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A new class of positive semi-definite tensors
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 |
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.
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. 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. |
[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. 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. |
[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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