-
Previous Article
Stability of ground state for the Schrödinger-Poisson equation
- JIMO Home
- This Issue
-
Next Article
Distributed convex optimization with coupling constraints over time-varying directed graphs†
Spectral norm and nuclear norm of a third order tensor
1. | Huawei Theory Research Lab Hong Kong, Hong Kong, 00852, China |
2. | Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou, 310018, China |
The spectral norm and the nuclear norm of a third order tensor play an important role in the tensor completion and recovery problem. We show that the spectral norm of a third order tensor is equal to the square root of the spectral norm of three positive semi-definite biquadratic tensors, and the square roots of the nuclear norms of those three positive semi-definite biquadratic tensors are lower bounds of the nuclear norm of that third order tensor. This provides a way to estimate and to evaluate the spectral norm and the nuclear norm of that third order tensor. Some upper and lower bounds for the spectral norm and nuclear norm of a third order tensor, by spectral radii and nuclear norms of some symmetric matrices, are presented.
References:
[1] |
S. Friedland and L. H. Lim,
Nuclear norm of high-order tensors, Mathematics of Computation, 87 (2018), 1255-1281.
doi: 10.1090/mcom/3239. |
[2] |
S. Hu,
Relations of the nuclear norm of a tensor and its matrix flattenings, Linear Algebra and Its Applications, 478 (2015), 188-199.
doi: 10.1016/j.laa.2015.04.003. |
[3] |
B. Jiang, F. Yang and S. Zhang, Tensor and its tucker core: The invariance relationships, Numerical Linear Algebra with Applications, 24 (2017), e2086.
doi: 10.1002/nla.2086. |
[4] |
Z. Li,
Bounds of the spectral norm and the nuclear norm of a tensor based on tensor partitions, SIAM J. Matrix Analysis and Applications, 37 (2016), 1440-1452.
doi: 10.1137/15M1028777. |
[5] |
L. H. Lim, Singular values and eigenvalues of tensors: a variational approach, 1st IEEE Internatyional Workshop on Computational Advances in MultiSensor Adaptive Processing, Puerto Vallarta, Mexico, 2005,129–132. Google Scholar |
[6] |
L. Qi, H. Chen and Y. Chen, Tensor Eigenvalues and Their Applications, Springer, Singapore, 2018.
doi: 10.1007/978-981-10-8058-6. |
[7] |
L. Qi, H. H. Dai and D. Han, Conditions for strong ellipticity and M-eigenvalues, Frontiers of Mathematics in China, 4 (2009), Art. no. 349.
doi: 10.1007/s11464-009-0016-6. |
[8] |
Q. Song, H. Ge, J. Caverlee and X. Hu, Tensor completion algorithms in big data analytics, ACM Transactions on Knowledge Discovery from Data, 13 (2019), Article 6. Google Scholar |
[9] |
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. |
[10] |
M. Yuan and C. H. Zhang,
On tensor completion via nuclear norm minimization, Foundations of Computational Mathematics, 16 (2016), 1031-1068.
doi: 10.1007/s10208-015-9269-5. |
show all references
References:
[1] |
S. Friedland and L. H. Lim,
Nuclear norm of high-order tensors, Mathematics of Computation, 87 (2018), 1255-1281.
doi: 10.1090/mcom/3239. |
[2] |
S. Hu,
Relations of the nuclear norm of a tensor and its matrix flattenings, Linear Algebra and Its Applications, 478 (2015), 188-199.
doi: 10.1016/j.laa.2015.04.003. |
[3] |
B. Jiang, F. Yang and S. Zhang, Tensor and its tucker core: The invariance relationships, Numerical Linear Algebra with Applications, 24 (2017), e2086.
doi: 10.1002/nla.2086. |
[4] |
Z. Li,
Bounds of the spectral norm and the nuclear norm of a tensor based on tensor partitions, SIAM J. Matrix Analysis and Applications, 37 (2016), 1440-1452.
doi: 10.1137/15M1028777. |
[5] |
L. H. Lim, Singular values and eigenvalues of tensors: a variational approach, 1st IEEE Internatyional Workshop on Computational Advances in MultiSensor Adaptive Processing, Puerto Vallarta, Mexico, 2005,129–132. Google Scholar |
[6] |
L. Qi, H. Chen and Y. Chen, Tensor Eigenvalues and Their Applications, Springer, Singapore, 2018.
doi: 10.1007/978-981-10-8058-6. |
[7] |
L. Qi, H. H. Dai and D. Han, Conditions for strong ellipticity and M-eigenvalues, Frontiers of Mathematics in China, 4 (2009), Art. no. 349.
doi: 10.1007/s11464-009-0016-6. |
[8] |
Q. Song, H. Ge, J. Caverlee and X. Hu, Tensor completion algorithms in big data analytics, ACM Transactions on Knowledge Discovery from Data, 13 (2019), Article 6. Google Scholar |
[9] |
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. |
[10] |
M. Yuan and C. H. Zhang,
On tensor completion via nuclear norm minimization, Foundations of Computational Mathematics, 16 (2016), 1031-1068.
doi: 10.1007/s10208-015-9269-5. |
[1] |
Braxton Osting, Jérôme Darbon, Stanley Osher. Statistical ranking using the $l^{1}$-norm on graphs. Inverse Problems & Imaging, 2013, 7 (3) : 907-926. doi: 10.3934/ipi.2013.7.907 |
[2] |
Hirofumi Notsu, Masato Kimura. Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity. Networks & Heterogeneous Media, 2014, 9 (4) : 617-634. doi: 10.3934/nhm.2014.9.617 |
[3] |
Caifang Wang, Tie Zhou. The order of convergence for Landweber Scheme with $\alpha,\beta$-rule. Inverse Problems & Imaging, 2012, 6 (1) : 133-146. doi: 10.3934/ipi.2012.6.133 |
[4] |
Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597 |
[5] |
Charles Fulton, David Pearson, Steven Pruess. Characterization of the spectral density function for a one-sided tridiagonal Jacobi matrix operator. Conference Publications, 2013, 2013 (special) : 247-257. doi: 10.3934/proc.2013.2013.247 |
[6] |
Hong Seng Sim, Wah June Leong, Chuei Yee Chen, Siti Nur Iqmal Ibrahim. Multi-step spectral gradient methods with modified weak secant relation for large scale unconstrained optimization. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 377-387. doi: 10.3934/naco.2018024 |
[7] |
A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044 |
[8] |
Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027 |
2019 Impact Factor: 1.366
Tools
Article outline
[Back to Top]