Orthogonal Equivalence, Singular Values and T-Rank of Third Order Tensors

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 a 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, the tensor tubal rank, T-rank, singular values and T-singular values of a third tensor are invariant when it was taking T-product with some orthogonal tensors. We make a conjecture that the sum of squares of the largest $s$ singular values of a third order tensor is greater than or equal to the sum of squares of any $s$ entries of that third order tensor. Kilmer and Martin showed that an Eckart-Young theorem holds for the tensor tubal rank of third order tensors. We show that our conjecture is true if and only if another Eckart-Young theorem holds for the T-rank of third order tensors.