三阶张量的一个正交等价定理
An Orthogonal Equivalence Theorem for Third Order Tensors
摘要：
2011年，Kilmer和Martin提出了三阶张量的张量奇异值分解（T-SVD）。此后，T-SVD在低秩张量近似，张量恢复，多视图聚类，多视图特征提取，张量素描等方面都有应用。通过离散傅里叶变换（DFT），矩阵SVD和逆DFT，将三阶张量映射到f-对角三阶张量。我们称之为Kilmer-Martin映射。证明了三阶张量的Kilmer-Martin映射是不变的，如果该三阶张量与某些正交张量取T-积。基于Kilmer-Martin映射，定义了三阶张量的奇异值和T秩.因此，当一个三阶张量与某些正交张量作T-积时，它的张量管秩、T-秩、奇异值和T-奇异值是不变的。讨论了奇异值、T秩和最佳T秩一逼近的一些性质。
英文摘要：
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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