# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2021010

## 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

* Corresponding author: Shenglong Hu

Received  August 2020 Revised  October 2020 Published  January 2021

Fund Project: Shenglong Hu: This author's work was supported by NSFC (Grant No. 11771328) and ZJSFC (Grant No. LD19A010002)

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.

Citation: Liqun Qi, Shenglong Hu, Yanwei Xu. Spectral norm and nuclear norm of a third order tensor. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021010
##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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