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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  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 & 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.  Google Scholar

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

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

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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.  Google Scholar

[8]

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

[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.  Google Scholar

[10]

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

[11]

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

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

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

[4]

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

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[10]

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

[11]

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

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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