Decompositions of third-order tensors: HOSVD, T-SVD, and Beyond

The higher order singular value decomposition, which is regarded as a generalization of the matrix singular value decomposition (SVD), has a long history and is well established, while the T-SVD is relatively new and lacks systematic analysis. Because of the unusual tensor-tensor product that the T-SVD is based on, the form of the T-SVD may be difficult to comprehend. The main aim of this article is to establish a connection between these two decompositions. By converting the form of the T-SVD into the sum of outer product terms, we compare the forms of the two decompositions. Moreover, from establishing the connection, a new decomposition which has a specific nonzero pattern, is proposed and developed. Numerical examples are given to demonstrate the useful ability of the new decomposition for approximation and data compression.