An orthogonal equivalence theorem for third order tensors

Liqun Qi; Chen Ling; Jinjie Liu; Chen Ouyang

doi:10.3934/jimo.2021154

Article Contents

2022, Volume 18, Issue 6: 4191-4200. Doi: 10.3934/jimo.2021154

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An orthogonal equivalence theorem for third order tensors

1.
Department of Mathematics, Hangzhou Dianzi University, Hangzhou, 310018, China
2.
School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, 200240, China
3.
School of Computer Science and Technology, Dongguan University of Technology, Dongguan, 523000, China

^* Corresponding author: Jinjie Liu
^* Corresponding author: Jinjie Liu

Received: March 2021

Revised: June 2021

Early access: September 2021

Published: September 2022

The second author's work was supported by Natural Science Foundation of China (No. 11971138) and Natural Science Foundation of Zhejiang Province (No. LY19A010019, LD19A010002). The third author's work was supported by Natural Science Foundation of China (No. 11801479, No. 12001366). The forth author's work was supported by Natural Science Foundation of China (No.11971106) and Guangdong Universities' Special Projects in Key Fields of Natural Science (No. 2019KZDZX1005)

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Abstract

In 2011, Kilmer and Martin proposed tensor singular value decomposition (T-SVD) for third order tensors. Since then, T-SVD has applications in low rank tensor approximation, tensor recovery, multi-view clustering, multi-view feature extraction, tensor sketching, etc. By going through the Discrete Fourier Transform (DFT), matrix SVD and inverse DFT, a third order tensor is mapped to an f-diagonal third order tensor. We call this a Kilmer-Martin mapping. We show that the Kilmer-Martin mapping of a third order tensor is invariant if that third order tensor is taking T-product with some orthogonal tensors. We define singular values and T-rank of that third order tensor based upon its Kilmer-Martin mapping. Thus, tensor tubal rank, T-rank, singular values and T-singular values of a third order tensor are invariant when it is taking T-product with some orthogonal tensors. Some properties of singular values, T-rank and best T-rank one approximation are discussed.

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Mathematics Subject Classification: Primary: 15A18, 15A69.

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References

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