Perron vector analysis for irreducible nonnegative tensors and its applications

Wen Li; Wei-Hui Liu; Seak Weng Vong

doi:10.3934/jimo.2019097

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doi: 10.3934/jimo.2019097

Perron vector analysis for irreducible nonnegative tensors and its applications

Wen Li ^1, , Wei-Hui Liu ^2,, and Seak Weng Vong ^2,

1.	School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
2.	Department of Mathematics, University of Macau, Macau, China

^* Corresponding author: Wei-Hui Liu

Received January 2019 Revised March 2019 Published July 2019

Fund Project: The first author is supported by NSFC grant 11671158, U1811464 and 11771159. The second author is supported by NSFC grant 11571124 and UM grant MYRG2016-00077-FST. The third author is supported by UM grant MYRG2017-00098-FST.

In this paper, we analyse the Perron vector of an irreducible nonnegative tensor, and present some lower and upper bounds for the ratio of the smallest and largest entries of a Perron vector based on some new techniques, which always improve the existing ones. Applying these new ratio results, we first refine two-sided bounds for the spectral radius of an irreducible nonnegative tensor. In particular, for the matrix case, the new bounds also improve the corresponding ones. Second, we provide a new Ky Fan type theorem, which improves the existing one. Third, we refine the perturbation bound for the spectral radii of nonnegative tensors, from which one may derive a comparison theorem for spectral radii of nonnegative tensors. Numerical examples are given to show the efficiency of the theoretical results.

Keywords: Nonnegative tensor, Perron vector, irreducibility, spectral radius, bound.

Mathematics Subject Classification: Primary: 15A69.

Citation: Wen Li, Wei-Hui Liu, Seak Weng Vong. Perron vector analysis for irreducible nonnegative tensors and its applications. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2019097

References:

[1]	K. Chang, K. Pearson and T. Zhang, Perron-Frobenius theorem for nonnegative tensors, Commun. Math. Sci., 6 (2008), 507-520. doi: 10.4310/CMS.2008.v6.n2.a12. Google Scholar
[2]	K. Chang, K. Pearson and T. Zhang, Primitivity, the convergence of the NQZ method, and the largest eigenvalue for nonnegative tensors, SIAM J. Matrix Anal. Appl., 33 (2011), 806-819. doi: 10.1137/100807120. Google Scholar
[3]	K. Chang and T. Zhang, On the uniqueness and non-uniqueness of the positive $Z$-eigenvector for transition probability tensors, J. Math. Anal. Appl., 408 (2013), 525-540. doi: 10.1016/j.jmaa.2013.04.019. Google Scholar
[4]	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
[5]	S. Friedland, S. Gaubert and L. Han, Perron-Frobenius theorem for nonnegative multilinear forms and extensions, Linear Algebra Appl., 438 (2013), 738-749. doi: 10.1016/j.laa.2011.02.042. Google Scholar
[6]	R. A. Horn and C. R. Johnson, Matrix analysis, Cambridge University Press, UK, 1991. Google Scholar
[7]	S. Hu and L. Qi, Algebraic connectivity of an even uniform hypergraph, J. Comb. Optim., 24 (2012), 564-579. doi: 10.1007/s10878-011-9407-1. Google Scholar
[8]	W. Li, L. B. Cui and M. Ng, The perturbation bound for the Perron vector of a transition probability tensor, Numer. Linear Algebra Appl., 20 (2013), 985-1000. doi: 10.1002/nla.1886. Google Scholar
[9]	W. Li and M. Ng, On the limiting probability distribution of a transition probability tensor, Linear Multilin. Algebra, 62 (2014), 362-385. doi: 10.1080/03081087.2013.777436. Google Scholar
[10]	W. Li and M. K. Ng, The perturbation bound for the spectral radius of a nonnegative tensor, Adv. Numer. Anal., 2014 (2014), 10pp. doi: 10.1155/2014/109525. Google Scholar
[11]	W. Li and M. K. Ng, Some bounds for the spectral radius of nonnegative tensors, Numer. Math., 130 (2015), 315-335. doi: 10.1007/s00211-014-0666-5. Google Scholar
[12]	L. H. Lim, Singular values and eigenvalues of tensors: A variational approach, in Proceedings of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, CAMSAP 05, vol. 1, IEEE Computer Society Press, Piscataway, NJ, 2005, 129-132. Google Scholar
[13]	Q. Liu, C. Li and C. Zhang, Some inequalities on the Perron eigenvalue and eigenvectors for positive tensors, J. of Math. Inequal., 10 (2016), 405-414. doi: 10.7153/jmi-10-31. Google Scholar
[14]	H. Minc, Nonnegative Matrices, John Wiley & Sons, New York, 1988. Google Scholar
[15]	M. Ng, L. Qi and G. Zhou, Finding the largest eigenvalue of a non-negative tensor, SIAM J. Matrix Anal. Appl., 31 (2009), 1090-1099. doi: 10.1137/09074838X. Google Scholar
[16]	K. Pearson, Essentially positive tensors, Int. J. Algebra, 4 (2010), 421-426. Google Scholar
[17]	L. Qi, Eigenvalues of a real supersymmetric tensor, J. of Symbolic Comput., 40 (2005), 1302-1324. doi: 10.1016/j.jsc.2005.05.007. Google Scholar
[18]	L. Qi, Symmetric nonnegative tensor and copositive tensors, Linear Algebra Appl., 439 (2013), 228-238. doi: 10.1016/j.laa.2013.03.015. Google Scholar
[19]	L. Qi and Z. Luo, Tensor Analysis: Spectral Theory and Special Tensors, Society for Industrial and Applied Mathematics, Pennsylvania, 2017. doi: 10.1137/1.9781611974751.ch1. Google Scholar
[20]	Z. Wang and W. Wu, Bounds for the greatest eigenvalue of positive tensors, J. of Indust. and Mgmt. Optim., 10 (2014), 1031-1039. doi: 10.3934/jimo.2014.10.1031. Google Scholar
[21]	Y. N. Yang and Q. Z. 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
[22]	Q. Z. Yang and Y. N. Yang, Further results for Perron-Frobenius theorem for nonnegative tensors Ⅱ, SIAM J. Matrix Anal. Appl., 32 (2011), 1236-1250. doi: 10.1137/100813671. Google Scholar

show all references

References:

[1]	K. Chang, K. Pearson and T. Zhang, Perron-Frobenius theorem for nonnegative tensors, Commun. Math. Sci., 6 (2008), 507-520. doi: 10.4310/CMS.2008.v6.n2.a12. Google Scholar
[2]	K. Chang, K. Pearson and T. Zhang, Primitivity, the convergence of the NQZ method, and the largest eigenvalue for nonnegative tensors, SIAM J. Matrix Anal. Appl., 33 (2011), 806-819. doi: 10.1137/100807120. Google Scholar
[3]	K. Chang and T. Zhang, On the uniqueness and non-uniqueness of the positive $Z$-eigenvector for transition probability tensors, J. Math. Anal. Appl., 408 (2013), 525-540. doi: 10.1016/j.jmaa.2013.04.019. Google Scholar
[4]	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
[5]	S. Friedland, S. Gaubert and L. Han, Perron-Frobenius theorem for nonnegative multilinear forms and extensions, Linear Algebra Appl., 438 (2013), 738-749. doi: 10.1016/j.laa.2011.02.042. Google Scholar
[6]	R. A. Horn and C. R. Johnson, Matrix analysis, Cambridge University Press, UK, 1991. Google Scholar
[7]	S. Hu and L. Qi, Algebraic connectivity of an even uniform hypergraph, J. Comb. Optim., 24 (2012), 564-579. doi: 10.1007/s10878-011-9407-1. Google Scholar
[8]	W. Li, L. B. Cui and M. Ng, The perturbation bound for the Perron vector of a transition probability tensor, Numer. Linear Algebra Appl., 20 (2013), 985-1000. doi: 10.1002/nla.1886. Google Scholar
[9]	W. Li and M. Ng, On the limiting probability distribution of a transition probability tensor, Linear Multilin. Algebra, 62 (2014), 362-385. doi: 10.1080/03081087.2013.777436. Google Scholar
[10]	W. Li and M. K. Ng, The perturbation bound for the spectral radius of a nonnegative tensor, Adv. Numer. Anal., 2014 (2014), 10pp. doi: 10.1155/2014/109525. Google Scholar
[11]	W. Li and M. K. Ng, Some bounds for the spectral radius of nonnegative tensors, Numer. Math., 130 (2015), 315-335. doi: 10.1007/s00211-014-0666-5. Google Scholar
[12]	L. H. Lim, Singular values and eigenvalues of tensors: A variational approach, in Proceedings of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, CAMSAP 05, vol. 1, IEEE Computer Society Press, Piscataway, NJ, 2005, 129-132. Google Scholar
[13]	Q. Liu, C. Li and C. Zhang, Some inequalities on the Perron eigenvalue and eigenvectors for positive tensors, J. of Math. Inequal., 10 (2016), 405-414. doi: 10.7153/jmi-10-31. Google Scholar
[14]	H. Minc, Nonnegative Matrices, John Wiley & Sons, New York, 1988. Google Scholar
[15]	M. Ng, L. Qi and G. Zhou, Finding the largest eigenvalue of a non-negative tensor, SIAM J. Matrix Anal. Appl., 31 (2009), 1090-1099. doi: 10.1137/09074838X. Google Scholar
[16]	K. Pearson, Essentially positive tensors, Int. J. Algebra, 4 (2010), 421-426. Google Scholar
[17]	L. Qi, Eigenvalues of a real supersymmetric tensor, J. of Symbolic Comput., 40 (2005), 1302-1324. doi: 10.1016/j.jsc.2005.05.007. Google Scholar
[18]	L. Qi, Symmetric nonnegative tensor and copositive tensors, Linear Algebra Appl., 439 (2013), 228-238. doi: 10.1016/j.laa.2013.03.015. Google Scholar
[19]	L. Qi and Z. Luo, Tensor Analysis: Spectral Theory and Special Tensors, Society for Industrial and Applied Mathematics, Pennsylvania, 2017. doi: 10.1137/1.9781611974751.ch1. Google Scholar
[20]	Z. Wang and W. Wu, Bounds for the greatest eigenvalue of positive tensors, J. of Indust. and Mgmt. Optim., 10 (2014), 1031-1039. doi: 10.3934/jimo.2014.10.1031. Google Scholar
[21]	Y. N. Yang and Q. Z. 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
[22]	Q. Z. Yang and Y. N. Yang, Further results for Perron-Frobenius theorem for nonnegative tensors Ⅱ, SIAM J. Matrix Anal. Appl., 32 (2011), 1236-1250. doi: 10.1137/100813671. Google Scholar

Figure 1. Comparison between two Ky Fan type Theorems

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Figure 2. The results of randomly constructed tensors

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Figure 3. The results of perturbation bounds (left) n = 5 and (right) n = 10

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Table 1. Comparisons with the upper bounds for the ratio

$ \omega_1 $ in (3)	$ \omega_2 $ in (4)	$ \omega_3 $ in (5)	$ \omega_4 $ in (11)
0.5575	0.5307	0.4855	0.5244

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$ \omega_1 $ in (3)	$ \omega_2 $ in (4)	$ \omega_3 $ in (5)	$ \omega_4 $ in (11)
0.5575	0.5307	0.4855	0.5244

Table 2. Comparisons with the lower bounds for ratio

	Example 2	Example 3	Example 4
Actual value of $ \frac{x_{\min}}{x_{\max}} $	0.9873	0.6402	0.6794
$ \kappa_0 $ in (2)	0.6300	0.3969	0.5000
$ \kappa_1 $ in (3)	0.7857	0.2083	0.3077
$ \kappa_2 $ in (4)	0.5848	0.5000	0.4642
$ \kappa_3^{(1)} $ in (13)	$ {\bf{0.9662}} $	0.2808	0.3445
$ \kappa_3^{(1)} $ in (13)	(t = -5.5602)	(t = -5.7276)	(t = -5.0250)
$ \kappa_3^{(2)} $ in (16)	0.9258	0.5000	$ {\bf{0.5539}} $
(also in (13))	(t = -5.1168)	(t = -2.2315)	(t = -2.2956)
$ \kappa_3^{(3)} $ in (13)	0.6300	$ {\bf{0.5724}} $	0.5503
$ \kappa_3^{(3)} $ in (13)	(t = -3.0000)	(t = -3.5887)	(t = -2.5208)
$ \kappa_3 $ in (13)	$ {\bf{0.9662}} $	$ {\bf{0.5724}} $	$ {\bf{0.5539}} $

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	Example 2	Example 3	Example 4
Actual value of $ \frac{x_{\min}}{x_{\max}} $	0.9873	0.6402	0.6794
$ \kappa_0 $ in (2)	0.6300	0.3969	0.5000
$ \kappa_1 $ in (3)	0.7857	0.2083	0.3077
$ \kappa_2 $ in (4)	0.5848	0.5000	0.4642
$ \kappa_3^{(1)} $ in (13)	$ {\bf{0.9662}} $	0.2808	0.3445
$ \kappa_3^{(1)} $ in (13)	(t = -5.5602)	(t = -5.7276)	(t = -5.0250)
$ \kappa_3^{(2)} $ in (16)	0.9258	0.5000	$ {\bf{0.5539}} $
(also in (13))	(t = -5.1168)	(t = -2.2315)	(t = -2.2956)
$ \kappa_3^{(3)} $ in (13)	0.6300	$ {\bf{0.5724}} $	0.5503
$ \kappa_3^{(3)} $ in (13)	(t = -3.0000)	(t = -3.5887)	(t = -2.5208)
$ \kappa_3 $ in (13)	$ {\bf{0.9662}} $	$ {\bf{0.5724}} $	$ {\bf{0.5539}} $

Table 3. Comparisons between (20) and (27)

Dimension	$ n = 5 $	$ n = 10 $	$ n = 15 $	$ n = 20 $
Lower bound	42.86%	64.02%	75.64%	81.37%
Upper bound	91.76%	94.50%	95.77%	96.83%

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Dimension	$ n = 5 $	$ n = 10 $	$ n = 15 $	$ n = 20 $
Lower bound	42.86%	64.02%	75.64%	81.37%
Upper bound	91.76%	94.50%	95.77%	96.83%

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