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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

Received  March 2021 Revised  June 2021 Early access September 2021

Fund Project: 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)

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.

Citation: Liqun Qi, Chen Ling, Jinjie Liu, Chen Ouyang. An orthogonal equivalence theorem for third order tensors. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021154
References:
[1]

Y. ChenX. Xiao and Y. Zhou, Multi-view subspace clustering via simultaneously learning the representation tensor and affinity matrix, Pattern Recognition, 106 (2020), 107441. 

[2]

G. H. Golub and C. F. Van Loan, Matrix Computation, 4$^{nd}$ edition, Johns Hopkins University Press, Baltimore, MD, 2013.

[3]

M. KilmerK. BramanN. Hao and R. Hoover, Third-order tensors as operators on matrices: A theoretical and computational framework with applications in imaging, SIAM J. Matrix Anal. Appl., 34 (2013), 148-172.  doi: 10.1137/110837711.

[4]

M. Kilmer and C. D. Martin, Factorization strategies for third-order tensors, Linear Algebra Appl., 435 (2011), 641-658.  doi: 10.1016/j.laa.2010.09.020.

[5]

M. Kilmer, C. D. Martin and L. Perrone, A third-order generalization of the matrix svd as a product of third-order tensors, Tech. Report Tufts University, Computer Science Department, 2008.

[6]

C. Ling, H. He, C. Pan and L. Qi, A T-Sketching Method for Low-Rank Approximation of Third Order Tensors, Manuscript, 2021.

[7]

C. Ling, G. Yu, L. Qi and Y. Xu, A parallelizable optimization method for missing internet traffic tensor data, arXiv: 2005.09838, 2020.

[8]

Y. MiaoL. Qi and Y. Wei, Generalized tensor function via the tensor singular value decomposition based on the T-product, Linear Algebra Appl., 590 (2020), 258-303.  doi: 10.1016/j.laa.2019.12.035.

[9]

Y. MiaoL. Qi and Y. Wei, T-Jordan canonical form and T-Drazin inverse based on the T-product, Commun. Appl. Math. Comput., 3 (2021), 201-220.  doi: 10.1007/s42967-019-00055-4.

[10]

L. Qi and G. Yu, T-singular values and T-Sketching for third order tensors, arXiv: 2103.00976, 2021.

[11]

O. SemerciN. HaoM. E. Kilmer and E. L. Miller, Tensor-based formulation and nuclear norm regularization for multienergy computed tomography, IEEE Trans. Image Process., 23 (2014), 1678-1693.  doi: 10.1109/TIP.2014.2305840.

[12]

G. Song, M. K. Ng and X. Zhang, Robust tensor completion using transformed tensor singular value decomposition, Numer. Linear Algebra Appl., 27 (2020), e2299. doi: 10.1002/nla.2299.

[13]

X. XiaoY. ChenY. J. Gong and Y. Zhou, Low-rank reserving t-linear projection for robust image feature extraction, IEEE Trans. Image Process., 30 (2021), 108-120.  doi: 10.1109/TIP.2020.3031813.

[14]

X. XiaoY. ChenY. J. Gong and Y. Zhou, Prior knowledge regularized multiview self-representation and its applications, IEEE Trans. Neural Netw. Learn. Syst., 32 (2021), 1325-1338.  doi: 10.1109/TNNLS.2020.2984625.

[15]

L. YangZ. H. HuangS. Hu and J. Han, An iterative algorithm for third-order tensor multi-rank minimization, Comput. Optim. Appl., 63 (2016), 169-202.  doi: 10.1007/s10589-015-9769-x.

[16]

J. Zhang, A. K. Saibaba, M. E. Kilmer and S. Aeron, A randomized tensor singular value decomposition based on the t-product, Numer. Linear Algebra Appl., 25 (2018), e2179. doi: 10.1002/nla.2179.

[17]

Z. Zhang and S. Aeron, Exact tensor completion using t-SVD, IEEE Tran. Signal Process., 65 (2017), 1511-1526.  doi: 10.1109/TSP.2016.2639466.

[18]

Z. Zhang, G. Ely, S. Aeron, N. Hao and M. Kilmer, Novel methods for multilinear data completion and de-noising based on tensor-SVD, IEEE Conference on Computer Vision and Pattern Recognition, (2014), 3842–3849. doi: 10.1109/CVPR.2014.485.

[19]

M. ZhengZ. Huang and Y. Wang, T-positive semidefiniteness of third-order symmetric tensors and T-semidefinite programming, Comput. Optim. Appl, 78 (2021), 239-272.  doi: 10.1007/s10589-020-00231-w.

[20]

P. ZhouC. LuZ. Lin and C. Zhang, Tensor factorization for low-rank tensor completion, IEEE Trans. Image Process., 27 (2018), 1152-1163.  doi: 10.1109/TIP.2017.2762595.

show all references

References:
[1]

Y. ChenX. Xiao and Y. Zhou, Multi-view subspace clustering via simultaneously learning the representation tensor and affinity matrix, Pattern Recognition, 106 (2020), 107441. 

[2]

G. H. Golub and C. F. Van Loan, Matrix Computation, 4$^{nd}$ edition, Johns Hopkins University Press, Baltimore, MD, 2013.

[3]

M. KilmerK. BramanN. Hao and R. Hoover, Third-order tensors as operators on matrices: A theoretical and computational framework with applications in imaging, SIAM J. Matrix Anal. Appl., 34 (2013), 148-172.  doi: 10.1137/110837711.

[4]

M. Kilmer and C. D. Martin, Factorization strategies for third-order tensors, Linear Algebra Appl., 435 (2011), 641-658.  doi: 10.1016/j.laa.2010.09.020.

[5]

M. Kilmer, C. D. Martin and L. Perrone, A third-order generalization of the matrix svd as a product of third-order tensors, Tech. Report Tufts University, Computer Science Department, 2008.

[6]

C. Ling, H. He, C. Pan and L. Qi, A T-Sketching Method for Low-Rank Approximation of Third Order Tensors, Manuscript, 2021.

[7]

C. Ling, G. Yu, L. Qi and Y. Xu, A parallelizable optimization method for missing internet traffic tensor data, arXiv: 2005.09838, 2020.

[8]

Y. MiaoL. Qi and Y. Wei, Generalized tensor function via the tensor singular value decomposition based on the T-product, Linear Algebra Appl., 590 (2020), 258-303.  doi: 10.1016/j.laa.2019.12.035.

[9]

Y. MiaoL. Qi and Y. Wei, T-Jordan canonical form and T-Drazin inverse based on the T-product, Commun. Appl. Math. Comput., 3 (2021), 201-220.  doi: 10.1007/s42967-019-00055-4.

[10]

L. Qi and G. Yu, T-singular values and T-Sketching for third order tensors, arXiv: 2103.00976, 2021.

[11]

O. SemerciN. HaoM. E. Kilmer and E. L. Miller, Tensor-based formulation and nuclear norm regularization for multienergy computed tomography, IEEE Trans. Image Process., 23 (2014), 1678-1693.  doi: 10.1109/TIP.2014.2305840.

[12]

G. Song, M. K. Ng and X. Zhang, Robust tensor completion using transformed tensor singular value decomposition, Numer. Linear Algebra Appl., 27 (2020), e2299. doi: 10.1002/nla.2299.

[13]

X. XiaoY. ChenY. J. Gong and Y. Zhou, Low-rank reserving t-linear projection for robust image feature extraction, IEEE Trans. Image Process., 30 (2021), 108-120.  doi: 10.1109/TIP.2020.3031813.

[14]

X. XiaoY. ChenY. J. Gong and Y. Zhou, Prior knowledge regularized multiview self-representation and its applications, IEEE Trans. Neural Netw. Learn. Syst., 32 (2021), 1325-1338.  doi: 10.1109/TNNLS.2020.2984625.

[15]

L. YangZ. H. HuangS. Hu and J. Han, An iterative algorithm for third-order tensor multi-rank minimization, Comput. Optim. Appl., 63 (2016), 169-202.  doi: 10.1007/s10589-015-9769-x.

[16]

J. Zhang, A. K. Saibaba, M. E. Kilmer and S. Aeron, A randomized tensor singular value decomposition based on the t-product, Numer. Linear Algebra Appl., 25 (2018), e2179. doi: 10.1002/nla.2179.

[17]

Z. Zhang and S. Aeron, Exact tensor completion using t-SVD, IEEE Tran. Signal Process., 65 (2017), 1511-1526.  doi: 10.1109/TSP.2016.2639466.

[18]

Z. Zhang, G. Ely, S. Aeron, N. Hao and M. Kilmer, Novel methods for multilinear data completion and de-noising based on tensor-SVD, IEEE Conference on Computer Vision and Pattern Recognition, (2014), 3842–3849. doi: 10.1109/CVPR.2014.485.

[19]

M. ZhengZ. Huang and Y. Wang, T-positive semidefiniteness of third-order symmetric tensors and T-semidefinite programming, Comput. Optim. Appl, 78 (2021), 239-272.  doi: 10.1007/s10589-020-00231-w.

[20]

P. ZhouC. LuZ. Lin and C. Zhang, Tensor factorization for low-rank tensor completion, IEEE Trans. Image Process., 27 (2018), 1152-1163.  doi: 10.1109/TIP.2017.2762595.

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