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An orthogonal equivalence theorem for third order tensors. (English) Zbl 1513.15048

Summary: In 2011, M. E. Kilmer and C. D. Martin [Linear Algebra Appl. 435, No. 3, 641–658 (2011; Zbl 1228.15009)] 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.

MSC:

15A69 Multilinear algebra, tensor calculus
15A18 Eigenvalues, singular values, and eigenvectors

Citations:

Zbl 1228.15009
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Full Text: DOI arXiv

References:

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