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March  2020, 16(2): 933-943. doi: 10.3934/jimo.2018186

A new class of positive semi-definite tensors

Yi Xu 1, , Jinjie Liu 2,, and  Liqun Qi 2,

1. 

Mathematics Department, Southeast University, 2 Sipailou, Nanjing, Jiangsu Province 210096, China
2. 

Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China

* Corresponding author: Jinjie Liu

Received  March 2018 Revised  September 2018 Published  March 2020 Early access  December 2018

Fund Project: The first author is supported by National Natural Science Foundation of China Nos. 11501100, 11571178 and 11671082. The third author is supported in part by the Hong Kong Research Grant Council Nos. PolyU 15302114, 15300715, 15301716 and 15300717

In this paper, a new class of positive semi-definite tensors, the MO tensor, is introduced. It is inspired by the structure of Moler matrix, a class of test matrices. Then we focus on two special cases in the MO-tensors: Sup-MO tensor and essential MO tensor. They are proved to be positive definite tensors. Especially, the smallest H-eigenvalue of a Sup-MO tensor is positive and tends to zero as the dimension tends to infinity, and an essential MO tensor is also a completely positive tensor.

Keywords: Positive (semi-)definite tensor, completely positive tensor, H-eigenvalue, MO-tensor, MO-like tensor, Sup-MO value.
Mathematics Subject Classification: Primary: 15A18, 15A69; Secondary: 15B99.
Citation: Yi Xu, Jinjie Liu, Liqun Qi. A new class of positive semi-definite tensors. Journal of Industrial and Management Optimization, 2020, 16 (2) : 933-943. doi: 10.3934/jimo.2018186
References:
[1]

J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer, New York, 2000. doi: 10.1007/978-1-4612-1394-9.

[2]

H. Chen, G. Li and L. Qi, SOS tensor decomposition: Theory and applications, Commun. Math. Sci., 14 (2016), 2073-2100.  doi: 10.4310/CMS.2016.v14.n8.a1.

[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, Wiley, New York, 2009. doi: 10.1002/9780470747278.

[4]

D. Hilbert, Über die Darstellung definiter Formen als Summe von Formenquadraten, Math. Ann., 32 (1888), 342-350.  doi: 10.1007/BF01443605.

[5]

C. J. Hillar and L. H. Lim, Most tensor problems are NP-hard, J. ACM, 60 (2013), Art. 45, 39 pp. doi: 10.1145/2512329.

[6]

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

[7]

C. Li, F. Wang, J. Zhao, Y. Zhu and Y. Li, Criterions for the positive definiteness of real supersymmetric tensors, J. Comput. Applied. Math., 255 (2014), 1-14.  doi: 10.1016/j.cam.2013.04.022.

[8]

Z. Luo and L. Qi, Positive semidefinite tensors (in Chinese), Sci. Sin. Math., 46 (2016), 639-654. 

[9]

Z. Luo and L. Qi, Completely positive tensors: Properties, easily checkable subclasses and tractable relaxations, SIAM J. Matrix Anal. Appl., 37 (2016), 1675-1698.  doi: 10.1137/15M1025220.

[10]

J. C. Nash, Compact Numerical Methods for Computers: Linear Algebra and Function Minimisation, CRC Press, 1990.

[11]

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

[12]

L. Qi, H$^+$-eigenvalues of Laplacian and signless Laplacian tensors, Commun. Math. Sci., 12 (2014), 1045-1064.  doi: 10.4310/CMS.2014.v12.n6.a3.

[13]

L. Qi and Z. Luo, Tensor Analysis: Spectral Theory and Special Tensors, SIAM, Philadelphia, 2017. doi: 10.1137/1.9781611974751.

[14]

L. Qi and Y. Song, An even order symmetric B tensor is positive definite, Linear Algebra Appl., 457 (2014), 303-312.  doi: 10.1016/j.laa.2014.05.026.

[15]

L. Qi, C. Xu and Y. Xu, Nonnegative tensor factorization, completely positive tensors, and a hierarchical elimination algorithm, SIAM J Marix Anal. Appl., 35 (2014), 1227-1241.  doi: 10.1137/13092232X.

[16]

L. Qi, G. Yu and E. X. Wu, Higher order positive semidefinite diffusion tensor imaging, SIAM J Imaging Sci., 3 (2010), 416-433.  doi: 10.1137/090755138.

[17]

L. Zhang, L. Qi and G. Zhou, $M$-tensors and some applications, SIAM J. Matrix Anal. Appl., 35 (2014), 437-452.  doi: 10.1137/130915339.

show all references

References:
[1]

J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer, New York, 2000. doi: 10.1007/978-1-4612-1394-9.

[2]

H. Chen, G. Li and L. Qi, SOS tensor decomposition: Theory and applications, Commun. Math. Sci., 14 (2016), 2073-2100.  doi: 10.4310/CMS.2016.v14.n8.a1.

[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, Wiley, New York, 2009. doi: 10.1002/9780470747278.

[4]

D. Hilbert, Über die Darstellung definiter Formen als Summe von Formenquadraten, Math. Ann., 32 (1888), 342-350.  doi: 10.1007/BF01443605.

[5]

C. J. Hillar and L. H. Lim, Most tensor problems are NP-hard, J. ACM, 60 (2013), Art. 45, 39 pp. doi: 10.1145/2512329.

[6]

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

[7]

C. Li, F. Wang, J. Zhao, Y. Zhu and Y. Li, Criterions for the positive definiteness of real supersymmetric tensors, J. Comput. Applied. Math., 255 (2014), 1-14.  doi: 10.1016/j.cam.2013.04.022.

[8]

Z. Luo and L. Qi, Positive semidefinite tensors (in Chinese), Sci. Sin. Math., 46 (2016), 639-654. 

[9]

Z. Luo and L. Qi, Completely positive tensors: Properties, easily checkable subclasses and tractable relaxations, SIAM J. Matrix Anal. Appl., 37 (2016), 1675-1698.  doi: 10.1137/15M1025220.

[10]

J. C. Nash, Compact Numerical Methods for Computers: Linear Algebra and Function Minimisation, CRC Press, 1990.

[11]

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

[12]

L. Qi, H$^+$-eigenvalues of Laplacian and signless Laplacian tensors, Commun. Math. Sci., 12 (2014), 1045-1064.  doi: 10.4310/CMS.2014.v12.n6.a3.

[13]

L. Qi and Z. Luo, Tensor Analysis: Spectral Theory and Special Tensors, SIAM, Philadelphia, 2017. doi: 10.1137/1.9781611974751.

[14]

L. Qi and Y. Song, An even order symmetric B tensor is positive definite, Linear Algebra Appl., 457 (2014), 303-312.  doi: 10.1016/j.laa.2014.05.026.

[15]

L. Qi, C. Xu and Y. Xu, Nonnegative tensor factorization, completely positive tensors, and a hierarchical elimination algorithm, SIAM J Marix Anal. Appl., 35 (2014), 1227-1241.  doi: 10.1137/13092232X.

[16]

L. Qi, G. Yu and E. X. Wu, Higher order positive semidefinite diffusion tensor imaging, SIAM J Imaging Sci., 3 (2010), 416-433.  doi: 10.1137/090755138.

[17]

L. Zhang, L. Qi and G. Zhou, $M$-tensors and some applications, SIAM J. Matrix Anal. Appl., 35 (2014), 437-452.  doi: 10.1137/130915339.

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