# 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
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