# An Orthogonal Equivalence Theorem for 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 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.

## Authors

• 20 publications
• 8 publications
• 1 publication
• 2 publications
• ### T-Singular Values and T-Sketching for Third Order Tensors

Based upon the T-SVD (tensor SVD) of third order tensors, introduced by ...
03/01/2021 ∙ by Liqun Qi, et al. ∙ 0

• ### ST-SVD Factorization and s-Diagonal Tensors

A third order real tensor is mapped to a special f-diagonal tensor by go...
04/12/2021 ∙ by Chen Ling, et al. ∙ 0

• ### A Fast Algorithm for Cosine Transform Based Tensor Singular Value Decomposition

Recently, there has been a lot of research into tensor singular value de...
02/08/2019 ∙ by Wen-Hao Xu, et al. ∙ 0

• ### Tubal Matrix Analysis

One of the early ideas started from the 2004 workshop is to regard third...
05/03/2021 ∙ by Liqun Qi, et al. ∙ 0

• ### An Optimal Inverse Theorem

We prove that the partition rank and the analytic rank of tensors are eq...
02/21/2021 ∙ by Alex Cohen, et al. ∙ 0

• ### Robust Tensor Completion Using Transformed Tensor SVD

In this paper, we study robust tensor completion by using transformed te...
07/02/2019 ∙ by Guangjing Song, et al. ∙ 3