Hot-SVD: Higher-Order t-Singular Value Decomposition for Tensors based on Tensor-Tensor Product

Abstract: 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)). \color{black}Such 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 $\sqrt{N}$ for an $(N+1)$-th order tensor. We provide numerical examples to validate Hot-SVD, truncated Hot-SVD, and sequentially truncated Hot-SVD.