: 10.1002/nla.1858.

[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, 129-132.

[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), 286-292. doi: 10.1016/j.cam.2010.06.002.

[17]

M. Ng, L. Qi and G. Zhou, Finding the largest eigenvalue of a nonnegative tensor, SIAM J. Matrix Anal. Appl., 31 (2009), 1090-1099.

[18]

Q. Ni, L. Qi and F. Wang, An eigenvalue method for the positive definiteness identification problem, IEEE Transactions on Automatic Control, 53 (2008), 1096-1107. doi: 10.1109/TAC.2008.923679.

[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), 1218-1229. doi: 10.1016/j.jmaa.2006.07.064.

[20]

C. L. Nikias and J. M. Mendel, Signal processing with higher-order spectra, IEEE Signal Process. Mag., 10 (1993), 10-37. doi: 10.1109/79.221324.

[21]

L. Qi, Eigenvalues of a Supersymmetric Tensor and Positive Definiteness of an Even Degree Multivariate Form, Department of Applied Mathematics, The Hong Kong Polytechnic University, 2004. doi: 10.2307/2152750.

[22]

L. Qi, Eigenvalues of a real supersymmetric tensor, Journal of Symbolic Computation, 40 (2005), 1302-1324. doi: 10.1016/j.jsc.2005.05.007.

[23]

L. Qi, Eigenvalues and invariants of tensors, Journal of Mathematical Analysis and Applications, 325 (2007), 1363-1377. doi: 10.1016/j.jmaa.2006.02.071.

[24]

L. Qi, W. Sun and Y. Wang, Numerical multilinear algebra and its applications, Front. Math. China, 2 (2007), 501-526. doi: 10.1007/s11464-007-0031-4.

[25]

L. Qi, F. Wang and Y. Wang, Z-eigenvalue methods for a global polynomial optimization problem, Mathematical Programming, 118 (2009), 301-316. doi: 10.1007/s10107-007-0193-6.

[26]

L. Qi, Y. Wang and E. X. Wu, D-eigenvalues of diffusion kurtosis tensors, Journal of Computational and Applied Mathematics, 221 (2008), 150-157. doi: 10.1016/j.cam.2007.10.012.

[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), 589-601. doi: 10.1002/nla.633.

[28]

Z. Wang and W. Wu, Bounds for the greatest eigenvalue of positive tensors, Journal of Industral and Management Optimization, 10 (2013), 1031-1039. doi: 10.3934/jimo.2014.10.1031.

[29]

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.

[30]

Q. Yang and Y. Yang, Further results for Perron-Frobenius theorem for nonnegative tensors II, SIAM. J. Matrix Anal. Appl., 32 (2011), 1236-1250. doi: 10.1137/100813671.

[31]

T. Zhang and G. H. Golub, Rank-1 approximation of higher-order tensors, SIAM J. Matrix Anal. Appl., 23 (2001), 534-550. doi: 10.1137/S0895479899352045.

show all references

References:
[1]

D. Cartwright and B. Sturmfels, The number of eigenvalues of a tensor, Linear Algebra and its Applications, 438 (2013), 942-952. doi: 10.1016/j.laa.2011.05.040.

[2]

K. C. Chang, K. Pearson and T. Zhang, Perron-Frobenius theorem for nonnegative tensors, Commun. Math. Sci., 6 (2008) 507-520.

[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, Chichester, 2009. doi: 10.1002/9780470747278.

[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), Selected papers presented at 4th IMA Int. Conf. on Mathematics in Signal Processing, Oxford University Press, Oxford, UK, 1998, 1-15.

[5]

L. De Lathauwer, B. D. Moor and J. Vandewalle, A multilinear singular value decomposition, SIAM J. Matrix Anal. Appl., 21 (2000), 1253-1278. doi: 10.1137/S0895479896305696.

[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), 1324-1342.

[7]

E. Deutsch, Bounds for the perron root of a nonnegative irreducible partitioned matrix, Pacific Journal of Mathematics, 92 (1981), 49-56. doi: 10.2140/pjm.1981.92.49.

[8]

S. Friedland, S. Gaubert and L. Han, Perron-Frobenius theorem for nonnegative multilinear forms and extensions, Linear Algebra and its Applications, 438 (2013), 738-749. doi: 10.1016/j.laa.2011.02.042.

[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), 508-531. doi: 10.1016/j.jsc.2012.10.001.

[10]

R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985.

[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), 863-884.

[12]

T. G. Kolda and J. R. Mayo, Shifted power method for computing tensor eigenpairs, SIAM Journal on Matrix Analysis and Applications, 32 (2011), 1095-1124. doi: 10.1137/100801482.

[13]

W. Ledermann, Bounds for the greastest latent root of a positive matrix, J. London Math. Soc., 25 (1950), 265-268.

[14]

C. Li, Y. Li and X. Kong, New eigenvalue inclusion sets for tensors, Numer. Linear Algebra Appl., 21 (2014), 39-50. doi: 10.1002/nla.1858.

[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, 129-132.

[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), 286-292. doi: 10.1016/j.cam.2010.06.002.

[17]

M. Ng, L. Qi and G. Zhou, Finding the largest eigenvalue of a nonnegative tensor, SIAM J. Matrix Anal. Appl., 31 (2009), 1090-1099.

[18]

Q. Ni, L. Qi and F. Wang, An eigenvalue method for the positive definiteness identification problem, IEEE Transactions on Automatic Control, 53 (2008), 1096-1107. doi: 10.1109/TAC.2008.923679.

[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), 1218-1229. doi: 10.1016/j.jmaa.2006.07.064.

[20]

C. L. Nikias and J. M. Mendel, Signal processing with higher-order spectra, IEEE Signal Process. Mag., 10 (1993), 10-37. doi: 10.1109/79.221324.

[21]

L. Qi, Eigenvalues of a Supersymmetric Tensor and Positive Definiteness of an Even Degree Multivariate Form, Department of Applied Mathematics, The Hong Kong Polytechnic University, 2004. doi: 10.2307/2152750.

[22]

L. Qi, Eigenvalues of a real supersymmetric tensor, Journal of Symbolic Computation, 40 (2005), 1302-1324. doi: 10.1016/j.jsc.2005.05.007.

[23]

L. Qi, Eigenvalues and invariants of tensors, Journal of Mathematical Analysis and Applications, 325 (2007), 1363-1377. doi: 10.1016/j.jmaa.2006.02.071.

[24]

L. Qi, W. Sun and Y. Wang, Numerical multilinear algebra and its applications, Front. Math. China, 2 (2007), 501-526. doi: 10.1007/s11464-007-0031-4.

[25]

L. Qi, F. Wang and Y. Wang, Z-eigenvalue methods for a global polynomial optimization problem, Mathematical Programming, 118 (2009), 301-316. doi: 10.1007/s10107-007-0193-6.

[26]

L. Qi, Y. Wang and E. X. Wu, D-eigenvalues of diffusion kurtosis tensors, Journal of Computational and Applied Mathematics, 221 (2008), 150-157. doi: 10.1016/j.cam.2007.10.012.

[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), 589-601. doi: 10.1002/nla.633.

[28]

Z. Wang and W. Wu, Bounds for the greatest eigenvalue of positive tensors, Journal of Industral and Management Optimization, 10 (2013), 1031-1039. doi: 10.3934/jimo.2014.10.1031.

[29]

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.

[30]

Q. Yang and Y. Yang, Further results for Perron-Frobenius theorem for nonnegative tensors II, SIAM. J. Matrix Anal. Appl., 32 (2011), 1236-1250. doi: 10.1137/100813671.

[31]

T. Zhang and G. H. Golub, Rank-1 approximation of higher-order tensors, SIAM J. Matrix Anal. Appl., 23 (2001), 534-550. doi: 10.1137/S0895479899352045.

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