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 \((N-1)\)]. Such a generalization is different from the t-SVD for tensors of order greater than three (Martin et al. in SIAM J Sci Comput 35(1):A474–A490, 2013). The decomposition is called Hot-SVD, since it can be recognized as a tensor–tensor product version of the celebrated higher order SVD (HOSVD). The existence of Hot-SVD is proved. To this end, the “small-t” 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 unfolding tubal matrices. We establish some properties of Hot-SVD, analogous to those of HOSVD. The truncated Hot-SVD and sequentially truncated Hot-SVD are then introduced, with \(\sqrt{N}\)-error bounds established for an \((N+1)\)-th-order tensor. We provide numerical examples to validate Hot-SVD, truncated Hot-SVD, and sequentially truncated Hot-SVD.