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An orthogonal equivalence theorem for third order tensors

  • * Corresponding author: Jinjie Liu

    * Corresponding author: Jinjie Liu 

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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  • 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.

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


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