2022/04/21 by Ying Wang, Wang, Ying, Yuning Yang +1
Mathematics · #FOS: Mathematics #Numerical Analysis (math.NA) #Optimization and Control (math.OC) #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.2204.10229
openalex publication_date 2022/04/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper considers a way of generalizing the t-SVD of third-order tensors (regarded as tubal matrices) to tensors of arbitrary order N (which can be similarly regarded as tubal tensors of order (N-1)). \colorblackSuch a generalization is different from the t-SVD for tensors of order greater than three [Martin, Shafer, Larue, SIAM J. Sci. Comput., 35 (2013), A474--A490]. The decomposition is called Hot-SVD since it can be recognized as a tensor-tensor product version of HOSVD. The existence of Hot-SVD is proved. To this end, a new transpose for third-order tensors is introduced. This transpose is crucial in the verification of Hot-SVD, since it serves as a bridge between tubal tensors and their unfoldings. We establish some properties of Hot-SVD, analogous to those of HOSVD, and in doing so we emphasize the perspective of tubal tensors. The truncated and sequentially truncated Hot-SVD are then introduced, whose error bounds are √(N) for an (N+1)-th order tensor. We provide numerical examples to validate Hot-SVD, truncated Hot-SVD, and sequentially truncated Hot-SVD.